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Optical spectroscopy of some simple free radicals Cheung, Allan Shi-Chung 1981-12-31

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OPTICAL SPECTROSCOPY OF SOME SIMPLE FREE RADICALS by ALLAN SHI-CHUNG CHEUNG Sc. (Hon.), University of Waterloo, 1977 A THESIS SUMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1981 0 Allan Shi-Chung Cheung, 1981 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of cHints TRY The University of British Columbia 2075 Wesbrook Place Vancouver, B.C. Canada V6T 1W5 Date %.L. tfjtfZ To my parents - i i -Abstract This thesis reports studies of the electronic spectra of some gaseous oxide molecules. The (0,0) band of the C4z" - X4i~ electronic transition of V0 has been recorded by intermodulated laser-induced fluorescence at a resolution of about 100 MHz over the range 17300 - 17427 cm"1. The hyper-51 fine structure caused by the V nucleus (I = 7/2) is almost completely resolved. Internal hyperfine perturbations between the and F^ elec tron spin components (where N = J - j and J + 1, respectively) occur in both electronic states; these are caused by hyperfine matrix elements of the type AJ = ±1. The C E state has many local electronic-rotational perturbations, and also suffers from large spin-orbit perturbations by distant electronic states, for which it has been necessary to introduce a second spin-rotation parameter, Ys> and the corresponding isotropic hyperfine parameter, b<.. The background theory for this new hyperfine parameter and the calculation of its matrix elements are described. 4 4 -The A n - H electronic transition of V0 in the near infra-red has been recorded at Doppler-limited resolution by Fourier transform spectros copy, and rotational analyses performed for the (0,0) band at 1.05yand the (0,1) band in 1.18 p. The hyperfine structure is prominent in the 4 4-n5/2 " x E subband, and in many of the spin satellite branches. As 4 shown by the value of the Fermi contact hyperfine parameter in the A n its electron configuration is (4SO)1 (SdS)1 HPTT)1 in the single confi guration approximation. Laser-induced fluorescence spectra of gaseous FeO have proved that the bands whose P and R branches have been analysed rotationally by Harris and Barrow (and which are known to involve the ground state) are = 4 -- i i i -ft"" = 4 transitions. The electron configuration (4s.a)' (..3d6)° (3dn)^ 5 A., is the only reasonable assignment for the ground state of FeO. The rotational structure of the 000-000 band of the 2490 K system of N02 (.2 B2 - \ A^) has been analysed from high dispersion grating spectrograph plates. The band is found to be slightly predissociated, exactly as in the ^N02 isotope, which suggests that it might be usable for laser separation of the isotopes of nitrogen. -i v-TABLE OF CONTENTS Chapter 1 Introduction Page 1 Chapter 2 Theory of Molecular Energy Levels of Free Radicals A. Introduction B. Hamiltonians and eigenfunctions (i) The general Hamiltonian (ii) Born-Oppenheimer separation of nuclear and electronic motions (iii) Rotational wavefunction (iv) Electron spin fine structure Hamiltonian (v) Nuclear spin hyperfine structure Hamilto nian (vi) Effective Hamiltonian and degenerate perturbation theory C. Evaluation of matrix elements (i) Angular momenta Irreducible spherical tensor Hund's coupling cases (ii) (iii) (iv) Matrix elements in case (b j) and case 5 6 9 9 10 13 15 23 28 35 36 40 45 53 Chapter 3 Higher Order Spin Contributions to the Isotropic Hyperfine Hamiltonian in High Multiplicity E Electronic States A. Introduction B. Isotropic hyperfine interaction in the third-order effective Hamiltonian C. Transformation to case (bgj) coupling D. Conclusion -v- • Page Chapter 4 Laser Induced Fluorescence Spectroscopy 90 A. Introduction 91 B. Saturation of molecular absorption lines 92 C. Saturated fluorescence spectroscopy 97 D. Intermodulated fluorescence spectroscopy 102 E. Resolved fluorescence spectroscopy 105 Chapter 5 Laser Spectroscopy of V0; Analysis of the Rotational and Hyperfine Structure of the C4z--X4E" (0,0) Band 109 A. Introduction 110 B. Experimental details 113 C. Rotational and hyperfine energy level expressions 118 D. Analysis of the spectrum 129 (i) General description of the band 129 (ii) Internal hyperfine perturbations 130 (iii) The band centre 141 4 -E. Electronic perturbation in the C E state 145 (i) The F4(26) perturbation 147 (ii) The F-](37) perturbation 152 F. Least square fitting of the line positions 157 (i) Deperturbation of the C^E~ F^ level positions 158 (ii) Least squares results 161 G. Hyperfine parameters 162 H. Discussion 168 Chapter 6 Laser-Induced Fluorescence and Discharge Emission Spectra of FeO; Evidence for a ^>A- Ground State A. Introduction 174 175 -vi-Page B. Experimental details 177 C. Results 179 D. Discussion 183 Chapter 7 Predissociated Rotational Structure in the 2490 A Band of 15N02 187 A. Introduction 8 B. Experimental details 18C. Analysis of the 2490 A band of 15N02 189 D. Conclusion 198 Chapter 8 Fourier Transform Spectroscopy of V0; Rotational Structure in the A^n-X^i- system near 10500 K 200 A. Introduction 201 B. Experimental details 202 C. Appearance of the spectrum 204 4 -D. Energy levels of n and i states 204 E. Analysis of branch structure 208 F. Least squares fitting of the data 213 G. Discussion (i) Spin-orbit coupling constants and indeterminacies 218 (ii) A-doubling parameters 220 (iii) Hyperfine structure of the A4n state 221 Bibliography 227 -vii-page Appendices I. Transformation between cartesian tensors and spherical tensors 240 II. A derivation of the nuclear spin-electron spin dipolar interaction matrix elements in case (bgJ) coupling 245 III. Derivation of the matrix elements of the ope rator E T1(I).T1[T3{T1(S),T2(s s.)},c2]/r. • .. . ~ ~ ~i j i j in case (b^j) basis 252 IV. Wigner 9-j symbols needed for I^.JS dipolar interaction and the third order Tsotropic hyperfine interaction 259 V. Molecular orbital description of the first-row transition metal oxides 262 VI. Rotational line assignments 265 -vi i i -LIST OF TABLES F4 levels page Table 3.1 The five types of term contributing to the third-order isotropic hyperfine interaction 75 3.2 Matrix elements of the third order isotropic hyperfine interaction in a Hund's case (a.) basis 82 p 5.1 Matrix elements of the spin and hyperfine Hamiltonian for a 4E state in a case (bgj) basis 127 5.2 Analysis of the C4E_, F4(26) perturbation 149 5.3 Analysis of the C4E-, F-|(37) perturbation 155 5.4 Calculated perturbation shifts in the VO C4£~ v=0 160 5.5 Rotational, spin and hyperfine constants for the C4E" and X4E- states of VO 163 5.6 Rotational and hyperfine energy levels of the X4E" v=0 state of VO for N < 5, calculated from the constants of Table 5.5 Values in cm-1 172 5.7 Ground state hyperfine combination differences, F2(N)-F3(N), in cm-1, for the X4E", v=0 state of VO in the range N=8-20 173 7.1 Rotational constants for the 2490 A band of 15N09 (cm"1) 1 8.1 Matrix elements of the rotational Hamiltonian for a 4n state in case (a) coupling 4 -8.2 Matrix elements for spin and rotation in a E state in case (a) coupling 207 8.3 Corrections applied to the obseryed F2 and F3 line positions to allow for the internal hyperfine perturbation shifts 215 8.4 Parameters derived from rotational analysis of the A4n-X4E" (0,0) and (P,l) bands of V0 in cm-1 217 197 206 -ix-Table I Rotational lines assigned in the C4Z~-X4E*" (0,0) band of V0 II Rotational lines assigned in the ST=4 - ft''=4 bands of FeO o 15 III Rotational lines assigned in 2491 A band of N02 IV Assigned rotational lines of the A4n-X4z" (0,0) and (0,1) bands of V0 page 267 280 282 289 -X-LIST OF FIGURES page Fig. 2.1 The step-wise development of the theory employed in the analysis and interpretation of molecular spectra 7 2.2 Hund's coupling cases (a) and (b) 47 2.3a Molecular coupling schemes including nuclear spin case (a ) and case (a0) 49 2.3b Molecular coupling schemes case (hgS) and case (.b N) 50 2.3c Molecular coupling scheme case (b^j) 51 4.1 Two level system 98 4.2 Molecular velocity distributions for both upper and lower levels under the action of a laser wave of frequency co 99 4.3 Lamb dip experiment 100 4.4 Velocity distribution curves 101 4.5 Total fluorescence intensity vs laser frequency 101 4.6 Experimental set up for intermodulated fluorescence 102 4.7 Origin of induced fluorescence lines 106 5.1 Experimental set up for intermodulated fluorescence spectroscopy | 114 5.2 Schematic diagram for intermodulated fluorescence detection system 116 5.3 Hyperfine structures of lines from the four electron spin components of the V0 C4£--X4 - (0,0) band 131 5.4 Electron spin fine structure of the V0 X4z.~ v=0 level plotted as a function of N. The rotational and hyperfine structures are not shown 133 5.5 Calculated hyperfine energy level patterns for the ?2 and F3 electron spin components, of the X4E" y=0 state of V0 in the range N""=9-22. The calculations are from the final least squares fit to the ground state hyperfine structures, and levels with the same values of F"-N" are connected 134 -XI-A A Page Fig. 5.6 The P3 branch lines of the VO C E"-X E" (0,0) band in the region N"=15-18, showing the hyperfine patterns near the ground state internal hyperfine perturbation. The F" quantum numbers for the hyperfine components are marked 136 5.7 The P2 and P3 branch lines of the MO C4E"-X4E~ (.0,0) band in the range W" =11-14. Numbers above the spectra are F-" values for the hyperfine components of the ?2 and P3 lines; other lines belonging to overlapping branches are indicated below the spectra 138 5.8 Calculated hyperfine energy level patterns for the F2 and F3 electron spin components of the C4z_ v=0 state of VO in the region of the internal hyperfine per turbation (N'=2-13). Levels with the same values of F"-N." are connected 139 5.9 The P2 and P3 branch lines of the VO C4E~-XV (0,0) band in the region N"=5-8; the F" quantum numbers of the hyperfine components are marked above the spectra. Overlapping high-N R lines and low-N Pi and P4 lines are indicated below the spectra. All four tracings are to the same scale 140 5.10 Two regions of the V0 C4z~-X4E~ (0,0) band near the band origin. Low-N lines are marked in roman type with hyperfine quantum numbers indicated as F'-F"; high-N lines are marked in italic type with only the ?" quantum numbers of the hyperfine components indi cated. Cross-over signals (centre dips) are marked 'cd1. The two tracings are at the same scale 142 5.11 Energy level diagram indicating the assignment of the four hyperfine components of the line Qef(3s) 144 4 -5.12 Rotational energy levels of the C E v=0 state of V0 (with scaling as indicated) plotted against J(J+1). Dots indicate rotational perturbations, and the perturbation matrix elements, H-]2, are given where they can be determined. The dashed line is probably a component of a 2n state (see text) with B ff=0.482 cm-"1 146 -xi i-page Fig. 5.13 Two regions of the intermodulated fluorescence spectrum of the C4£--X4z- (.0,0) band of VO. Upper tracing: the P3C38) and perturbed Pi(38) lines. Lower tracing: the unperturbed P-j(27) line 153 5.14 Upper-state term values (cm~^) of the hyperfine levels of the perturbed F-|(37) rotational level of the C4i", v=0 state of VO plotted against F(F+1) 154 5.15 Residuals (Obs-calc) for the R branch lines of the VO C4£--X4£- (.0,0) band, as compared to the positions predicted from the constants of Table5.5,plotted against N". 'Raw' data have been used, so that the R4 lines from N"=9-22, which were deperturbed for the least squares treatment (thick lines) have non-sero residuals. Vertical bars indicate the spread of the residuals for the hyperfine structure 170 6.1 Upper print: Head of the 5819 ft band of FeO. Lower print: Head of the 6180 ft band of FeO 8.3 High N main branch Q and P lines in the tail of the VO A4n-X4z- (0,0) band 180 6.2 Two regions of the intermodulated fluorescence spectrum of FeO: (a) the two A-components of the R(15) line of the 5819 ft band, (b) the Q(4) and R(10) lines of the 58:19 A band; the A-doubling is not resolved for these lines 181 6.3 Resolved fluorescence spectra of FeO produced by excitation of various lines of the 5819 A: excitation of Q(4), Q(5), Q(6) and R(16). The intensity of the excited line is anomalously high as a result of scattered laser light 182 7.1 Heads of the 2490 A bands (2ZB?-ft2A-|, 000-000) of 14N02 (above) and 15NO2 (below). The line assignments refer to the T5NO2 spectrum 193 7.2 K=7 and 8 subbands of the 2490 ft band of 15N02 (in the region 2502-2508 ft 195 8.1 Fourier transform spectrum of VO in the region 9410-9570 cm-"1 showing the heads of the A4n-X4z-(0,0) band of VO 203 8.2 Rotational structure in the A4n-X4z~ (0,0) band of VO. showing the F4" branch structure (4n5/2).. 210 212 -xii i-Fig. 8.4 Reduced energy levels of the A^n state of ¥0 plotted against J(J+1). The quantity plotted is the upper state term value less (0.50865 + 0.00365ft) {j+h)2 - 6.7 x 10-7 (J+^)4 cm-l 214 8.5 Hyperfine widths:, AEh.fS=EnfsCF = J-I), of the four spin components of the A4n state of VO, plotted against J. Points are widths calculated from the ground state hyperfine structure and the observed line widths, without correction for the Doppler width 223 -xiv-ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my research director, Professor Anthony J. Merer, for his guidance, in valuable advice and constant encouragement throughout my work and the prepara tion of this thesis. I am grateful to Dr. David M. Milton, Dr. Marjatta Lyyra, Dr. Yoshiaki Hamada, Miss Rhoda M. Gordon, Mrs. Rhonda C. Hansen and Mr. Alan W. Taylor for the many useful discussions and the friendly atomsphere created by them. I wish to thank Professor Robert F. Snider for stimulating discussions. I would also like to thank Professors Giacinto Scoles, Donald E. Irish and Terry E. Gough (University of Waterloo) for their constant interest and encouragement. Technical assistance from the Mechanical, Electronic and Glass-blowing Workshops is also much appreciated. I am also indebted to Mrs. Tilly Schreinders for kindly helping with the formidable task of typing this thesis. Appreciation is extended to Mr. Rob Hubbard for his technical assistance in obtaining the Fourier transform spectra, and also to Kitt Peak National Observatory for the used of the Fourier transform spectrometer. Finally, receipt of a University of British Columbia Graduate Fellowship (78-81) is gratefully acknowledged. -1-Chapter 1 Introduction -2-Free radicals are molecular fragments or unstable molecules which usually possess one or more unpaired electrons. This definition of C.H. Townes and his coworkers Cl) includes closed-shell unstable molecules such as CS, CF2, etc. as well as open-shell stable molecules like C^, NC^, etc. Most spectroscopists would enlarge the definition to include any open-shell or paramagnetic species regardless of stability. In this thesis we shall be dealing with molecules with open-shell ground and excited electronic states. Free radicals are of importance in almost every branch of chemistry, even though most branches are not directly concerned with their study. A chemical reaction involves the breaking and/or making of covalent bonds while the redistribution of electrons involved in either of these processes can result in species with open-shell electron configurations. Thus the intermediates in chemical reactions are often free radicals, which is why radicals are of such importance in chemistry. Radicals can undergo various types of reaction, such as decomposition, abstraction and combin ation, and consequently a wide variety of end products can result. A knowledge of these reaction pathways is important for both kinetic and photochemical studies, although for different reasons - whereas a kineti-cist determines reaction rates for individual steps involved in the total reaction, a photochemist is concerned with the way in which these inter mediates, when formed in excited states, lose their excess energy. Nevertheless, the information provided by such studies is complementary in the sense that both indicate the reaction mechanism. An understanding of the processes Involved in these complex gas phase reactions is also -3-essential to those interested in the chemistry of the upper atmosphere. As a further example of the role played by radicals in gas phase reactions, it has been suggested that many reactions occurring in inter stellar gas clouds proceed via free radical intermediates. The evidence for such reactions is supplied by the detection of absorption and emission signals by radioastronomers. The interstellar radicals detected so far have been mainly organic molecules, but in view of the high cosmic abun dances of transition metals and oxygen, transition metal monoxides are possible interstellar molecules. These radicals are, therefore, of great astronomical significance, since in any case several of them are important constituents of the atmospheres of cool stars (2). A rather different aspect of open-shell molecules arises from the presence of electron spin and/or orbital angular momenta within the molecules. Interactions between these angular momenta, although often making the analysis of the spectra more complicated, ultimately yield far more information about the electronic structure of the molecule than can be obtained for closed-shell systems. An accurate determination of the parameters describing the intra molecular interactions is invaluable in evaluating theoretical models for the electronic structure: the experimental parameters are compared with those computed using ab initio wavefunctions. The magnetic hyperfine parameters are particularly useful in this respect since they are sensiti ve to the distribution of unpaired electrons, and hence provide a rigorous test for proposed wavefunctions. Measurements of this order of accuracy call for high resolution experimental techniques such, as sub-Doppler laser spectroscopy (3), molecular beam methods (4), or microwave-optical double -4-resonance (.5). Our particular interest lies in high resolution studies giving information on the electronic structures of gas-phase radicals. The ex periments described here employ the high resolution techniques of conven tional grating spectroscopy and sub-Doppler laser spectroscopy in the visible region, and also Fourier transform spectroscopy in the near infrared region. The parameters obtained in these experiments are inter preted through an effective Hamiltonian which is restricted to operate only within the particular electronic and vibrational state from which the spectra arise. Chapter 2 deals in detail with the theory of molecular energy levels, including the construction of an effective Hamiltonian. The derivation of the third order isotropic Fermi contact interaction (a higher order effect appearing in the energy levels of high multiplicity states) is detailed in chapter 3. Chapter 4 describes briefly the technique of laser induced fluorescence spectroscopy. Chapters 5, 6 and 7 describe studies of free radicals by means of different experimental 4 - 4 -techniques. Laser-induced fluorescence studies of the C E - X z system of VO and the ground state of FeO are presented in chapters 5 and 6; chapter 7 2 ~2 is concerned with conventional grating spectroscopy of the 2 B2 - X A-| system of N02- Finally chapter 8 gives the analysis of the Fourier 4 4 -transform spectra of the A n(b) - X £ system of VO. -5-Chapter 2 Theory of Molecular Energy Levels of Free Radicals -6-A. Introduction The energy levels of a molecule are given hy the eigenvalues of the time-independent Schrodinger equation where H is the total Hamiltonian which may be written as H=V H™t + Hel +Hhfs (2-2) HQ represents the nonrelativistic Hamiltonian of the non-rotating mole cule,i .e. the kinetic and potential energies of the electrons and nuclei other than the nuclear rotational energy, HrQt symbolizes the rotatio nal motion of the nuclei, He-j contains magnetic terms that cause the electron spin fine structure and Hnfs includes all nuclear spin and nuclear moment terms that cause the hyperfine structure, v is the eigenfunction associated with a stationary state and the eigenvalue E is the energy of this state. It is impossible to solve Eq. (2.1) analytically. In practice, one chooses a convenient finite basis set <j>^ and expands the eigen-functions in terms of <j>^ ¥ =E.ai4>i (2-3) This then reduces the solution of Eq. (.2.1) to finding the roots of the secular determinant | Hfj-E^l-O (2.4) -7-where the quantities H. . are the matrix elements of H, defined as H ij=y*iH*o dT (2.5) The choice of basis set is, of course, arbitrary and any complete basis set would suffice provided the calculations are carried out to sufficient accuracy. However by making a wise choice of the initial basis set, H may be roughly partitioned into diagonal blocks (sub-matrices) between which there are only small off-diagonal matrix ele ments. In the general case the diagonal blocks refer to Born-Oppenheimer or adiabatic states. This thesis is concerned with the rotational and spin structure of individual vibronic states. A convenient way to obtain the required energy level expression is illustrated in Fig. 2.1. In the first step degenerate perturbation theory is used to include matrix elements linking a particular vibronic state with nearby vibrational and/or electronic states. This leads to an effective Hamiltonian operator which operates only within the rotational sub-space of that vibronic state. HTot Degenerate perturbation Heff Matri x element r <H > eff Matrix diaqonalizationr Ei gen-values + Ei gen-functi ons theory evaluation by computer Fig. 2.1 The step-wise development of the theory employed in the analysis and interpretation of molecular spectra. -8-The second step in Fig. 2.1 is the evaluation of a matrix repre sentation of th.eeffective Hamiltonian. Irreducible tensor methods have been used in these evaluations because they are particularly convenient for dealing with coupling of three or more angular momenta. The final step is the diagonalization of the Hamiltonian matrix and the determination of the eigenvalues and eigenfunctions. Electronic computers make this nowadays a routine matter. In section B(i) the general Hamiltonian HQ is discussed. This is followed by a description of the Born-Oppenheimer separation of nuclear and electronic motion in section B(ii). Section B(iii) is concerned with the eigenfunctions of the rotational Hamiltonian, ^r0^- Sections B(iv) and B(v) derive the operators for the electron spin fine structure and the nuclear hyperfine structure from physical principles. Finally, B(vi) deals with the important concepts and derivation of an effective Hamiltonian operator. Section C describes the evaluation of matrix elements of the effec tive Hamiltonian. In sections C(i) and (ii), some important results from the theory of angular momentum and some useful relationships in spherial tensor algebra are stated. These will be needed in the subse quent calculations. Hund's coupling cases(a) and (b), and their inter-conversion, are discussed in detail in section C(iii). Matrix element expressions are listed in section C(iv), for both case (bgj) and case (a.), in terms of Wigner's 3-j, 6-j and 9-j symbols. The absence of the third Euler angle in linear molecules, which leads to problems in the computation of "matrix elements, is also discussed. -9-B. Hamiltonians and eigenfunctions (i) The General Molecular Hamiltonian A molecule is an assembly of nuclei and electrons, which is, in Carrington's words(l), "prepared to coexist in a certain configuration with considerable stability". To understand the energy levels of a molecule it is necessary to begin with the Hamiltonian operator corres ponding to the total energy (2,3). it has been shown by Howard and Moss (4,5) that, starting from a relativistic many-body Hamiltonian, with terms correct to order c , it is possible to obtain all the familiar energy level expressions used by experimental spectroscopists , and in addition, some not readily detectable new terms such as mass polarization and spin-vibration interactions. For our purpose, however, a non-relativistic many-body Hamiltonian, in which spin is added as an additional hypothesis, is adequate to describe the system. We therefore omit the relativistic effects, and define the potential energy of the Hamiltonian as depending only on the positions of the nuclei (Q) and the positions of the electrons (q). The Hamiltonian operator for the total energy of a molecule consisting of N nuclei and n electrons is the given by H (electron kinetic energy) (nuclear kinetic energy) + V(q,Q) (potential energy) (2.6) -10-where V(q,Q)=-Z Z a l ~ia (.electron-nuclear attractions) N N , +z. z a (nuclear-nuclear repulsions) (2.7) a b>a r -ab n n 2 +z z f-i j>i ~ij (electron-electron repulsions) In eq. (2.6) P and P denote the linear momenta of electron i (mass electron charge, Ze is the charge on nucleus a and r is the radius from particle x to particle y. The magnetic interactions, such as electronic and nuclear depen dent terms, are much smaller then the electrostatic interactions which characterized V, and for the present purpose may be neglected. They will be added later in a perturbation treatment. (ii) Born-Oppenheimer Separation of Nuclear and Electronic Motion Just as the classical problem of the relative motions of three bodies cannot be solved exactly, the quantum mechanical problem of finding the exact solutions of the full SchrOdinger equation is also impossible for anything except the hydrogen atom. The approxi mation introduced by Born and Oppenheimer (6), in which the electronic part of the problem is solved first, forms a good basis for finding m ) and nucleus « (mass Mp) respectively, and in eq. (2.7) e is the (.2.8) -n-approximate solutions. The total wavefunction v is assumed to be expand able in a complete set of functions which are products of an electronic part ^ei.(..q,Q) and a nuclear part T(.Q), that is W(q,Q)=^ei(q'Q)4r(Q) (2'9) where ijje1-(q»Q) is an eigenfunction of the "electronic Hiamiltonian" ViT^Pe2 + (2.10) e e according to He^ei(q,Q)=Ee\ei(q,Q) (2.11) For eq. (2.9), which implies that the total energy £ is the sum of the electronic energy, Eg, and the nuclear energy, Eyr, that is £ = E + E e e vr (2.12) on substituting eq. (2.9) into eq. (2.8) one has P 2 (izPp2+V(q.0>lj:^-)z*e1(q,Q)*vr1(Q) -g^el^Kr'W (2.13) e e n n i i or i( e e 1 Pn2 . v(rq3Q)^pi(q,Q)+^ei(q»Q)(^:ir)^vr ei n n -12-* -(q.QHkJ-TP tq.Q)][Pn E^ei(q,Q)^vr1(Q) (2.14) Since the ^e1-(q,Q) are eigenfunctions of Hg, the first two terms give ^vr1(Q)Ee1(Q)^ei(q,Q)- Multiplying from the left by ^ek*(q,Q) and inte grating over the electron coordinates q (which essentially picks out the state k from the electronic manifold) the equation becomes 2 2 (\ n n' N n n/ ) (Q)+,iJ^ek(^^ (^ei^'Q)dcl^n]vi(Q) =^C(Q) {2J5) k Neglecting the cross-term which contains ^vyt1(Q), eq. (2.15) is a new differential equation that defines the vibrational and rotational functions of the electronic state k. The equation has the form of a Schrbdinger equation where the Hamiltonian consists of the nuclear kinetic energy, the electronic energy as an effective potential energy, and a small mass-dependent term which contributes a small electronic isotope effect. The cross-terms neglected in eq. (2.15) are those which couple the electronic and vibrational motions in different electronic states. If the electronic states are degenerate (meaning that there are two or more orthogonal electronic functions corresponding to states of the same energy) severe breakdowns, of the Born-Oppenheiraer approximation can occur: these were first described for linear molecules by Renner (7), and for symmetric top molecules (possessing a 3-fold or -13-higher axis of symmetry) by Jahn and Teller (_8). If the electronic states are non-degenerate the cross-terras give rise to the phenomenon of vibrational momentum coupling, which is one of the principal causes of the complexity of the electronic spectra of N02 and S02. (iii) Rotational wavefunctions The separation of vibration and rotation is in general difficult because certain combinations of vibrations produce motions indistin guishable from rotations, with accompanying vibrational angular momen tum. This separation has been discussed in many excellent texts (9,10), and it is not necessary to repeat it here. The success of this separa tion depends of course on the magnitude of the vibration-rotation coupling terms. If these can be neglected one has essentially indepen dent Hamiltonians which describe the vibrational and rotational motions separately. We restrict ourselves in this thesis to an extremely brief discussion of the rotational Hamiltonian and its eigenfunctions. The classical rotational Hamiltonian (10) is 02 J2 J,2 Hrot=2r + 2f + 217 <2'16' x y z where J , J and J are the components of the rotational angular momen-x y z turn J along the principal axes of the molecule, and I , I and I are the ~ x y z. principal moments of inertia (that is, where the axis system is chosen such that the off-diagonal elements of the moment of inertia tensor I vani sh). -14-If the moments of inertia do not change during the rotational motion eq. (2.16) is known as the rigid rotator Hamiltonian. The corresponding Schrbdinger equation H ip = E IJJ (2.17) rot yr ryr can only be solved analytically for spherical and symmetric tops. The motions of a rotating body (a 'top') are usually described by specify ing the Euler angles (CX,3,Y) (.11) about the principal moments of inertia which are angles describing how much the body has rotated from an initial reference configuration. The symmetric top eigenfunctions of "th eq. (2.17) are simply related to the elements of the J rank rotation matrix (12) h (J)* 2 J+1 8TT Dm (a.e.v) (2-18) where J is the rotational angular momentum quantum number, K is the projection of the rotational angular momentum vector on the molecule-(J)* fixed z-axis, M is its projection on the space-fixed Z-axis, and ' is an element of Wigner's rotation matrix. The choice of phase factor implicit in eq. (2.18) is equivalent to defining the matrix element of J , the component referred to the molecule-fixed x-axis, as real and -X positive (13) . The eigenfunctions of a rigid asymmetric top molecule (where no axes of symmetry higher than 2-fold are present) are more complicated than those just considered; they may nevertheless be expressed as -15-linear combinations of symmetric top wavefunctions (14) where <Jj^ is an appropriate numerical coefficient, the subscript K -j and K-j are the values of the component K for the limiting prolate symme tric top and the limiting oblate symmetric top, respectively. (iv) Electron Spin Fine Structure Hamiltonian The effects of spin in molecular spectra arise from the inter action of each of the electron spin magnetic moments with: (a) the magnetic moments generated by the orbital motions of the elec trons (interaction with its own orbital motion being the most im portant); this is known as spin-orbit interaction. (b) the magnetic moments generated by the rotational motions of the nuclei; this is called spin-rotation interaction. (c) the spin magnetic moments of the other electrons which is known as spin-spin interaction. Therefore the electron spin fine structure Hamiltonian operator, He-|, is the sum of these interactions, Hel = Hso + Hsr + Hss <2-20> where HsQ, Hsr and Hgs are respectively the energy operators for spin-orbit, spin-rotation and spin-spin interactions, (a) Spin-orbit Interaction An electron possesses an intrinsic spin angular momentum, s^. This gives rise to a spin magnetic moment y^, whose value is -16-~m = "^B s (2.21) In this equation g is the relativistic "g-factor", 2.0023, is the unit of magnetic moment (the Bohr magneton, efi/2m, which equals -24 -1 % 9.274 x 10 JT in SI units), and s is the spin angular momentum such that <s2>* = (sfs+l))3* fi (2.22) According to Maxwell's equations (15) an electron moving with uni form linear velocity v relative to a coordinate system where a static electric field E exists, experiences a magnetic field, B, given by B = E ^ ~ (2.23) cc as a result of its motion. Now electric field is the gradient of po tential, so that if the electric field results from the presence of a charged nucleus we have E = -vV = - (£\ dV (2.24) dr where ~ is the unit vector from the position of the nucleus towards r the position of the electron, and dV is the rate of change of the dr (Coulomb) potential of the nucleus with distance. Then B = - -4— (&\r*v (2.25) r~ L \dr/~ ~ c r \ / -17-The operator for the interaction energy of a magnetic dipole with a magnetic field is H = -JJ.B. C2.26) This becomes the operator for the spin-orbit interaction, on substituting the values of v and B; the result of the substitution is H,„ = "9PB /dV\ r„v.s (2.27) When the explicit meanings of r, and v are considered, r becomes r , the distance of the electron from the nucleus, and v becomes vgn, the veloci ty of the electron with respect to the nucleus. For completeness the electron spin s is written s . When the further relativistic correction introduced by Thomas (16) is included in the expression, v^ equals ^Xe " Xn' ^nis 'Thomas precession' represents the fact that time appears to be slowed down for the fast-moving electron as seen by the nucleus, and it happens that it appears (to the nucleus) to be spinning only half as fast as if it were static. As a result we have hi 9JJB /dV \r„,fe -vj.s; (2.28) ("en) so " 1^—i-en^^e ~n' ~e c r en The spin-orbit interaction is additive for the various electrons and nuclei, so that for a molecule H 0 = E E _J— / dv \ ^-C^ - vn) . se (2.29) • ° " n ren lfrenj -18-If eq. (2.29) is written as the difference of two terras, the second term may be taken as the interaction between the rotation of the nuclei, represented by v , and the electron spin: it is the spin-rotation interaction. The second terra will be dropped at this point and consi dered again in the next section. The first term becomes recognisably the spin-orbit interaction when we substitute l = r , * m v„ (2.30) *en ~ev ~e as the orbital angular momentum vector operator for electron e moving around nucleus n: HS0 = "^B EZj_/dLkn 'le (2'31) ° " n ren Znrnc2 6 This may be approximated as Hso = E ^e(r) £e • ie » (2-32) e where 2*mc2 n ren Kn, is a parameter for electron e, which suras over all n nuclei. Eq. (2.32) is the so-called "microscopic spin-orbit Hamiltonian" (17,18), describ ing the interaction of electron spins with the field due to electrons and nuclei, and containing spin-orbit, spin-other orbit and electronic screening effects. Using either eq. (.2.31) or (2.32) the correct para metric dependence of the interactions is obtained but the interpretations -19-of the constants differ. A convenient simple isotropic form of eq. (.2.32), (2.33) where A(.r) is an r-dependent parameter, is often used for the calcula tion of matrix elements diagonal in S. In eq. (2.33) L is the total electronic orbital angular momentum of the molecule or radical, and S is the total electronic spin angular momentum of the molecule or radical, The microscopic spin-orbit operator, Ea.(r)£..s. , is needed if matrix elements of the spin-orbit operator off-diagonal in S are to be calculated. (b) Spin-Rotation Hamiltonian The so-called spin-rotation interaction arise from two causes. One of these is the direct interaction of the electron spin with the magnetic field of the molecular rotation. It can be shown (.18) that the second term in eq. (2.29) is equivalent to (2.34) S = (2.35) E (2.36) -20-where the vector £ denotes the nuclear rotational angular momentum, and the coefficients e' are complicated functions of the .moments of inertia, the internuclear distances, and the average distances of the electrons to the various nuclei of the molecule. Besides the direct coupling between electronic spin and molecular rotation, the combination of off-diagonal spin-orbit and orbit-rotation matrix elements, treated by second order perturbation theory, gives rise to much larger effective operators with the same angular momentum dependence (19,20). The two contributions cannot be distinguished, and are therefore added to produce the determinable spin-rotation coefficient the tensor z, so that, strictly speaking, it should be written as the hermitian average: (c) Spin-Spin Hamiltonian Since electrons possesses magnetic moments, all the electrons of a molecule must interact with each other through Coulomb's law of mag netic interaction. This mechanism gives rise to the dipolar electron spin-spin interaction. A spinning electron at a point j in space can be considered as a bar magnet with magnetic moment y.. The magnetic vector potential (15) that is produced at coordinate i is given by Eq. (2.36) relates two non-commuting operators, 1^1 and S, through H (2.37) sr (2.38 -21-By Maxwell's equations this corresponds to a magnetic field at point i B. = V ^ A. (2.39) The Hamiltonian for the interaction of a second electron placed at i (with magnetic moment Vi) and the electron at j (with magnetic moment v5) is v A s . r.. Using the relations 6K (2.40) (2.41) and v /s v ~ V = v (v.V) - v V (2.42) for any vector V, this can be written .2 ~2 n The first term of eq. (2.43) can be further simplified with Gauss' diver gence theorem which gives H .. = g2pB [s, . s.) v2 - (srv)(Sj.v (2.43) 4u 5(Tj|) (2.44) -22-where <5(.r...) is a Dirac delta function. It is defined such that it picks out the square of the amplitude of the electron wavefunction for electron j at the coordinate origin, i.e. at the position of electron i <«(rji)> = 1^(0) |2 (2.45) Therefore, the first term becomes Hss'ij (contact) = -g2pB2 4^ | ^.(o) |2 ^ (2.46) The second term in eq. (2.43) is more involved, but, it can be shown that, after some algebra, it becomes 2 2 J4f U.(o)|2 + £(s..s.)r..2 - 3(s..r.,)(s..r..)]r.,~5i (2.47) ft2 | 3 |Vjv 1 L ~i ~j' ji ~ji ~j ^ji'J ji \ ' Finally, collecting terms and summing over all electrons, the spin-spin interaction Hamiltonian is (2.48) The first term turns out to giye a constant contribution to the energy, and is included with the Born-Oppenheimer potential; the second term is the dipolar electron spin-spin interaction. -23-(v) Nuclear Spin Hyperfine Structure Hamiltonian Hyperfine interactions, caused by non-zero nuclear spins, are the last terms we consider in the molecular Hamiltonian. Hyperfine structure results from the interaction of the magnetic and electric moments of the nuclei with the other electric and-magnetic moments in the molecule. We shall write Hhfs =Hmag. hfs + HQ (2-49) where Hmag ^ is the magnetic hyperfine Hamiltonian, which arises from the interaction of the nuclear-spin magnetic moments with other magnetic moments in the molecule, and HQ is the electric quadrupole Hamiltonian, which arises when nuclei with spin 151 (which possess electric quadrupole moments) interact with the non-spherical electron charge distribution in the finite volume of each nucleus. In molecules with electronic angular momentum, the magnetic hyper fine structure is usually much larger than that due to electric quadrupole moments. We begin by discussing the magnetic hyperfine Hamiltonian for a single spinning-nucleus. (a) Magnetic hyperfine structure hamiltonian The theory of magnetic hyperfine structure in diatomic molecules has been given by Frosch and Foley (21), who derived the Hamiltonian from the Dirac equation for the electron. An alternative and simplified derivation of the same hyperfine interaction expression is given by Dousmanis (22). The mechanism giving rise to magnetic hyperfine structure is exactly -24-the same as the electron spin interaction, so with the replacement of one of the electron spin magnetic moments, JJ , by the nuclear spin mag-~s netic moment, jjj, Hi - Vnl (2'5°) Eq. (2.48) becomes 2 2 H u.r = 9" P" £ j 8TT U,(o)|2 I,.S. ma9-hfs IT »>J ! ^ 1 ~J (2.5u--[(lrsj)rj12 - 3Ut.Ijl)<.,.rj1)]rj(-5| In this case the first term does not give a constant contribution; it is called the Fermi contact interaction and is a measure of the extent to which the unpaired electrons have non-zero probability amplitude at the spinning nucleus. For the amplitude to be non-zero an unpaired electron must occupy a molecular orbital derived from an atomic s orbital. Unlike the dipolar coupling, the Fermi contact interaction is isotropic, and is represented by a term of the form c,i 1 (2.52) where a .is the isotropic coupling constant, c, 1 The second term is the dipolar (I,s) interaction between the nuclear magnetic moment and the magnetic field produced at the nucleus by the valence electrons. Eq. (2.51) gives the contact and dipolar parts of the magnetic inter--25-action; on addition of the term z a. (23), which gives the inter-action of the nuclear magnetic moment with the orbital angular momentum of the unpaired electron or electrons, the total magnetic hyperfine Hamiltonian is obtained as H i^ =z a. I.£. +z a .. I .s. raag.hfs . l ~ ~i . c,i ~ ~i 1 (.2.53) -E Wn [(I.s.)r2 - 3(I.r)(s..r)]r"5 . 7.— ~ ~i ~ ~ ~i ~ It is usually sufficient to collect the sums over electrons into a single parameter, giving H ., = a I.L + b I.S + c I_S7 (2.54) mag,hfs ~ ~ ~ ~ z z where a = Wn /Jv (2.55-t2 \ „ 3 / r. l a = Sri gVByn U (o)|2 (2.56) C 3 *2 c =39WV(3 cosVlK (2.57) and b=ac-lc (2.58The terms a, b and c are determinable coefficients in the magnetic hyper fine Hamiltonian; a is the nuclear spin-orbit interaction, b is a com bination of c with the Fermi contact interaction, ac, and c is the dipo lar electron spin-nuclear spin interaction. -26-(b). Electric quadrupole Hamiltonian Although the magnetic dipole interaction is responsible for the largest contribution to the observed hyperfine structure of a molecule in a multiplet electronic state, electric quadrupole effects are present whatever the multiplicity is, provided I L 1. These are caused by the interaction between the nuclear electric quadrupole moment and the electric field gradient produced by the surrounding electrons(24). The electrostatic interaction between a nucleus at point r. and an electron at the point is given by Coulomb's law as Hc = ^ze2, = -ze£ (2.59) r. . Ir.-r. I ~i j '~i ~j 1 Recalling that electrons are point charges but nuclei have a finite size for their electric charge distribution, we shall neglect any electronic charge lying within the nuclear radius, then r* > r. and we can expand eq. (2.59) in ascending powers of r-/r- giving (25) 2 00 r k H = -ze' z li_ Pjcos e. .) (2.60) c k=o k+1 K 1J ri where P. (cos e..) is the Legendre polynomial of order k and e.• is the angle between r, and r.. The first term in the summation of eq. (2.60) represents a monopole interaction and when summed over all electrons gives the familiar coulomb interaction. The second term represents a nuclear electric dipole inter action which, by application of parity and time reversal symmetry arguments (26), can be shown to be identically zero, as are the higher electric multipole moments of odd order. Finally, the term with k=2 -27-corresponds to an electric quadrupole interaction. The separation of electric and nuclear coordinates in eq. (2.60). can be completed By applying the spherical harmonic addition theorem: k z 2k+l q=-k k where Yq (e,<|>) is the qtn component of the spherical harmonic of order k, and the angles e and <t> are spherical polar coordinates. The electric quadrupole interaction then becomes Pk(cos 6i.) - l (-Dq Yk (.V*f) Y.k (6(2.61) (2.62) We observe that the above expression has the form of the scalar product of a nuclear electric quadrupole tensor and an electric field gradient tensor, each of rank two. (The properties of spherical tensors will be discussed in section (2.G)). Therefore: HQ = e T2(Q) . T2(/VE) (2.63) For a linear molecule the tensor T (vE) has one independent component, so that there is only one "quadrupole coupling constant" which will be defined in section C2.C). -28-(vi) Effective Hamiltonian and Degenerate Perturbation Theory The analysis of molecular spectra using the true JDOIecu!ar eigen-functions is impractical, since it would require the diagonalization of an infinite matrix. Even if this matrix were suitably truncated the problem would still be very difficult to handle. Ideally a matrix re presentation is required that contains no terms off-diagonal in vibra tional or electronic state; although the matrix representation is still infinite it consists only of submatrices, each containing only elements pertaining to a single electronic-Vibrational level. Eigenvalues can be obtained for the various submatrices, from which it is possible to determine the transition frequencies. The construction of such sub-matrices requires that the effects of all elements of the full Hamilto nian that are off-diagonal in vibrational or electronic state be reduced to a negligible level. A simple practical way to set up these matrices employs an effective Hamiltonian that only operates within the manifold of a particular electronic-vibrational state. There are two commonly-used methods for deriving the effective Hamiltonian, namely contact transformations and perturbation theory. In the contact transformation method (27) a carefully-chosen unitary transformation is applied to the Hamiltonian to eliminate specific off-diagonal elements; since this method has not been used in this thesis it will not be discussed further. The technique of degenerate perturba tion theory has been used extensively in the next chapter and will therefore be discussed in more detail . Messiah (25) has described the techniques of non-degenerate and degenerate perturbation theorymost comprehensively, but a rather more -29-readable account of the derivation of an effective Hamiltonian has been given by Soliverez (28), using the formalism set up by Bloch (.29). The eigenfunctions |i> , of the total Hamiltonian which operates over all vector space, form a complete orthonormal set. We want a Hamiltonian that operates only within a particular manifold of the total Hilbert space. In other words we wish to project the effects of the total Hamiltonian operator onto a chosen vector space which is of dimension less than that of the total vector space, and hence to construct an effective Hamiltonian that operates only within this chosen vector space, and with the equivalent operator form within this manifold of the total Hamiltonian. The operator which brings about this projection of the total Hamiltonian is a projection operator, P . Soliverez (28) shows that it is possible to set up an effective Hamiltonian which has the following properties: (a) It operates only within a manifold of dimension less than that of the total vector space. (b) Its eigenvalues are identical to those corresponding eigenvalues of the total Hamiltonian. (c) Its eigenvectors are simply related to the corresponding eigen vectors of the total Hamiltonian. (d) It can be expanded as a power series in terms of a perturbation V, and is Hermitian to all orders of the expansion. We shall indicate briefly here how such a Hamiltonian is set up. The total Hamiltonian of the system under study is split into two parts -30-H = HQ + V (2.64) where the eigenvalues and eigenvectors of H are known: H0lJ>0 = E. |j>o (2.65) and the eigenfunctions |j>Q from a complete orthonormal set over all vector space. V is a perturbation to this Hamiltonian and we are interested in its effects on the eigenvectors and eigenvalues of HQ. In particular, we are concerned with the eigenvectors spanning the particular manifold onto which the total Hamiltonian is projected; these will have a particular eigenvalue, EQ. The projection operator is defined as P0 = S |1><i| (2.66) where the eigenvectors | i> span the manifold under consideration. It follows at once that H„p„ = P~ H. = EJ> (2.67) 0 0 0 0 0 0 I't is supposed that EQ can be degenerate, and that the perturbation y lifts this degeneracy. The eigenvalues corresponding to the perturbed energy levels are given by (HQ + M) |k> = (E0 + Ak) |k> (2.68) -31-which can be rearranged to give (HQ - EQ) |k> = (Ak - M) |k> (2.69) AK are the shifts in the energy levels caused by the perturbation y. Using eq. (.2.67) it can easily be shown that PQV |k> = AK PQ |k> = AK |k>0 (2.70) where the |k>Q are eigenfunctions of HQ, and in particular are those eigenfunctions |i> spanning the vector space under consideration. There is a complementary projection operator QQ which follows from the closure relationship Ho o = l\a><i\ (2.71) where the eigenvectors \i> have been excluded from eq. (2.66) since they do not span the manifold we are interested in. QQ also has the property "0 =(^K - Eo> -Uo-vtr) (2-72) \ 3 / WH6RE -IJ^lL (2.73) From eq. (2.69) and eq. (.2.72) it follows that k> =(%\(U - V) |k> 12.74) Q0 |k> =(^)(Ak -32-We are now able to find a relationship for | k> in terras of the un perturbed eigenvectors |k>Q: |k> = (P0 + Q0) |k> = |k> +/QOUA, - V) |k> (2.75) 0 W • The terms can be eliminated from eq. (.2.75) by repeated use of eq. (2.70) to give an expansion of | k> in terms of |k>0> V, PQ and^c^, that is substituting L.H.5. of eq. (2.75) into R.H.S. The eigenvectors for the perturbed and unperturbed Hamiltonians are thus related by an identity of general form |k> = U |k>Q (2.76) where U is an operator involving V, Pn and _°_ which can be expanded as a infinite series in terms of these operators. Substitution of eq. an (2.76) into eq. (2.70) leads to the following eigenvalue expression Po VUlk>o = Ak lk>o (2'77> If we identify (PQ V |J) with the effective Hamiltonian we see that it does possess the properties listed by Soliverez (28), although its hermitian nature has not been demonstrated. Note that the A^'s denote the energy shifts from an origin EQ. Eq. (.2.77) is also particularly convenient in that it uses eigenfunctions of the unperturbed Hamiltonian as basis function and these are by definition known. -33-We can now return to the derivation of an effective .molecular Ha miltonian. The basis functions Ik> are taken to be the electronic-.vi.bra-1 o tional states of interest, which are eigenfunctions of HQ, and V is the perturbation that mixes them. As described above (28,30) the total Hamiltonian is divided into two parts: H , = HQ + A V (2.78) The parameter A is a small number because V is assumed small compared to V The projection operators P0 and QQ are given the more explicit definitions Po = E l4o i><£o (2'79) i -)= E I l&ixfti 1 (2.80) a7 lH0 i (Eo - EA)n where I refers to the electronic-vibrational state of interest, l refers o to the electronic-vibrational states other than £Q and i refers to the set of quantum numbers within the manifold such as or K etc. The effective Hamiltonian is given by Heff=AP0VU (2.81) As has already been noted, U can be expanded as an infinite series -34-U = £ A" U" (2.82) n=o where Un is given by the general formula Un = zVl V SK2 V . . . SKn V Pn (2.83) o except that U° = P (2.84) o K can take the values 0, 1, 2 ... such that n K1 + K2 + . . . Kn = n K1 + K2 + . . . Kjlj (j = 1,2 ...n-1) (2.85) Y." is the condition that all Ki are non-negative. In addition n s" for n f o (2-86) As noted by Freed (31), certain terms in the expansion may be non-Hermitian, but by taking the Hermitian average of such terms the effec tive Hamiltonian is made Hermitian to all orders. By the use of eq. (2.82) and (2.83),eq. (2.81) can be expanded as follows: •35-Heff - * P0 V U = * Po V Po + A PQ V (Q0/a) V P0 ,3, +. r(P0 V (Q0/a) V (Q0/a) V PQ - IP0 V (0/a2) V P0 V P/} + A 0 4 where the dagger means that the Hermitian average of the term ia square brackets is to be taken. The coefficient of An represents the nth order contribution to the effective Hamiltonian. The expansion of the effec tive Hamiltonian is expected to converge fairly rapidly, although the rate of convergence will depend on how the Hamiltonian was originally partitioned. In practice the total Hamiltonian is partitioned in such a way that the dominant interactions arise in first order of perturbation theory. Smaller interactions are included in the effective Hamiltonian by appealing to higher orders until the required precision of the eigen values, a limit usually imposed by experiment, is reached. The effective Hamiltonian written in terms of operator equivalents will be considered in the next section after a brief discussion of angular momentum operators and some standard spherical tensor techniques. C. Calculation of Matrix Elements The concepts of angular momentum and rotational invariance play an important part in the analysis of molecular spectra. Using the general theory of angular momentum 01,12), expressions which depend only on the rotational properties of various operators and state vectors can be separated from quantities which are invariant under rotations. -36-It is worth noting that the structure of these expressions is primarily a function of the complexity of the system being studied such as, for instance, the number of angular momenta in the coupling scheme. When these ideas are to be carried out mathematically, spherical tensor algebra has proved extremely useful and also offers great physical in sight. It is not intended to give a through account of angular momentum and spherical tensor theory in this section j the aim is merely to give some important results and useful relationships that will be called upon in subsequent calculations. (i) Angular Momenta Angular momentum operators are defined as those quantum mechanical operators that obey the commutation rules (.11, 12). [Px, PYJ = i PZ (2.88) (and cyclic permutations of X, Y, Z) where Px, Py and P^ are cartesian components of the operator £. Since these components do not commute it is not possible to determine them all simultaneously. However, the an gular momentum and the energy are both constants of the motion, so that [P, Hj = 0 (2.89) where H is the Hamiltonian operator. Because they commute P and H must possess simultaneous eigenfunctions. This is an important point because -37-matrix elements of the Hamiltonian operator can be calculated using the angular momentum eigenfunctionsas basis functions. For this purpose it is convenient to write the Hamiltonian operator in terms of angular mo mentum operators and their components, rather than in terms of differen tial operators. The basis functions are defined in terms of quantum numbers relevant to the individual angular momenta rather than the ex plicit forms of the wavefunctions. There are various kinds of angular momenta that can arise in a molecule. Firstly, there is the electronic orbital angular momentum L. which is the sum of the orbital angular momenta of each of the electrons where r. and p. are respectively the position and momentum operators for the individual electrons. S and I are the electronic spin and nuclear spin angular momenta respectively. Finally we shall consider the angular momentum due to rotation of the nuclei, R. According to the rules of vector coupling1 resultants £ and £ can be constructed: L = ? *i = ? ^ £i (2.90) J = R + L + S (2.91) F = J + I (2.92) 1 The possibility of different coupling schemes and the different sets of well-defined quantum numbers that emerge will be dealt with in section C (iii). -38-J is the total angular momentum In the absence of nuclear spins, while F is the grand total angular momentum. A partial sum N=J-S (2.93) '^w rs^i r++s will also be considered. Conservation of angular momentum applies to the total angular momentum (F or J) but not necessarily to the component angular momenta. This is equivalent to saying that only the conserved angular momenta possess well defined eigenfunctions. In general, for any conserved angular momentum P we have the well known relations P2 | P M > = P(P+l)fi2 | P M > (2.94) r r Pz | P Mp > = Mp n | P Mp > (2.95) P is the quantum number of the angular momentum^, and can take integral or half-integral non-negative values. Mp is the quantum number referring to the projection of the operator JP along the z-axis (as yet undefined) and takes the (2P+1) values P, P-l, -P. The symbol "n is Planck's constant divided by 2-rr; however in what follows it will be assumed that the angular momenta are dimensionless, and the in units will be mainly dropped. It is possible to define two new operators P+ and P_ for which P+|PMp> = (Px+iPY)|PMp> - exp Ci*) [P(.P+l)-^p(Mp+1):i55 lpMp+1> (2-96> p.|pMp> - (px-ipY)|py = exp (i*) lP(P+l)-MpCMp-l)]35 |PMp-l> (2.97) -39-where P+ and P_ are called raising and lowering or shift operators, as they raise and lower by unity the projection quantum number Mp. In eqs. (.2.9.6) and (.2.97) $ is an arbitrary phase angle: the commutation rela tionships of eq. (_2.88) do not ftx the magnitude of Condon and Shortley (24) take $=0 which fixes the relative phases of the C.2P+1) states | P Mp> of different^. This phase convention is used throughout this thesis. It is well known that the operator P2 is the "generator of infinite simal rotations about the z-axis"(32), because by successive infinitesimal rotations about this axis it is possible to generate an operator for rotation through a finite angle a about this z-axis. This operator will be called D(a), and defined as D(a) = 1 - ia P - a_ P 2 4n2 = exp (-ia P2.) (2>98) In general, for a rotation of a physical system in which the coordinates of points after rotation are related to the original coordinates by the E uler angles a,B,y (11). D(CI,S,Y) = exp(-iaPz/'h)exp(-i6Py/'n)exp(-iYPz/'n) (2.99) The matrix elements of this rotation operator are defined as <P M | D(a,B,Y) I P '>= 4"Pi'k»B'T} (2'10°) P P P Unfortunately the phase conventions used by various authors (Rose (33), -40-Brfnk and Satchler (.12), and Edmonds OU) differ; the definitions for DCa.B.-y) given by Rose will be followed in this thesis. It has been stated in section B(iii) that the rotation matrix (P) elements #M M ' Ca,6,y) are the eigenfunctions of a symmetric top, and P P it can also be shown that they are eigenfunctions of the angular momentum operators (11).; therefore the angular momentum ei genfunctions |P ,M > in general can be defined in terms of rotation matrix elements. (ii) Irreducible Spherical Tensors An irreducible spherical tensor operator of rank k will be defined as consisting of (2k+l)functions T (£) (q = -k, -k+1, ...,-k) which trans-form under the 2k+l - dimensional representation of the rotation group according to D(a3y) T^(P) D"1 UY)= I (P) D[}J (aBy) (2.1C1) q ~ q q q This means that under a coordinate rotation D(aBy) the operator T (£) is transformed into linear combinations of the 2k+l operators Tqip)» where the expansion coefficients are the elements of the Wigner rotation matrices, D^(a&y). This definition can be shown (32) to be equivalent q q ' to Racah's original expression (34) . Tk(£)] = Tk , (P) [(k+qHkiq+OJ32 (2-102) and [J- , iqv^;j - iq±1 Uz. T$(£>] - q TjCP) (2-103) -41-which are the alternative definitions of irreducible tensor operators. A simple illustration of Racah.'s expressions is provided by the set of angular momentum operators; for instance, if k=l then the sphe rical components T^i.P). are related to the cartesian components of a vector P (a vector is a first rank tensor) according to Tl,{P) = + — (P-y ± i PY) (-2.104) ±1 ~ JI Y T\P) = P7 (2.105) Angular momentum operators are vectors, and can therefore be written in spherical tensor form. Tensor operators of rank 2 or more can also arise in the Hamiltonian operators, for instance, the dipolar spin-spin interaction operator and the electric quadrupole operator both involve second rank tensors. We choose to write all the terms in the Hamiltonian operator in spherical tensor notation. We are then able to use the extremely use ful, and powerful, spherical tensor technique in the calculation of matrix elements. Products of spherical tensor operations can be treated without much difficulty, which is of particular value when products of matrix elements are to be written in an equivalent operator form. Some particular equations that will be of importance are as follows; these and other standard expressions can be found in standard texts on angu lar momentum (11, 12, 32, 33, 35). (a) Tensor product of two tensor operators with |krk2|<k<|k1+k2| (2-106) -42-(b) Scalar product of two tensor operators of the same rank Tk(A).Tk(B) = Z (-l)q Tk(A) Tk(B) (2.107) q4~' -qx~/ q=-k M (c) Wigner-Eckart Theorem <P M |Tk(P) | P'M;>= (-1)P"MP/ P k P-\<P||TR(P)||P'> (2.108) / P  P\< The symbol <P||T (P)||P'>is called a reduced matrix element because it contains no reference to a coordinate system. It is the Wigner-Eckart theorem that enables terms which depend on the orientation of the coor dinate system, mainly terms involving Mp, to be factored out. In this thesis Edmonds' definition (11) of reduced matrix elements is used, which implies explicitly <P| |T(P) | |P> = [P(P+1)(2P+1)]!2 6pp^ (2.109) (d) Relation between tensor operators in different coordinate systems -these systems can be transformed into each other by means of rotations through the appropriate Euler angles. If p and q are the components of the tensor in the two different coordinate systems, where D^*{a8y) is the complex conjugate of the (p,q) element of the ktn rank rotation matrix"/3^k^(aBy) • The phase convention implicit in eq. (2.110) is that adopted by Brink and Satchler, and Rose (opposite to Edmonds). The symbol /J-, J2 J3\ in equations (2.106) and (2.108) is the I m-j m2 .m-j -43-Wigner 3-j symbol, which is a coefficient relating the eigenvectors corresponding to the angular momenta J-j and jJ2 to those correspondi ng to the angular momentum that results from coupling with ^ Wigner 3-j symbols are simply related to the Clebsch-Gordan coefficients that arise in the coupling of two angular momenta. We shall also have occasion to use Wigner's 6-j and 9-j symbols, which are needed to describe the coupling of three and four angular momenta, respectively. Wigner symbols are used here because they have greater symmetry than the corresponding Clebsch-Gordan and Racah coefficients; in addition, they are easier to manipulate. The symmetry properties of these symbols are given in standard texts (11,12). The relations of particular relevance in subsequent calculations are those by which 3-j (and 6-j) symbols can be reexpressed in terms of 6-j(and 9-j) symbols; a product of two 3-j symbols can be contracted in the following manner = E £ | J-jm^l J2m2>(-1)J1"J2 m3 (2.111) E (2 J.+1) J3 (2.112) -44-where P = £-| + £2 + £3 + y-j + y2 + y3' and tne last collect'ion of symbols in braces is a 6-j symbol. Similarly, by the use of the Biedenharn-Elliot relationship (11), a product of two 6-j symbols can be rewritten 1 J2 J12) jJ23 Jl J123| = j 3 J123 J23S <J4 0 J14 ' E (-l)b (2J124+1) 124 lJ3 J2 J23 (J14 J J124 iJ2 Jl J12 J4 J124 J14 J3 J12 J123 J4 J J124) (2.113) where S = J-, + J2 + J3 + J4 + J12 + J23 + J14 + 23 + J124 + J A product of four or five 3-j symbols summed over appropriate indices can be contracted as follows: E /B E H defghi\b e h l\ /C FI\/AD G\/D E F\ /G H I ./\c f ij\a d g/Vd e f) \g h i /A B C\ \a b cj ABC DEF G H I S /CFIWAD G\ /D E F\ /G H l\ df91 \c f ijVa d gHd e fJVg h 1/ = E (2B+1) B /A B C\ /B E H\ \a bc/\b e f) IA B C DEF G H I (2.114) (2.115) The symbols containing a 3 x 3 array of letters are Wigner 9-j symbols. In this section, only the properties of irreducible spherical tensors have been discussed. The transformation from cartesian tensors T^j -45-to irreducible spherical tensors is considered in Appendix I. In the next section, we will consider different Hund's coupling cases and the different sets of well-defined quantum numbers that are implied. After that we can calculate the matrix elements we require by applying irreducible spherical tensor techniques. (iii) Hund's Coupling Cases Provided we use basis functions that are combinations of eigen-functions of the appropriate angular momentum operators, we have essen tially complete freedom of choice in how to set up these combinations when we calculate matrix elements of the Hamiltonian. Sometimes it may be advantageous to use a basis where the calculation of the matrix ele ments is easy but the matrix itself is far from diagonal, since digital computers make the diagonalizations routine. At other times we may wish to use a basis that gives the most nearly diagonal representation, since the diagonal elements will already be a good approximation to the observed energy level pattern. The basis giving the most nearly diagonal repre sentation depends on the relative magnitudes of the couplings of the various angular momenta. In all known cases, coupling between the nuclear spin and other angular momenta by hyperfine interactions is much smaller than other couplings, so it is reasonable to begin the considerations by excluding nuclear spin. Hund investigated the various coupling schemes for elec tronic motion and showed that there are five possibilities, known as Hund's couplings cases (a) to (e) (36). Light diatomic and symmetric triatomic molecules (free radicals), such as described in this thesis, -46-normally belong toeithercase (a) or case (b), although case (c) is occasionally met. Figure 2.2 illustrates coupling schemes (a) and (b). Case (a) applies when the spin-orbit interaction is quite large relative to the rotational energy. The orbital angular momentum JL (which results from the circulation of the electrons around the internuclear axis) precesses about this axis, and I . the projection of L onto the internuclear z-axis, remains a constant of the motion but L itself is not conserved. S is coupled strongly to L by the spin-orbit interaction. The quantum num bers E and A refer to the projections of £ and JL along the internuclear -lar axis, respectively, j. and £ are coupled to the nuclear rotational angular momentum, IR, to produce the total angular momentum excluding nuclear spin, R + L + S (2.116) The quantum number for the projection of J along the molecular axis is called ft, and is given by A + E = ft (2.117) The basis set in this case is completely defined in terms of the above-mentioned quantum numbers; | n A ; S E ; J ft > (2.118) The symbol n refers to the other electron coordinates needed to describe the electronic state fully. -47-(ii) case (b) N + S = J ~ ~ n A N (K) S J > K „ A Fig. 2.2 Hund's coupling cases (a) and (b) -48-Th e case (b) scheme is important when spin-orbit coupling is small, so that there is no reason for JS to be coupled to the internuclear axis. JL has the same significance as for case (a); however in case (b) no distinction is made between the angular momenta for the motions of the nuclei and the electrons in obtaining the total rotational angular momen tum of the molecule, JN. Formally one can write R + L = N (2.119) This is then coupled with £ to form the total angular momentum excluding nuclear spin J: N + S = J (2.120) <-s^ />J I-*/ The quantum numbers ft and z are undefined in this scheme; the only 'good' projection quantum number is Kj corresponding to the component of along the z axis. K is called A for a linear molecule, where R is perpendicu lar to the molecular axis so that the projection of N, along the axis is due to the electron orbital motion only. The basis set for case (b) takes the form |nA;NKSJ> (2.121) The transformation of basis functions from case (b) to case (a) is given by Brown and Howard (13): |nA;NKSJ>= E (-1) N ~ S + ^ (2N + 1 )Vj S N\|nA;SZ;Jft> E,ft \ft -I -K/ (2.122) -49-z z A M— Z —• InASzJfil F> Fig. 2.3a Molecular coupling schemes including nuclear spin Case (a ) and Case (aa) -50-Case (bgS) K .r A I+S = G & G + N = F A (I S) G N F > N + I = Fn & F-, + S = F | n A (N I) F1 S F > Fig. 2.3b Molecular coupling schemes Case (bgS -51-I z | n A (N S) J I F > Fig. 2.3c Molecular coupling scheme Case (b5j) -52-We now consider the couplings involving nuclear spin angular momen tum. It will be assumed that only one spinning nucleus is present. The nuclear spin may be coupled with varying strength to the several molecu lar angular momenta, providing additional coupling possibilities. The commonly expected coupling schemes are shown in Fig. 2.3 . They are classified according to Hund's scheme, with the subscript a indicating that the nuclear spin is most strongly coupled to the molecular axis (as is £ in Hund's case (a)) and a subscript 3 indicating that the nuclear spin is not coupled to the molecular axis but to some other angular mo mentum (as in case (b)). For Hund's case (a), one may expect the nuclear spin to be coupled either to the molecular axis (case (a^)) or to J (case (a^)). Fig. 2.3a. However, for Hund's case (b), where the electron spin is not coupled to the molecular axis, it is very unlikely that the nuclear spin will be coupled to the molecular axis since the interaction of its small nuclear magnetic moment with the molecular fields should be considerably less than that between the electron moment and the molecular fields. Hence only the various (bD) cases are expected to occur. The vector p models and basis sets for different coupling schemes are written explicit ly in Fig.2.3 b&c.Case (bpN) where the nuclear spin is coupled to the molecular rotation more strongly than the electron spin is, is also not expected to occur. Case (bgS) is found when the Fermi contact interaction a I.S is larger than any of the electron spin interactions, but case c ~ ~ (bfi1) is the most common situation. -53-iv) Matrix Elements in case (b^j) and (a£) coupling Quantum mechanical calculations for systems having symmetry may usually be divided fairly completely into two parts. One part consists of deriving as much information as possible from the symmetry alone. The other is the evaluation of certain integrals, the estimation of parameters, or the solution of equations which have no symmetry or for which symmetry considerations can provide no information. The irreduci ble tensor methods described above are designed to separate these two parts and then to provide a well developed and consistent way of calcu lating matrix elements involving the angular momentum operators. Since the free radicals discussed in this thesis are diatomic and triatomic, matrix elements for both Hund's cases (a^) and (b^j) are given. Case (a) expressions are particularly useful in dealing with diatomic or linear free radicals. In the derivation, we follow Brown and Howard's procedures (13). They do not use Van Vleck's reversed angular momentum method (18) but instead evaluate matrix elements directly in space-fixed (p) components, rather than in a molecule-fixed coordinate system (q). Operators that are naturally defined in the molecule-fixed axis system, such as the electron orbital angular momentum I, are referred back from the space-fixed axis system by the use of the rotation matrix eq. (2.110). In this way the anomalous commutation relations are completely avoided and spherical tensor methods can be applied in their standard form. Matrix elements in the case (bD1) coupling scheme will be presented first. The details of the derivations are mainly omitted, but as an -54-illustration the matrix elements of the dipolar hyperfine interaction are derived in Appendix II. The rotational Hamiltonian for a near-prolate asymmetric top can be written in the familiar cartesian form of eq. (2.16 ), where the Ir association (10) of the inertial angular momentum axes has been made: Hrot = A Nz2+ B Nx2 + C Ny2 {2J23) The equivalent irreducible tensor form of HrQt is 2 k k Hrot = E T (B) • T (M) (2-124) k=o where the T (B) are irreducible tensors of rank k. There are only three o 2 2 2 distinct non-zero components, namely TQ(B), TQ (B) and T2(B) = T_2(B) (Appendix I). T (N,N) is an irreducible tensor operator of kth rank obtained by coupling T"1(N) with itself, according to eq. (2.106). The matrix elements of the rotational Hamiltonian HrQt in eq. (2.124) are given by < n N K' S J I F | Hrot | n N K S J I F > 2 N k / N k N\ . = I I (-DN'K C(B) ( ^ n J< N||TK(N,N)||N> (2.125) k=o q q \-K q K/ where the reduced matrix elements are < N| |Tk(N,N)||N > = (-l)k I(2k+1)]55 [N(N+1)(2N+1)] (N N 1) Ik 1 N (2.126) -55-Th e quartic centrifugal distortion terms for an asymmetric top can be described by an operator of the form (37) Hcd = -V^SK N2N2-AKN4V2(Nx2"Ny2) -6K{(Nx2-Ny2)Nz2+Nz2(Nx2-Ny2)} (2.127) This expression is valid provided that the effective Hamiltonian is soundly based (i.e. there are no strong interactions between vibrational states) and that the molecule is not an accidental symmetric top. When cast in irreducible form the general expression for is Hcd = " k=o kl!k2 Tk(v2> • <2-128> k^k2) is an irreducible tensor operator formed by coupling the vector T'(N) with itself four times Tp<4lk2> • (-I»P,KRK2 {2M)\y\ TP!(W) il^ G £ -p) (2.129) The five determinable centrifugal distortion parameters can be chosen in several different ways; eq. (2.127) represent just one such choice. The matrix elements of the centrifugal distortion Hamiltonian are -56-< n N K S J I F | Hcd | n. N K S J I F > = -AN N2(N+1)2-ANKN(N+1)K2-AKK4 (2.130) and <n N K+2 S J I F | H , | n N KS J I F > = { - 6N N(N+1) - \ 6K [K2+(K±2)2]} (2.131) [N(N+1 )-K(K±l )2h [N(N+1)-(K±l)(K±2)]% The electron spin fine structure Hamiltonian consists of spin-orbit, spin-rotation and spin-spin dipolar interactions. We will first consider the spin-orbit interaction for a non-linear molecule. The spin-orbit Hamiltonian takes the general form so where A" is a 3x3 matrix containing the spin-orbit coupling coefficients. Since molecules do not possess spherical symmetry, L is far from being a good quantum number. This makes it necessary to consider the product L+. AS° as a vector V (19). The interaction Hamiltonian is therefore a scalar product of this vector with !S, or, in irreducible tensor notation, H so S (2.132) so H so = T^V) . T^S) (2.133) The matrix elements (38) are < n' N' K' S J I F | Hso | n N K S J I F > = (-1)N+S+J [S(S+1)(2S+1)]1'£ I(2N+1)(2N'+1)]1'S (J S IT) (_!)N'-K- I I N' 1 N\ < n' 1 T1 (J (2.134) -57-The spin-rotation Hamiltonian in cartesian tensor notation (39) is Hc = 1 E' e a(N S _ + S _ N ) (2.135) sr 2 a,B aB ~a ~B ~B ~a v '• where a and g run separately over the molecule-fixed coordinates. In irreducible tensor notation, this Hamiltonian is 2 E k=o H =A lr lTk(e).Tk(N,S)+Tk(N,S).Tk(e)] (2.136) in which T (N,!S) is the tensor operator obtained by coupling the two first rank tensors T^jJ) and T^S,) in a manner defined in eq. (2.106). T (e) is an irreducible tensor (k= 0,1,2 ) and in general all nine compo nents can be non-zero; these components are usually defined in the mole cule fixed inertial axis system. The relationships between these nine components and t^, van Vleck's spin-rotation parameters (18) as extended by Raynes (40)), and Curl and Kinsey's parameters (41) are given by Bowater et_ al_ (39). The matrix elements of the spin-rotation Hamiltonian are given by the expression < n NVs JIF|Hsr|nNKSJIF> = Z (2k+l;% [S(S+1)(2S+1)]35 [(2N+1)(2N^+1)]J5 k=o (-1)J+S+N(NS J) 1 I(-Dk [N(N+1)(2N+1)]3 IsN-ir2 •1 1 kl + [N'(N'+l)(2N'+l)]35jl 1 k j i K/ N' k N\ V-K' q K / I (-1)N "K/  N\Tk(e) (2J37) -58-Th e spin-spin dipolar interaction given in eq. (.2.48) can be ex pressed in tensor notation either as the tensor product (.32) Hss=-vT0g2 yB2 i R.f ^(^(T^s.),!1^)^2) (2.138) or the scalar product H =/f0 g2 y2 l R, -3 T^sJ.T1^., C2) (2.139) " i > j ~ J where R. • is the distance between the unpaired electrons and C (e,<f>) is closely related to the second rank spherical harmonic, C2(e,<J>) = (^j2 Y2(e,d>) (2.140) e and <f> are the spherical polar angles defining the position of one electron relative to the other. The matrix elements of the spin-spin interaction are r 2 2 < TI N' K' S J I F I Hss | n N K S J I F > = -/6 g y£ (-1)S+J+N )J S N| I < S M T^.s,) ||S > J2Nsti>J E(-1)N "K [(2N+1)(2N'+1)]^K- q KJ r(C) (2.141) where the reduced matrix element z <S || T2(^- \ | S > has the i>j J value listed in ref. (42) and Appendix HI for different spin multiplici ties. The components -59-(2.142) are the parameters describing the dipolar interaction. The nuclear spin terms consist of two kinds of interactions, namely the magnetic hyperfine and the electric quadrupole. We shall confine ourselves with the case in which only one of the nuclei has non-zero spin. Thus the magnetic hyperfine interactions in eq. (2.54) can be written as (39) where the third term is the nuclear spin-electron spin dipolar interaction and R is distance between the unpaired electron and the nucleus with spin I. The matrix elements of H hfs are H mag.hfs = a T^D.T^L) + ar T^.D.T^S) (2.143) < n N' K"S I F | H mag.hfs n N K S J I F > = (-1) J+I+F F I J] I ia+l)(2I+l)(2J+l)(.2j'+l)J^ 1 J li i {2n+m.zn'+mh (-D N +S+J+1 N J S J N 1 -60-+ yN VK is(s+i).(.2s+i)]% c-nN+s+J'+1 IS J Nj a{ J S 1 C30)3sgnBgrpaN [S(S+1 )(.2S+1 )(2N+1 }(.2N'+1 )]' ,'N N 2 S S 1 j' J 1 q N'-K7n 2N\ \-K q Kj TJ<C> (2.144) where the quantities • <n|Tq(L)|n>.a and ac are experimentally determinable rameters. The components T (C) are the parameters describing the dipolar interaction. In general, there are five independent components of T (C) but for a planar molecule these are reduced to three (39). A detailed derivation of the nuclear spin-electron spin dipolar interaction matrix elements is worked out in Appendix II. If the nucleus has a spin greater than we must include the nuclear quadrupole interaction. The operator for this interaction has been written in tensor notation in eq. (2.63). The matrix elements of HQ are given by (39) . < n N' K' S J' I F | HN | n. N K S J I F > eg 2 I 2 IV1 (-1)J+I + F ji J' F / Ij I 2J (-l)r+S+J \N' J Sj [(2J+l)(2J%l)(2N+l)(2N'+l)]i2 J N 2 z (-1)N "K / H2 N\ T*(.vE) q \-K q K/1 (2.145) -61-where the electric quadrupole moment Q is defined by lQ = < HI T?, CQ) | I I > 2 I Q I 2 I\ < I || T2 CQ) || I > (2.146) /I2I\ \-I 0 I ) and T2(vE) is the electric field gradient tensor. Finally, the interaction of the nuclear spin with the magnetic field of the molecular rotation gives rise to an operator HIN cT T](I) . T^N) (2.147) 1 ^ ~ where Cj is the nuclear spin-rotational coupling parameter. The matrix elements are <nNKSj'lF|HIN|r,NKSJIF> - (-1)J+I + F (F I j'| (-1)N+S+J+1 \S N J' (1 J I ) jl J N [I(I+1)(2I+1) (2J+1)(2J'+1) N(N+1)(2N+1)]^ Cj (2.148) This completes the matrix element expressions for case (bgJ) coupling For non-linear molecules all three Euler angles are needed to specify the orientation of the molecule, but this is a problem for linear molecules because the orientation round the molecular axis is undefined so that one of the Euler angles is missing. The 'absence' of the third Euler angle leads to problems later on in the theoretical treatment when one comes to -62-compute and use matrix elements, since much of the theory, and in parti cular irreducible tensor .methods, require there to be three rotational coordinates. Hougen (43) and later Watson (44), have provided a solution to this problem by introducing the third Euler angle as a redundant coordinate in an isomorphic Hamiltonian. Consider a linear molecule which is modified by the addition of an off axis, nearly massless particle, which is bound to the molecule, but which does not affect the motion of the nuclei and electrons of the mole cule. The missing third Euler angle must be restored in order to specify the orientation of this non-linear pseudo-molecule. The visualization of such a molecular system immediately suggests that the formalism developed earlier will be applicable to linear molecules. This isomorphic Hamil tonian can be handled in the normal way, except that only certain of its eigenvalues and eigenfunctions are acceptable, the other solutions being a consequence of the redundancy introduced into the Hamiltonian. Linear molecules or free radicals where A and S are greater than zero have first order spin-orbit effects. The diagonal elements in Hund's coupling case (a) are then a closer approximation to the energy level pattern. We now discuss the operators and matrix elements for linear molecules or free radicals in case (a^) coupling. First of all, the rotational kinetic energy and centrifugal distor tion energy are described by the operators (45) HrQt =B T2(.R) (2.149) and -63-H.cd = " D T (S) • T'CR) (2.150) where R = J - L - S cv <*>^ rsw> Their matrix elements are respectively. < n A S £ J ft' I F | HrQt | n A S Z JBl F> = B|VE V, U(^)-^^(S+1)-E2 + <L2 + L2>] -2 E (-1) q=±l J-ft +S-E \-ft q ft / \-E q S q E [J(J+1)(2J+1) S(S+1)(2S+1)] (2.151) and < n A s J a' I F I HCD h A s U a I F > "D{Vl Vft [J(J+D - ft2 + s(s+i) - E2 ]2 + 4 E E / J 1 J \2 / S 1 S \' q=±1 fi"'E"U q n'7 W q z'V rJ(J+D(2J+l) S(S+1).(2S+1) ] 1 J\ / S 1 s\ q ft/ \-E' q E/ -2 * , (-1)J " / J q=±l \-ft q U(J+1)(2J+1) S(S+1)(2S+1)]35 [2J(J+1) - (ft')2 - ft2 + 2S(S+1) - (E')2 - E2] -64-+4 E E (-1) q=±l fi"E" Q +E +Q"+E" /J1J\/S1S\/J1J\ / s'is\ U(J+1)(2J+1) S(S+1)(2S+1)J | (2.152) The spin fine structure Hamiltonian consists of three different con tributions . Hc = A TV|_) T^S) represents the spin-orbit SO 0 o ^ interactions (2.153) 2 7 4 /6 A T„(S , S) the spin-spin interaction (2.154) Hss " 3 u ~ ~ H = YT^J - S) . T^S) the spin-rotation interaction (2.1 55) where A, \ and y are spin-orbit, spin-spin and spin-rotation parameters respectively. Their matrix elements are < n A S E J n I F | HSQ | r, A S E J fi I F > = A A E (2.156) <nASEJfiIF|Hss |nASEJnIF>=|x [3E2 - S(S+1)] (2.157) < n A SZ'JQ'I F I Hsr | n A S £ J n I F > = VJ6E.Z6^[,E-S(S+1)]+^ (-1)^^ J ,1J\ / S 1S\ £J(J+1)(2J+1) S(S+1)(2S+1)J^(2-158) / J .1 J\ / s 1 S\ q ft/ q E/ The magnetic hyperfine interaction terms appropriate to Hund's case (ag) coupling are H , . = a T1 (I) . T1 (L) + a T](l) . T](S) mag.hfs q=o ~ q=o ~ c ~ ~ + c T2 (I , S) (2.159) -65-where a and c are the hyperfine parameters defined by prosch and Foley (21) and ac is the Fermi contact parameter. The matrix elements are given by < n AS J:' j' n' I F | Hraag>hfs | n AS E J ft I F > = (-DJ+I+F JF J I j [1(1+1) ('21+1). (2 J+1 ).(2j'+l)]* jl I J ) E (-1)^/ j'l J\ [a A 6^ 6fl.n q q ft y-ft q + a, (-1)S"Z/ S 1 S\ [S(S+1)(2S+1)J1 7  1 S\ [S(S+1 + 1 (30)h c (-l)q (-1)S_E [S(S+1)(2S+1)]!2 / S 1 S\ / 1 2 1 \ y-E' q E ) \-q 0 q / (2.160) In a linear molecule, the charge distribution is symmetric around the molecular axis; the electric quadrupole interaction becomes HQ = e T2(Q) . T2(vE) (2.161) and the matrix elements are -66-< n, A S E J ft I F | HQ | n. A S E J ft I F > = 1 eq„Q / I 2 (-1)J+I+F JF J I 0 1/ 2 I J l(2j+-i)(2j%i);]Jl c-i)J "n / j' 2 J\ where qQ = e q = e <VEZZ> which is the expectation value of the zz com ponent of the electric field gradient tensor at the nucleus produced by the electrons. -67-Chapter 3 High Order Spin Contributions to the Isotropic Hyperfine Hamiltonian in High Multiplicity I Electronic States -68-A. Introduction The nuclear hyperfine structure of an open-shell molecule :is domina ted by interactions between the nuclear spin magnetic moments and the electron spin and orbital magnetic moments (1). The two principal inter actions between an electron spin £ and a nuclear spin I are the isotropic, or Fermi contact, interaction, which has the operator form I-S (2), and 2 -5 the dipolar interaction, with operator form [3(Ij-r)(£•£)-(I-S)r ]r , which corresponds to the interaction between the two particles treated as tiny bar magnets whose separation is given by the vector r. The Fermi contact interaction is proportional to ^ (0), the probability of finding the electron at the nucleus, and is therefore particularly large when unpaired electrons occupy a m.o.'s derived from s atomic orbitals in the l.c.a.o. description. When the unpaired electrons possess orbital angu lar momentum their orbital magnetic moments also interact with the nuclear spin magnetic moments, producing an operator of the form l^-l (1,3). The much smaller hyperfine effects that are familiar in closed shell molecules, and which do not depend on the presence of unpaired electrons, are of course still present in open-shell molecules; these include electric quadrupole interactions for nuclei with I 5-1, nuclear spin-rotation interactions and couplings between two or more nuclei. So far the hyperfine structure in gaseous open-shell molecules has only been studied extensively for doublet and triplet electronic states. States of quartet and higher multiplicity are quite rare, and high resolution spectra have only been obtained for such states in diatomic molecules; as a result not much information on their hyperfine structure is available (4). The aim of this chapter is to point out that higher -69-order magnetic hyperfine interactions are required for a full description of states of quartet and higher multiplicity, and that the largest of these is a third-order cross term between the spin-orbit interaction and the isotropic hyperfine operator. A second independently-determinable isotropic hyperfine parameter arises, whose existence is required by group theory arguments. Recent sub-Doppler optical spectra of the C E state of VO (4) have shown the need for this second isotropic parameter, and it should obvious ly be included in accurate work on all z states where S >y 3/2. There is a close parallel between the new hyperfine parameter and the second spin-4 rotation interaction parameter, originally introduced for E states by Hougen (6), and discussed in more detail by Brown and Milton (7); it will be shown that the mechanisms for their appearance are very similar, and that their qualitative effects on the level structure are analogous. B. Isotropic hyperfine interaction in the third-order effective Hamil  tonian When unseen electronic states are causing perturbations that affect every level of a vibronic state whose structure is to be analysed, it is often convenient to set up an effective Hamiltonian (8) which has matrix elements acting only within the state of interest. All the parameters determined by least squares are effective, but the problem of determin ing them is separated from the problem of interpreting them. A conve nient procedure for setting up the effective Hamiltonian, based on dege nerate perturbation theory, has been described by Miller (9). The Hamil -70-tom'an is •divided into a zero-order part that is independent of the spin contributions, and a perturbing Hamiltonian, which for the purposes of this work can be taken as (7). V=H +H +H +H rotation spin-orbit spin-spin spin-rotation + H magnetic hfs (3.1) The effective Hamiltonian consists of the zero-order part, plus additio nal terms, which, up to third order, from eq. (2.87), read Heff (1) = PoV Po Heff U) = PoV <Va>Vpo (3.2) Heff = PoV <Va>V <Va)V Po - ?LP0V (Q0/a2)V PQV PQ + PQV PoV (Q0/a2)V PQ] where the perturbing Hamiltonian V is that part of the total Hamiltonian giving matrix elements off diagonal in vibronic state, and P = I II k><! kl o ,o o 1 k (3.3) (Q /an) = z z llkxlkj o""l In eq. (3.3) the symbol 1 refers to any vibronic state, including the state of interest (which is given the special symbol 1Q), and k stands for all the rotational and spin quantum numbers for the sub-levels making up a vibronic state . -71-Since this study is concerned with hyperfine effects in E electronic states we shall write the perturbing Hamiltonian as v-B(a)2 + »ili-*iM>l3s«2-s2)+ Yt*'y'4 • + (3'4> 1 1 where the terms correspond to the way in which eq. (3.1) is written. All the coefficients in eq. (3.4) are assumed to be functions of internuclear distance, and it is further assumed that i electrons are present, but only one spinning nucleus, which has spin I. When the additional terms in the effective Hamiltonian, eq. (3.2), are computed, the largest of the higher order terms are centrifugal distortion corrections to the rotational energy and the spin energies, and corrections to all of the main parameters resulting from transforma tion of off-diagonal elements of the spin-orbit interaction. For the hyperfine structure the largest corrections can be shown, by order-of-magnitude considerations, to be cross-terms between the spin-orbit inter action and the isotropic hyperfine interaction, because in general the spin-orbit parameters a^ for the various electrons are larger than the rotational constant B, and the isotropic hyperfine parameters b^ are larger than the dipolar parameters c^ when s electrons are considered. From here on we can omit the spin-spin and spin-rotation interactions (the terms in \ and y, respectively) because they are very much smaller than the effects of the spin-orbit interaction. -72-Qualitative arguments based on perturbation theory show that the first cross-terms that give additional hyperfine interactions in i elec tronic states arise in third order. For instance, if the off-diagonal elements of V are treated directly by second order perturbation theory, and the spin-orbit and isotropic hyperfine operators are approximated as L.-!S and S-J. respectively, the second order cross-term essentially has the form o ~ ~ 1 — —^' o ' 11 0 using the notation of eq. (3.3). Even without writing explicit matrix elements it can be seen that this is equivalent to the results of having (2) an effective operator Hg^fv ' of the type J_-^ acting within the state of interest, 1 . An operator J.-^, or l^-l, must have zero matrix elements within a I electronic state, because the value of A is zero. In third order one of the cross-terms given by perturbation theory has the form <1 k|S-L|lk'xlk>|L-S|l'k"><l'k"|S-I|l k'">/AE2 (3) which has the same effect as if an operator ' of the type S/I^ (or J^-Sj were acting within the state of interest, 1 . It will be shown below that this third order effective operator Hgffv ' contains a part which is exactly equivalent to the first order isotropic hyperfine inter action, and a part which has a slightly different dependence on the spin quantum numbers; in qualitative terms the difference between the two parts is connected to the relationship between k and k"'. The first part of Hgf^3^ is incorporated into the first order hyperfine interaction, -73-but the second part gives, the new hyperfine effect which appears for states with S>3/2. The exact form of this third-order interaction is most easily derived using spherical tensor algebra. We choose a Hund's case (a^) basis, |nASZjftIF>, because it is an advantage to have as many of the electronic angular momenta as possible with well-defined eigenvalues for their mole cule-fixed z-components. The symbols in this basis are well-known; n stands for the vibronic state, and the others are all familiar diatomic molecule quantum numbers (10). Translating the operators of eq. (3.4) into spherical tensor form, we have Hspin-orbit = f^ili)- T1^.) (3.5) "isotropic hfs = JV U>- t1^-) Their matrix elements, in Hund's case (aD), are P <nASUnlF|Hsp.n_orb.t|n'A'S'i'JnlF> = z(-l )q(-l )s"^j s-Z/S 1 S q i" X Z<S| iT^.HIS-xnAlT1 (a.l.)|n'A'> (3.6) i -q ^nASlJQlFlH. . . , , |n'AS'Z'J'fi'IF> = (-1)I+J +F i F J I } 1 isotropic hfs1 ( 1 1 J ) X [I(I+1)(2I+1)(2J+1)(2J'+1)]* Z (-l)J_n/ J 1 jA q ^-ft q P.J X (-DS"Z/s 1 sA Z<S| IT1 (s .) J |S'><nAS|b .|n'AS'> ^-Z q Z J i (3.7) -74-The final terms in each of these expressions are parameters that in prin ciple can be evaluated experimentally, and can be computed by ab initio methods. Both expressions obey the selection rule AS = 0, ±1, as a result of the microscopic form of the electron spin operators; in addition the isotropic hyperfine operator is diagonal in A, but the spin-orbit opera tor follows AA = 0, ±1. When we substitute the matrix elements of eq. (3.6) into the ex-(3) pression for Hgf^ given in eq. (3.2) we get nine terms, because there (3) are three parts to He^f , and three ways of permuting the operators of eq. (3.6) remembering that H . ,.. must be taken twice. There is no ^ 3 spin-orbit need to write out any of these terms because the substitution is entirely straightforward. Many common factors occur in all nine because they are constructed similarly, and closer examination shows that they can be collapsed to five different types of term, which must be evaluated sepa rately. Table 3.1 summarizes the properties of these five types. It is apparent that the quantum numbers S and z for the distant per turbing electronic states occur in the matrix elements of the third order effective Hamiltonian, but they must not appear explicitly in the final expressions because the effective Hamiltonian is assumed to act only within the vibronic state of interest. It is therefore necessary to use relationships between the Wigner coefficients to sum over these quantum numbers as far as possible, and to cast them into the form of an experi mental parameter or parameters. We follow Brown and Milton (7), who en countered a similar problem in their discussion of higher order spin-rotation interactions in multiplet I states, and solved it by applying the relation in eq. (2.112). Table 3.1 The five types of term contributing to the third-order isotropic hyperfine interaction. First Second Initial Operator intermediate Operator intermediate Operator Final Energy state state state state denominator 1 SA S .0. S'A' iso S'A' S .0. SA S .0. S'A' iso SA' S .0 SA S .0. SA' i so S'A' S .0 . SA 2 SA S .0. S'A' s .0 . S'A' iso SA iso S'A' S .0 . S'A' S .0 . SA 3 SA S .0. SA' S .0. S'A iso SA iso S'A s.o. SA' S .0. SA 4 SA S .0. S'A' S .0. SA iso SA 5 SA iso SA s.o. S'A' S .0. SA (E0-E1><E0-E25 (E0"E1> -0-t2j The operators are s.o. = spin-orbit and iso = isotropic hyperfine (see eq (3.6)) -76-For example, in the terms of type 1 from Table 3.1 we haye the product / s i s'\ (-i)s'-Ws- i s '\(-i)s'-E'/ sis \ where the single and double primes on S and I refer to the intermediate states, and 1"' refers to the final state. After two applications of eq. (2.112) this product becomes (-1)S" _S"C« I (2k+l)(2K+l) JS S K j/S S K\/ 1 k K\ k'K ii k s"f\i -E"-q7\-q q+q'-'q/ j S" S kI / 1 1 k \ X 1 1 1 S't\-q -q' q+q7 In this expression k and K are tensor indices that arise in the successive applications of eq. (2.112). It will be seen that nowhere do the inter mediate state E values (E' and 1") appear. Eventually the general matrix element of the third order effective Hamiltonian can be obtained as <nASEjftIF|H.so(3)|nASE"'j^'s?'"lF>= (-1)J'"+I+F)F J I ( 1 I J'" XLKI+1X2I + 1)]11 [(2J+1)(2J'-+1)J15 I(-l)J~r/Jl J'"\y(-DS"Z/SK S q' Uq- -'ik U x\ £(-l)S'~S(-l)q(2k+l)(2K+l)/ 1 1 k \/l k K\{1 k K j (J} Vq -q' Q+q7\q -q-q' q'/ls S S"( -77-X I<^|T]q(aili)|n'A'><n'A'|Tq(ai^i)|nA><S'1|T1(si)||S> 1 JS S" S'j x (E^,-E^.fl.^J•1{<nA'S1b.|n'^^'S->(-l)Uk+K+<n''AS-|biinAS>[l + (-^)K+1 *• nAS n A b 'i 1 [S(S+1)(2S+1)]J5 y<S|!T1(si)||S"> jl 1 kj(E°AS-E°_A.s_) 1 Js S" s( -2 x <nAS!b.|nAS>xin+(-l) 1 (3.8) The separate contributions of the five terms from Table 3.1 can be distin guished in the bracket forming the second half of eq. (3.8). The triangle rules from the 3-j symbols limit the values that k and K can assume, such that k can be 0, 1 and 2, and K is then restricted, according to the value of k, to k = 0 1 2 K = 1 0,1,2 1,2,3 -78-K+1 It is evident that the coefficients 1+C.-1). in eq. (_3.8) cause most of it to yanish for even K values, and only the term with coefficient (-1). coming from the type 1 terms in Table 3.1 is left. However it can be shown, by arguments similar to those used by Brown and Milton (7), that this also vanishes for K even. The procedure, in essence, is to prove, by the Biedenharn-Elliot relationship (eq. 2.113), that the terms 1+K with q=l and -1 in the sum over q in eq. (3.8) differ by a factor (-1) ; therefore they are equal and opposite for even K. The q=0 term is easily shown to be zero for even K, so that the whole sum vanishes. One of the steps in the proof requires the equality <nA|Tq(aili) |n'A'> = <n'A'|T^(aili) h A > (3.9) which therefore limits the results to A=0, i.e. z states only. The value of K can be 1 or 3 only, in consequence. Consider K=l first. The two 3-j symbols involving k in eq. (3.8) can be contracted: 2 I I 1 1 k \/l k 1 \ l(\ 1 k \ n-q-q-q+q7W^q7= "WW (3J0) and, using the orthogonality properties of 3-j symbols [11], become W1 1 k \? = 4- (3.11) q \q q- -q-q'/ The general matrix element reduces to -79-<nASE0nIF|His^3),K=1|nASZ'"J'-n'"IF> = (-l)J'"+I + F JF J I }[I(1+1)(21+1)]' 1 I J ( *y(-i)J"7ji j-\(-i)s"7s 1 S \l I I (-i)s"-s(-Dq(2k+i) X HMT] In'A-xn-A'lTjta^^ |nA> \<S | 1T1 (si) | |S"> x jl k 1 is S S" I <S I IT1 (^ ) 1 IS - > <S -1 IT1 (^ ) 1 IS - > jl 1 k j U°AS-En'A'S'rl S S" S'\ X (E°AS-E°_A.s.J'1{<nA'S'|bi |nA'S">(-l)k+1+2<n"AS"|bi|nAS>) (3.12) - lS(S+l)(2S+l)]l5I<S!|T1(sui)]|S-> jl 1 k|(E°AS-E°_;i.s.J_2<nAS|biJnAS> 1 IS S" s which can be seen, by comparison with eq. (3.7), to be exactly similar in form to the normal isotropic hyperfine matrix elements. For K=l the third order spin-orbit interaction therefore gives a higher order contribution to the Fermi contact parameter. The final remaining term in eq. (3.8) has K=3 and k=2. Using eq. (2,112) again, the pair of 3-j symbols involving k in eq. (3.8) can be recast into the form -80-'3 1 2 vq' q -q-q 11 2 j = (-l)q_q' •q -q' q'+q (5) f3 1 2)C '2\ j312( (3.13) Vq'-q' 0/\q -q 0/ h 1 2J (-1)' ( 1 2\(-l)q -q' 07 C 1 \q -q 2 q 0 (3.14) where we have substituted the value ^3 1 2)1 ( «- (3.15) 112 b It turns out that the 3-j symbol with q' in it is important when we carry out the transformation from case (a^) coupling to case (bgj)> s° "that it must not be incorporated into the sum over distant states comprising the experimental parameter. The K=3 term then becomes <nASUfiIF|HisJ)3),K=3!p1ASl"'J"Tr"IF>= (-1)J'"+I+F JF J I j |l I J"-i n(i+D(2i+i)]-X [(2J+1)(2J'"+1)]^ I (-l)J"n/ J 1 J'"\(-1)S-E / S 3 S q '-!) q' fl"7 \-E q7 E S"~S / t T 0\ , I Tl x (-DH/3 1 2^ W -q' o, 35 I U-lf ° /I 1 2\ <nA|Tq(aili) |n'A'> q -q oj \ 35 I I(-1)J "J / j |_ S'A'S" q \^ <T>'A'|T]q(a.l(i)|nA> J <S||T1(^i)||S-> jl 2 3 ( [J<S | | T1 ) | |S '> 1 Is S S"! 1 -81-•1 /^O ro \-l x<S-MT1(li)iiS-> jl 1 2 j (E°AS-E:-A-S-» <Cs-En-Is S" S'j {<nA'S'|bi | n "AS ' ->+2 <n "AS " | bi |nAS>] - [S(S+1) (2S+1) ]** 1<S | IT1 (s.) | |S "> (3.16) Eq. (3. 6 ) represents a new type of hyperfine interaction matrix element, which, as can be seen from the properties of the second 3-j symbol, is non-vanishing for electronic states where S ^ 3/2. The form of eq. (3.16) is somewhat similar to the isotropic hyperfine matrix element given in eq. (3.6); the differences are that the reduced spin matrix element appears in another place, and that the simple matrix element of b^ is replaced by the complicated expression between the large brackets which becomes the experimental parameter. Our definition of the new experimental parameter has been chosen with the analogy between it and Brown and Milton's second spin-rotation para meter -ye- (7) in mind. Not surprisingly, since the mechanisms for their appearance are similar, the analogy between the two parameters is very close. To make the analogy as close as possible we name the new parameter b^, and define it as bs=-4( 3/35)35[(2S-2)(2S-l)2S(2S+l)(2S+2)(2S+3)(2S+4)]~^t (3.17) where t is the complicated expression in large brackets in eq. (3.16). The reason for the peculiar numerical factor will become apparent when we con sider the matrix elements in case (b^j) coupling in the next Section. Table 3.2 Matrix elements of the third order isotropic hyperfine interaction in a Hund's case (ag) basis <T)ASEJMF|H. ^ |nASEjftIF>= -bcflE[F( F+l)-!(I +1) - J( J+1) ][3S(S+1)-5E2-1 ] iso i 2J(J+1) <nASE JftIF|H. SJ)3) |DASE , J-l ,«I F>= M( J2-Q2)hl( F+I+J+1)(I+J-F) ( F+J-I) ( F+I-J+1) ]h [3S(S+1 )-5g2-T]_ 2J(4J2-l)ia <nASEJnIF|H. ^ |nAS,E±l ,j,n±l ,IF>= -bP[J( J+1)-^(^±1)]^[s(S+1)-E(E±1)]^ iso b 4J(J+1) x [F(F+l)-I(I+1)-J(J+1)][S(S+l)-5E(E±1)-2] <nASEjfiIF|H.s^J; |nAS,E±1 ,J-1 ,n±1 ,IF>= ± bs [( JT£>) ( JTB-1 ) ] rc(£j i) £(z±i)]'/z 4J(4J2-!)'5 x [( F+I+J+1) (I+J-F) ( F+J-I) ( F+I-J+1) ]'5[S(S+1 )-5E(E±1 )-2] The phase choice for the rotational wave functions follows that of Brown and Howard (14) or Carrington, Dyer and Levy (l9), based on Condon and Shortley's conventions (20). -83-On substituting eq. (.3.17) into eq. (.3.16), and writing explicit ex pressions for the Wigner coefficients, the matrix elements of the third order isotropic hyperfine interaction in case (aD) coupling can be obtained; P they are listed in Table 3.2. C. Transformation to case (b^j) coupling Of the various states known, only two, those in GeF (12) and SnH (13), show marked departures from case (b) coupling. Therefore, despite the logical preference among diatomic spectroscopists for calcu lating energy levels in a Hund's case (a) basis, it is instructive to look at the form of the matrix elements in case (b) coupling, because the diagonal elements show directly how the parameter affects the level struc ture in a real situation. With the case (b) functions given in terms of case (a) functions (14) by |NASJ>= I (-l)N"S+n(2N+l)'a S |ASEJQ> (3.18) •n Vft -1 -A we obtain <NASJIF|Hiso(3),K=3|N'A'SJ'IF>= t.(-l)I+J'+F JF J I j [I (1+1) ( 21+1) l*5 x I I (-l)N"S+n+N'-S+fi[(2N+l)(2N'+l)]lj /J S N\/J'S N'\ i,n r$r \p -z -A] yr-r A'/ -84-X q [(2j+i)(2j'+i)],s (3-19) / J 1 JA(-1)S-Z/S 3 S i(-Dq73 1 2\ (where we have replaced the triple primes of eq. (3.16) by single primes) After some rearrangement (which by happy chance eliminates virtually all the phase factors) we can contract the sum over the product of five 3-j symbols to an expression involving a 9-j symbol (15); remembering that A and A' are restricted to the value 0 we finally get <NSJIFiHis|)3),K=3|N^J'IF>= 1(35/3)%. (-1)I+J' + F j F J I ! Il I J') X [I(I+1)(2I+1).(2J+1X2J'+D .(2N+l)(2N'+l).(2S-2)(2S-l)2S(2S+l)(2S+2) (2S+3)(2S+4)r (-ir /N 2 N'\ |N N' « (320) .1)N /N \ |N N \p o o Ms s which is the desired result. These 9-j symbols are unfortunately not listed in standard tabulations (16), and give very cumbersome algebraic expressions, but vast amounts of cancelling occur in the evaluation of actual matrix elements, so that quite simple expressions are finally ob tained. Our algebraic forms for the 9-j symbols are given in ref. (5). -85-We do not list general forms for the matrix elements of eq. (.3.20), but give merely the diagonal elements for a 4i state. It is useful to include the diagonal elements, of the isotropic hyperfine interaction b I.S^ in these expressions: F1(J=N+|) F2(J=N+^) F3(J=N4) F4(J=N-|) C[b-b.N/(2N+3)]/(2N+3) ic[b(2N+9)+3bs{(3N+2)+3/(2N+3))]/[(2N+l)(2N+3)] jC[b(2N-7)+3bs{(3N+l)+3/(2N-l)}]/[(2N-l)(2N+l)] (3.21) •|C[b-b.(N+l)/(2N-l)]/(2N-l) where C=F(F+1)-I(I+1)-J(J+1) (3.22) It can be seen how when N is large, so that similar powers of N can be cancelled, these expressions simplify so that there is one effective b para meter for the F-| and F^ levels and another effective b parameter for the F2 and F3 levels: F1 and F4: beff^-2bc; F2 and F3: beffb+2bS (3.23) The 3-j symbol and its phase factor in eq. (3.20), if the values of A and A' are left unspeci fied, ire actually (-1)N"A/N 2 N\, which, when the {-A q K') normalization factor [(2N+1)(2N'+1) is included, is the reduced matrix element of the second rank rotation matrix (17): -86-<NA||P(2)*(a))||N'0=(-l)N-A[(2N+1)(2N'+l)]15 / N 2 N'\ (g \-A q A'j This suggests that it is possible to devise an equivalent operator, acting within the manifold of a given vibrational level of a multiplet l electronic state, which has the same matrix elements as eq. (3.20) but which gives a different perspective on how the new hyperfine interaction operator is con structed . After some experimentation the equivalent operator was found to be Hi,i3).^1j/U).T,[T3n,(y.T^tl^,).c2]/r13 (3.25) 2 where i and j are electrons, r^ is their separation and C is related to the spherical harmonic giving their relative polar coordinates, ^=(4^/5)^^(6^) (3.26), The matrix elements of eq. (3.25) are identical to those of eq. (3.20) 2 except that they are given in terms of a parameter TQ(C), which must be expressed in terms of b^, according to bs=(3/10)T2(C)/( 14)32 (3-27) The derivation of the matrix elements of eq. (3.25) is interesting because it involves a number of widely-occurring electron spin reduced matrix elements, several of which appear not to have been given in general form (though some explicit expressions have been given by Brown and Merer (18). -87-In order to interpret the parameter according to eq. (3.16), or to ob tain expressions for the A-doubling parameters, o, p and q, in terms of the spin-orbit matrix elements for high multiplicity states (18), it is useful to have these general forms. Accordingly the derivation of the matrix elements of eq. (3.25) is given in Appendix HI. The operator form of eq. (.3.25) shows exactly how the effective opera tor for the new third-order cross-term is constructed. In Cartesian tensor notation it consists of a sum of terms of the type I S.S S •, the advantage of the spherical tensor form is readily appreciated. Exactly similar expressions to eq. (3.21) are found to hold for Brown and Milton's Y$ parameter (7). The transformation of the case (a) matrix given in ref. (7) to case (b) coupling is rather more messy than the trans formation of eq. (3.19) to eq. (3.20) because now there is only a partial sum over the index q" (which cannot take the value zero since the spin-un coupling operator is -2B(Jxsx+Jysy) "rather than -2B J.S so that the q'=0 component is missing). After some algebra we find <NSJiHspin-rotation'K=3lN'SJ^ X [(2S-2)(2S-l)2S(2S+l)(2S+2)(2S+3)(2S+4)]ls X I (22+1) / 3 z 1 \ (-1)N /N zN'\ N z j 2=2,4 1-10 1/ \0 0 0 / S S 3 (3.28) J J 1 1 As explained in ref, (5) the matrix elements are more easily obtained by an algebraic transformation of the case (a) matrix rather than directly from eq. (3.28) because of the complexity of the 9-j symbols. Correspond ing to eq. (3.21) we have -88-F1(0=N+|): |[Y-YS(N+1)/(2N+3)]N F2(J=N+2-): |[Y(N-3)+3YSN(3N+5)/(2N+3)J F3(J=N-^) : -J[Y(N+4)+3YS(N+1)(3N-2)/(2N-1)] F4(J=N+|): -|[Y-YSN/(2N-1)](N+1) which simplifies for high N to F] and F4: YFIFF ^ Y-^$; F2 and F3: Yfiff , Y+1Y$ (3-30) The points we make in this section are (i) that by choosing the numerical factor as in eq. (3.17) we can define the new hyperfine parameter bs so that eqs. (3.23) and (3.30) have exactly the same form, and (ii) that the case (b) expressions show how the third order isotropic hyperfine term and the third-order spin-rotation term both act in the same way, which is to give the F-| and F^ levels different effective parameters from the F2 and F3 levels in a 4£ state. In addition it is possible to derive the form of the effective operator, acting entirely within the manifold of the £ electronic state, which is equivalent to the third-order isotropic hyperfine term. D. Conclusion This chapter gives the background theory for the new hyperfine para meter bs which had to be introduced by Cheung et aj_ (.5) to explain the -89-4 -hyperfine structure of the C z state of VO ltieasiired by sub-Doppler laser-induced fluorescence. The new. parameter is a third-order cross-term be tween the spin-orbit interaction and the familiar isotropic hyperfine interaction, and, like the corresponding spin-rotation effect (7), must be included in accurate work on all electronic z states of quartet and higher multiplicity. The new effect will be especially large if there are nearby states that interact strongly through the spin orbit operator with the state of interest; therefore it will probably be more important in the excited electronic states of high multiplicity molecules than in their ground states, since ground states are often well separated from other interacting electronic states. The new term in b^ is in fact required for al1 electronic states of quartet and higher multiplicity, not just Z states. The reason is that no approximations have been made in its derivation which limit the value of A (or K for polyatomic molecules), so that eqs. (.3.16) and (3.17) for case (a) coupling, or eqs. (3.20) and (3.24) for case (b) coupling, are entirely general. The only restriction to z electronic states is in eq. (3.9), which was invoked to prove that the terms involving the tensor ranks K=0 and 2 vanish for A=0. We have not investigated the consequences of not invoking eq. (3.9), but qualitatively it seems that the K=2 terms should give rise to a higher order contribution to one of the other hyperfine operators, probably a J^.J., which is non-vanishing when A/0 -90-Chapter 4 Laser Induced Fluorescence Spectroscopy -91-A. Introduction The advent of high power monochromatic tunable laser sources has stimulated important advances in optical spectroscopy as documented by several recent reviews (1 - 4). Laser induced fluorescence (which will be taken here to mean the process whereby a molecule absorbs laser light at one wavelength and emits some fraction of the energy as light at the same and other wave lengths) is a much more sensitive technique than absorption spectroscopy (5). The ratio of (S/N)abs and (s/N)fiuor> where S/N means signal-to-noise ratio, is proportional to the noise equivalent power of the detector, NEP, and the fourth power of the fluorescence wavelength. As the fluorescence wavelength decreases, fluorescence detection is increasing ly favored. In regions where photomultipler tubes can be used, the NEP drops considerably and fluorescence detection becomes even more favorable. Consequently, most laser experiments done in those regions use fluorescence detection techniques. The highest resolution in optical spectroscopy is achieved by eliminating the Doppler broadening of atomic and molecular spectral lines. The high intensity of laser light has led to the development of a variety of new nonlinear spectroscopic techniques which permit Doppler-free observations of a simple gas sample. Saturated absorption spectroscopy or Lamb dip spectroscopy (6,7) was the earliest developed and perhaps the most widely used of these methods; here the spread of atomic velocities along the direction of observation is effectively reduced by velocity-selective bleaching and probing with two counter-propagating monochromatic laser beams. Saturated fluorescence spectros--92-copy (8) and in particular the sensitive technique of intermodulated fluorescence (9) extended the potential of thisjoethod to optically very thin fluorescent samples. In this chapter, various effects related to laser experiments will be discussed; they include (i) non-linear interactions of mole cules with a very intense laser beam to produce saturation effects, (ii) observation of excitation spectra (which are essentially absorption spectra provided no radiationless process is occurring) by monitoring the total fluorescence, and (iii) the use of saturation effects to study atomic or molecular lines without Doppler broadening. Moreover, the experimental techniques of intermodulated fluorescence and resolved fluorescence will also be considered; the former yields line positions p to very high precision (1 part in 10 ), while the latter gives relation ships between lines which permit unambiguous rotational assignment. B. Saturation of Molecular Absorption Lines In light absorption experiments at "conventional" low power levels Beer's law states that the absorbed power is a constant fraction of the incident power. This linear relationship holds only if the incident power is low. In contrast, intense coherent light sources, which are capable of supplying very high power densities, and hence a very strong optical electric field, can generate a wealth of non-linear phenomena as a result of the dielectric polarization of the absorbing medium by the intense field. The order of magnitude of the optical electric field £ required to produce non-linear effects can be obtained from Heisenberg's uncertainty pri.nci.pl e -93-jje-T»f|. (.4.1) In this expression JJ is the electric dipole matrix element for the transition, T the relaxation time and fi (= Planck's constant h, divided by 2-n) ^ 10"34 J.s. A laser power of 1 W through a cross-section of 2 5-2 0.1 cm (typical beam size) gives a power density of 10 Wm and 4 -1 produces an electric field of e ^ 10 NC • , so that a power of 1 pW gives E ^ 10 NC"^ (since power density is proportional to the square of the electric field). Therefore, a laser power of 10"^ to 1 W is adequate to observe non-linear effects in molecules with p = 1 D and a relaxation time of 10"^ to 10"8 s. A rotational relaxation time of -3 this magnitude is typical at a gas pressure of 10" to 1 torr (10). Let us consider the behavior of an isolated system with two energy levels, E-| and Eg, under the action of an electromagnetic field. The behaviour of the system, that is its wavefunction is described by the time-dependent Schrodinger equation, ifi dv_ = (4.2) 3t where H is the total hamiltonian of the system, composed of the unper turbed hamiltonian HQ and the energy of the quantum system-field inter action, or, more specifically in this case, the electric dipole inter action between particle and field, which has the form H' H = -p "E = H + H' 0 = H - p-e cos IDt (4.3) -94-In eq. (4.3) JJ is the component of the jnol ecular dipole moment along the direction of the field, e is the,strength of the electric field of the light wave, and u its frequency. The wavefunction ¥ can be expressed in term of eigenfunctions of the HQ operator, * = £an(t)<|>n (4.4) i.e. as a superposition of the wavefunctions $ of the quantum system without a light field, where $ is defined as ifi ^n = H0 <j»n (4.5) "at Then the equation to determine the coefficients in the expansion i s It3n= tk H'nk3k 6XP [fT(En"Ek)t] (4-6) from eq. (4.2) . In a two-level system, n = 1 and 2, and the single transition frequency is (JJq = (E 2 - E -j) /ti. Eq. (4.6) gives a pair of coupled equations for a-j(t) and a^(t) da-i= i_ K E a? {exp[i(w-u> )t] + exp[-i(w+w )t]} dt 2 L 0 0 (4.7) dao= i K e a, {exp[-i( to-w )t] + exp £i(w+w )t]} dT 2 1 0 0 where K = 2 y-^/ti, in which p-|2 is tne dl*pole matrix element between the states 1 and 2. As long as the Rabi frequency ^ = Ke is very much -95-less than w , we may neglect the high-frequency terras, exp£ i(o)+co0)t], in the rotating wave approximation; then eq. (4.7) gives d2a? + i(a)-aij dao + (Ke)2a = 0 (4.8) dt2 ° dt 4 2 The general solution to eq. (4.8) is a2(t) = [A exp(iftt/2) + B exp(-iftt/2)] exp(-iAt/2) M*) = - ]_[(A-n)A exp(iftt/2) + (A+ft)B exp(-iftt/2)] exp(iAt/2) ' Ke with A = 03-a>o and ft = .[A + (KE) the constants A and B are determined from the initial condition of the system. Assuming that the molecule is initially in the lower state 1, al^to^ = exP(ie) (i-e- one multiplied by an arbitrary phase factor) and a2(tQ) = 0. This gives the coefficient a,(t) = fcos ft (t-t)-i A sin ft (t-t )'lexp Ei e+iA(t-tJ/2] 1 L2°ft2°J a2(t) = i Ke sin p(t-t0) J exp[ie-iA(t-tQ)/2] (4.10) The squares of the coefficients which correspond to relative populations of the two states are Nn(t) = |a.(t)|2 = A2 + (Ke)2 COS2 ft(t-tj ft^ ^ d (4.11) No(t) = |a9(t)|2 = (Ke)2 sfn2ft(t-tn) ft C In a gas at low pressure, the coherent oscillation of the molecular di pole moment is interrupted by collisions between molecules and by the -96-Kfe time of the eigenstate. The effect of collisions can be incorporated in this treatment by averaging eq. (.4.11) over a Poisson distribution of dephasing collisions with relaxation time T. The probability that the molecule has survived under coherent interaction with the field in the interval t = tQ is dN(t) = 1 exp[-(t-t0)/x] dt (4.12) T The transition probability for an ensemble of molecules in the gas is then obtained by taking an average over tQ, to give <|a^i"> = i i ia„it.i: II exDi"-(t-t.)/x] dt. i2|2> = 1 f l^'VI exp[-(t-t0)/x] 0 = 1 (Ke)2 (4.13) 2 (w-o>o)2 + T"2 + (.Ke)2 This average increases monotonically with the intensity of the radiation field and approaches 0.5 as the limit e ->-«>. This means that a very intense field will eventually equalize populations between upper and lower levels of a transition. The power absorbed, which is the observable in this system, can be obtained as AP = dW =(NrN2) fiw (Ke)2 dt 2T L>-U)o)2 + x-2 + (Ke)2] where N-j and N2 are the numbers of molecules in the states 1 and 2, respec tively. The power flow per unit area in SI units P = l«„.ce2 (4.14) 2 -97-where €c is the dielectric constant and c is the speed of light. As E -><*>, AP becomes a constant, the power absorption coefficient of a gas of two-leyel molecules is given by a = AP = (N,-NJ 2fiwK2 -> 0 (4.15) p ! r. €„ T I(o)-a.o) + T~2 + (KE)2] and the medium saturates. With co = ID , this can be rewritten as the phenomenological expression a = o a- (4.16) 1 + I/Is where all the appropriate factors are incorporated into aQ and 1^. The saturation parameter 1^ is the power per unit area that a wave on reso nance must carry in order to reduce the population difference to one-half its unsaturated value (11). With a moderate intensity (I < Is), we have a = an (1 - I + ...) (4.17) T A similar derivation, by the use of density-matrix equations, is discussed by Letokhov and Chebotayev (3). Saturation of Doppler-broadened absorption lines has been considered by Shimoda and Shimizu (10). C. Saturated Fluorescence Spectroscopy A Doppler-broadened spectral line is a sum of a great number of much narrower lines corresponding to molecules with different thermal -98-velocity, v. This is why the Doppler effect on spectral lines is often called inhomogeneous broadening. A coherent light wave of wave vector k interacts only with particles it resonates with, that is, with parti cles for which the Doppler shift in the absorption frequency, k.v^, com pensates precisely for the detuning of the field frequency, w, with The excitation of particles with a certain velocity changes the equilibrium distribution of particle velocities in each level of the transition. In the lower level there is a lack of particles whose velo city complies with the resonance condition, that is, a hole appears in the velocity distribution, an effect i- known as Bennett hole burning (12), Fig.4.2. By contrast, in the upper level there is an excess of particles with resonance velocities or a peak in the velocity distribution. The hole depth and peak height depend on the degree of absorption saturation by the light field. The width of the hole deter mines the homogeneous line width, which can be thousands of times less than the Doppler width. respect to the transition frequency, w , of a fixed molecule, (Fig. 4.1) 0) = w + k.v (4.18) o 7\ co =(E2-Eil/fi Fig. 4.1 Two level system -99-Fig. 4.2 Molecular yelocity distributions for both upper and lower levels under the action of a laser wave of frequency u>. A related phenomenon known as the Lamb dip (7) forms the basis for many experiments in saturation spectroscopy (6). Although Lamb has shown, in his gas-laser theory (7), that the interaction of a Doppler-broadened line with a standing wave produces this phenomenon, in fact, the light wave need not be a standing wave: a strong travelling wave is sufficient to produce the same effect (9). Also this signal can be detected by monitoring the total fluorescence, which is just a constant fraction of the total absorbed power. Consider the situation that two strong travelling waves from the same laser source pass through a cell in opposite directions. A photo-multiplier tube is mounted next to the cell so that fluorescence light -100-frora the cell, perpendicular to the laser propagation direction, can be monitored, Fig. 4.3. Fig. 4.3 Lamb dip experiment Each travelling wave burns its own hole in the velocity distribution. Because these two waves run in opposite directions, there arise two holes symmetrical about the centre of the Doppler profile, fig. 4.4a In this case, the total fluorescence intensity is the sum of the contri butions from each beam. As the laser frequency is tuned near to the centre of the Doppler profile, the two holes get closer and closer; also because of the gaussian distribution of the molecular velocities, the total fluorescence intensities increases. When the laser frequency is tuned to the centre of the Doppler profile, those two holes coincide and the travelling light wave interacts with only one group of particles, Fig. 4.4b. This results in a resonant decrease of absorbed power, which in turn, decreases the total fluorescence intensity, Fig. 4.5. This -101-effect is known as a 'Lamb dip'. Experimental observations of this effect were reported in (6,8). (a) (b) Pop. Pop. 0 V V Fig. 4.4 Velocity distribution curves Fig. 4.5 Total fluorescence intensity vs laser frequency -102-D. Intermodulated Fluorescence Spectroscopy The decrease in fluorescence intensity from the Lamb dip phenomenon has been used to detect saturation peaks and is particularly useful when the total absorption is small. Sorem and Schawlow (9) developed a sen sitive modulation method for isolating a small change in fluorescence intensity. Consider an experimental set up as Fig. 4.6. The molecules are exposed to light of two oppositely-directed beams from the same laser which are chopped at different frequencies, w-j and u^. The modulated fluorescence signal is detected by phase sensitive detection with reference signal set at the sum frequency, + u^. Io PMT Fig. 4.6. Experimental set up for intermodulated fluorescence -103-The basic concepts behind this saturated fluorescence technique can be understood in terms Df a very simple two level model. In the limit of the Doppler width being much greater than the homogeneous linewidth and I < Is, from eq. (4.17) the absorption coefficient at frequency co is given by a = a (1 - I ) (4.18) is The total power of the beam is Al where A is the cross-section of the beam and the transmittance is given by Al exp[-aL] where L is the sample length. Since the fluorescence power, F, is some fraction of the ab sorbed power, F = KAI { 1 - expI-aL] \ (4.19) where K is a proportionality constant. When aL < 1, the exponential term in eq. (4.19) becomes 1-aL, such that F = KAIuL (4.20) Since the beams are modulated at w-, and u2> the powers contained in beams 1 and 2 are I.| = IQ COS co-jt, *2 = 7 *o C0S "2*' resPectivelv> with I = I-] + I2 = - I (cos Wlt + cos u9t) (4.21) 2 0 I £• where I„ is the total power before splitting. Substituting eq. (4.17) -104-and eq. (4.19) into eq. (.4.20) with rearrangement gives •i F = -5-KALI„a„-(cos to, t + COS 0)ot 2 o 01 1 2 -I0 II + COS (w-| + a)2)t + COS (u-| - u2)t + j cos 2(0^ + |- cos 2a>2t]| (4.22) Therefore the fluorescence power at the sum frequency w-j + w2 is KALIo2°o cos ^ w-| + oo2)t (4.23) It should be noted that there will also be narrow resonant terms modu lated at the frequencies w-j, co2, w-j - w2 and zero, but that each of these will be accompanied by a large background because of low frequency amplitude noise in the laser power. Equations (4.22) and (4.23) show that the ratio of the power at the reference frequency to the d c fluorescen ce background is independent of aQL. Comparing to a similar calculation for the saturated absorption (.13) shows that this ratio is proportional to aQL in the limit of aQL<l. Thus this method, which is called inter-modulated fluorescence, has a strong advantage for experimentalists working with very weak transitions, very low particle densities, or a poorly populated lower state. This technique has been employed by a few workers in spectroscopy to study hyperfine splittings (9,14 - 16). The highest resolution achievable by this method, where the signals have a full width at half maximum (FWHM) of a few hundred kilohertz is obtained when the gas pressure is less than 0.1 torr. The signal suffers from sizeable pressure broadening if the pressure is substantially higher than 0.1 torr. -105-E. Resolved Fluorescence Spectroscopy Heavy molecules with, high spin multiplicity always exhibit severely blended spectra in the optical region. The overlap of differ ent subbands or hot bands makes the analysis difficult simply because it is no longer possible to recognize the patterns of branch structure, and some "lines" have unexpected intensity due to the blending of many lines. In addition, rotational perturbations by different electronic states cause shifts and splittings of the lines. Under such circumstan ces, the rotational analysis would be completely impossible without some knowledge of the quantum numbers involved in the individual lines or their relationship to other lines. Resolved fluorescence is of very great value in solving this problem. If a molecule is irradiated with laser light having the wavelength of a single rotational line, the absorbing molecules will be brought into the upper state of this particular absorption line only. The excited molecules can then emit light, falling to different rotational levels in the ground state according to the selection rules for elec tronic transitions (17) with emission of radiation, and giving rise to other fluorescence wavelengths besides the exciting wavelength. The first step in the experiment is to tune the laser to a parti cular line of an electronic transition. Fluorescence, induced by the pump laser, is monitored, perpendicular to the laser propagation di rection, by a monochromator with a photomultipler tube. A resolved fluorescence spectrum is obtained by scanning the monochromator. There are two important pieces of information concerning the line assignment that can be obtained from a resolved fluorescence spectrum: -106-i) the AJ selection rule for the line which is excited by the laser. When a single energy level is optically excited the fluorescence spectrum consists of two or three lines. If X^ is the exciting wave length, one of the lines in the fluorescence spectrum will always appear at X^. If only two lines appear and the other line is at a shorter wavelength, the exciting line belongs to an P branch; if the other line is at a longer wavelength the exciting line belongs to an R branch. A Q line may appear, in between the RP doublet, depending on the select ion rules and the type of transition. When a Q line is being excited in a parallel transition, most of the intensity is re-emitted in the P and R lines, Fig. 4.7. 7^ J' = J RJ"+I ) Q(J") GO R(J"-1) J"+1 • i J ± J"-1 Fig. 4.7 Origin of induced fluorescence lines -107-ii). The separation between P(_J"+1) and R(J"-l) fluorescence lines, called A2F"(J"), is given for a linear molecule by A2F"(.J") = (4BJ - 6D;)(J" + - 8DJ(.J" + (4.24) where B" and D" are the effective rotational and centrifugal distortion constants for the lower state (17). Measurement of the various sepa rations allows the assignment of the J numberings to be made and simul taneously gives a rough B^ value for this ground state. In an actual case, because the resolution of the monochromator is an order of magni tude lower than that of the laser excitation spectrum, it is often not possible to identify exactly which line in a crowded excitation spectrum goes with the line being excited, to within 1 cm-1. It may be necessary to determine this by measuring the resolved fluorescences for various lines in the range indicated by the first experiment. Furthermore, the excited molecules can also emit to different vi brational levels of the lower state. We could thus obtain in fluorescen ce a progression with V = constant but different v" in the lower state, each band consisting of a P line, an R line and possibly a Q line. Vibrational assignment of the upper state can sometimes be made by counting the minima in the intensity pattern of a vibrationally resolved fluorescence spectrum (18); the number of minima corresponds to the number of nodes in the vibrational wavefunction. Combining the powerful techniques of intermodulated and resolved fluorescence, it becomes possible in principle to analyse any spectrum of any complexity - for instance high resolution spectra where small -108-hyperfine splittings, are present can be analysed to giye the details of the electron spin and hyperfine coupling constant with great precis ion, and perturbed electronic band systems can be unambiguously assigned no matter how fearsome the perturbations may be. -109-Chapter 5 Laser Spectroscopy of VO; Analysis of the Rotational and Hyperfine Structure of the cV - xV (0,0) Band -110-A. Introduction Vanadium monoxide, VO, is an important constituent of the atmos pheres of cool red stars, its band systems being used for the spectral classification of stars of spectral types M7-M9. (1). There are three band systems of VO in the-visible and near infra-red. Near 1.05 y is the A-X system discovered by Kuiper, Wilson and Cashman (2), and later studied in the laboratory by Lagerqvist and Selin (3); following recent Fourier transform work by Cheung, Taylor and Merer (4) this is 4 4 - 4 assigned as A n--X Z , where the A n state has quite small spin-orbit coupling. At 7900 A is the B4n-X4Z~ system (5,6) where the B4n state 4 - ° is extensively perturbed by an unseen z state (4), and near 5700 A 4 - 4 - /v is the C z -X z system (6-9) whose detailed analysis is described in this chapter. A partial rotational analysis of bands of the C-X system was first performed by Lagerqvist and Selin (8). Their spectra were obtained using an arc between vanadium electrodes, which gives a very high tem perature, and correpondingly wide lines. Their line assignments were correct, but they were only able to analyse parts of the R2, R3, P2 and 2 2 P3 branches, and they suggested that the transition was possibly A- A. 2 4-The ground state was later established as 06 z by the e.s.r. work of Kasai (10), and shortly afterwards Richards and Barrow (6), reinves tigating the B-X and C-X systems, found that the two systems contain lines of different widths as a result of hyperfine structure in the ground state caused by the 51V nucleus, which has 1=7/2. Richards and Barrow could not resolve the hyperfine structure in their furnace spectra, but they discovered a very unusual internal hyperfine perturb-ation in the ground state. What happens is that the F2 and F-j levels (J=N±3s) with the same N value happen to cross near N=15 because of the particular values of the rotational and electron spin parameters. Matrix elements of the hyperfine Hamiltonian of the type AN=0, AJ=±1 act between them, and cause an avoided crossing of the hyperfine levels making up the two rotational levels. At medium resolution the perturb ation appears as a small doubling of the lines near N">=15, and the minimum separation, which is related to the isotropic hyperfine para meter b (11), was found to be consistent with the e.s.r. work. The hyperfine structures of the R-|, R^, P-| and P^ lines were re solved by Hocking, Merer and Milton (12) in high resolution grating emission spectra. The hyperfine patterns give the difference of the b parameters in the C and X states, and it was found that there are sizeable hyperfine splittings in the C state as well. Another internal hyperfine perturbation, similar to that in the ground state, was dis covered at N"=5 in the C4Z~ state. A rough value could be obtained for the isotropic parameter b, but the dipolar interaction parameter c could not be extracted from their Doppler-limited spectra. In this chapter we describe an analysis of the C4E~-X4Z" (0,0) band from sub-Doppler spectra recorded by the technique of intermodula-ted fluorescence (13). The line width is limited by pressure broadening effects to about 100 MHz, but this is sufficient for the hyperfine structure to be essentially completely resolved, barring the region at the R2 head. Except for the places where the upper state suffers from electronic perturbations the lines can be fitted by least squares with -112-a standard deviation of better than 0.0008 cm-1. Accurate values for the rotational, electron spin and nuclear spin constants have been ob tained for both states. -113-B. Experimental Details VO was prepared in a flow system by passing VOCI3 mixed with argon through a 2450 MHz electrodeless discharge operating at 75 W. The mixture was pumped through a stainless steel fluorescence cell fitted with quartz windows, and VO fluorescence was excited by light from a Coherent Inc. Model CR-599-21 tunable dye laser. The strongest fluores cence occurred when the microwave discharge was a purplish pink colour with a pale blue 'tail'; the fluorescence induced by excitation of the C4E~-X4E~ (0,0) band (at 5738 A) was a yellow orange colour. Spectra of VO were recorded at sub-Doppler resolution by intermo dulated fluorescence (13). Figure 5.1 illustrates the optical arrange ment for this experiment. A Coherent Radiation CR-10 Ar+ laser operat ing at 514.5 nm with 2.2 W output is used to pump a Coherent Radiation CR-599-21 dye laser with rhodamine 6 G dye. Dye laser output is typi--5 1 cally 30 mW single frequency (Av^^ ^ 3x10 cm ), and is monitored using a 1.5 GHz free spectral range (FSR) spectrum analyzer, a 299-MHz FSR fixed length semiconfocal Fabry-Perot interferometer, and an I2 cell. I2 fluorescence excited by the dye laser is detected perpendi cular to the laser propagation direction by an RCK IP28 photomultiplier tube (PMT 1) operated at -870 VDC. The dye laser beam was split by a 50-50% beam splitter (B S), and the resulting two beams were chopped mechanically at 582 Hz and 784 Hz. The laser power was about 15 mW in each beam, and the fluorescence signal was recorded through a sharp-cut yellow filter using an RCA C31025C (PMT 2). A narrow-band electrical filter selected the sum of the chopper frequencies, and the intermodul ated signal was extracted with a Princeton Applied Research model 128A PMT 2 12 cell 299 MHz FSR Interferometer 99% Refl 1.5 GHz FSR CR599-21 Laser Dye Laser Spectrum Analyzer Fig. 5.1 Experimental set up for intermodulated fluorescence spectroscopy. -115-lock-in-amplifier. All the necessary electronics were connected as in Fig. 5.2 to minimize the ground loop problem. The results were dis played on a three-pen chart recorder. As the laser frequency was scanned one pen plotted the intermodulated signal, the second pen gave frequency markers spaced at 299 MHz intervals from the semi-confocal Fabry-Perot interferometer, and the third pen recorded the fluorescence spectrum of \^ for absolute calibration. This system is similar to that used by Field et al (14) in their intermodulated fluorescence ex periments, except that we use a narrow band electrical filter rather than a second lock-in-amplifier (15). For intensity reasons we were not able to run the microwave dis charge such that the total pressure in the fluorescence cell was less than about 1 mm Hg if we were to record intermodulated fluorescence. As a result the linewidths in our spectra are pressure broadened, and were never less than about 80 MHz even though the laser linewidth is 1-2 MHz. For the weaker high N lines we had to increase the total pressure; the linewidths rose to about 130 MHz, but fortunately these lines are only rarely blended so that the only adverse effect was lower precision in their calibration. We encountered no problems with relative calibration of the spectra over a 1 cm"^ scan of the dye laser; for example the ground state hyperfine combination differences in the P^ and P3 lines were routinely reproduced to within ±0.0005 cm-1 (15 MHz); this is because the inter ferometer markers are sharp compared to the V0 lines, and because the temperature of the room (which affects the positions of the markers Ref. signal PD Ammeter PMT Expt. J High Voltage Power Li ne Fig. j Interferometer r . 5.2 Schematic diagram for intermodulated fluorescence detection system. \, HR i CTl BS Laser Beam BS -117-though not their spacing) remained sufficiently constant during the few minutes required for a scan. However the absolute calibration was always much less certain. As explained we used Doppler-limited I2 fluorescence lines, excited by a portion of the laser beam picked off by abeam-splitter. The wavenumbers of the I2 lines have been listed to 0.0001 cm"1 (3 MHz) by Gerstenkorn and Luc (16), but, since the I2 lines are about 1 GHz wide because of unresolved hyperfine structure, their absolute uncertainty is about 0.002 cm-1 (60 MHz). Fortunately the V0 spectrum is sufficiently dense near the band head that we were able to establish the relative shifts of the "ladders" of interfero meter markers between successive 1 cm-1 laser scans, using lines duplicated in the overlap regions. In this way we could plot a cali bration graph for the I2 lines relative to the interferometer markers over ranges of up to 10 cm-1. This gave us the marker spacing with great accuracy, and enabled us to use 40-50 I2 lines to establish the absolute calibration of the "ladder". The calibration graphs consisted of a scatter of points, one for each I2 line, spread randomly over ±0.002 cm-1 along a straight line. This procedure is very laborious, but we consider it worthwhile because it improves the standard deviation in a least squares fit to the line positions by nearly a factor of two: the final standard deviation for 1300 low N lines calibrated in this way was 0.00076 cm"1 (23 MHz). -118-C. Rotational and hyperfine energy level expressions Since this work on VO is the first detailed study of the hyperfine structure in a electronic state, we did not know at first which terms to include in the Hamiltonian. After some experimentation we found it necessary to vary 12 parameters for each electronic state. The Hamiltonian was taken (17,18) as H = H . + H , + H. ,c + H -j . + Hp), (5.1) rot el hfs el ,c.d. s.o. where Hrot = *^ Hel = TN.S+fAOS^-S2) (5.2) HhfQ - bl.S + cl S + e'qq(3I2fc-Ifc) + ln hfs ~ ~ z z 41(21-1) !~ ~ Hel,c.d. = + TV(3Sz2-i2)N2 + H2^z2-iZK (3) which are basically described in chapter 2, and H^ ' represents third-order spin-orbit contributions to the parameters y and b, which are described later. The rotational energy, given by H t, requires no explanation; the terms in Hg^ are the electron spin-rotation inter action and the electron spin-spin dipolar interaction, while the terms in ^Y\fs are tne direct electric and magnetic hyperfine interactions. The terms in b and c are the determinable coefficients in the magnetic hyperfine Hamiltonian for a i state; c is the dipolar electron spin-nuclear spin interaction, and b is a combination of c with the Fermi contact interaction, ac> A more fundamental way of writing these I ,S magnetic hyperfine terms (19), which is convenient for the calculation -119-of matrix elements, is Hma„ hfc = ar I.S + c(I,S, - h.S) (5.3) mag.hfs c ~ ~ z z 3^ ~ The electric quadrupole interaction (e Qq) and the nuclear spin-rotation interaction (c^) are familiar from microwave spectra of singlet states (18,20), while the centrifugal distortion corrections to H -j should be sel f-explanatory. In the z states of VO under discussion the electron spin-spin interaction •(x) is large compared to the hyperfine effects, so that the basis giving the most nearly diagonal representation is case (b^j) coupling (11,18) where N + S = J-, J+i = F (5.4) The basis functions are then |NASJIF>, where A can be suppressed be cause it is equal to zero. However the matrix elements in case (b^j) coupling are much more complicated algebraically than those in case (a^) coupling so that the use of case (b^j) is logical for VO only because of the internal hyperfine perturbations mentioned in the Intro duction. The matrix elements responsible for the internal hyperfine perturbations are off-diagonal in both basis sets, but in case (a^) the spin-uncoupling is also off-diagonal. The spin-uncoupling matrix elements, which arise from the x and y components of the operator -2B J.S, are very much larger than the internal hyperfine perturbation elements, and we found that they gave trouble with the energy ordering -120-of the eigenvalues in the regions of the internal hyperfine perturbations. This led to disaster in our attempts to fit the line positions by least squares, so that we returned to case (b^j), though not without misgi vings. In retrospect a two step diagonalization starting from case (a^) might have saved much lengthy algebra: the first step would have been essentially a numerical transformation to case (b^j), and the second step would have completed the diagonalization. We quickly discovered that it was necessary to use a full matrix treatment for a correct description of the magnetic hyperfine effects; for instance the AN=±2, AJ=±1 elements of the dipolar interaction have a significant effect on the course of the energy levels at the internal hyperfine perturbations. Therefore the only simplification we have made was to omit the AJ=±2 elements of the electric quadrupole inter action, after calculating that these were less than 1 MHz. The matrix elements of the terms in B, D, Y and yn are diagonal in case (b^j): <NSJIFlHrot + spin-rotlNSJIF> = BN(N+1)-DN2(N+1)2 ^ -)S[N(N+1)+S(S+1)-J(J+1)J[Y+YI/I(N+1)] Matrix elements of the other terms in eq. (5.2) are most conveniently calculated using spherical tensor formalism as shown in chapter 2 (19,21). For electronic Z states of any multiplicity, where a single spinning nucleus is present, we have •121-2,, -, NN+S+J <N-SJIF|H IN_ IN|NSJIF> -fxt-ir-"(JSN-2 N S x[S(S+l)(2S+l)(2S-l)(2S+3)]J5 (-1)N' /N' 2 N\[( 2N+1) (2N'+1) ] /N' 2 N\l \0 o oj (5.6) <N'SJ'IF|H |NSJIF> = (-1)J+I+F (F I J')[(2J+1)(2J'+1) 1(1+1) 1 mag.hfs1 J ( , 4L I J I (21+ur (_1}N+S+J'+1 - ^ N. [s(S+l)(2S+l)]32 a( JS 1 -±c[30(2N+l)(2N'+l) S(S+1)(2S+1)J35 ,N' N 2\ (-1)N'/N' 2 N' S S 1 J' J 1 /N' 2 \ \0 0 0/ (5.7) +(-D N+S+J+l N J' S) [N(N+1)(2N+1)]^ Cj J N l( where a (the true Fermi contact interaction) is b +-jc, and <rsj^F|Hquadnjpole NSJIF> = he2 -1 Qq / I 2 I\ (-1) VI 0 1/ J+I + F F I J' 2 J I (5.8) c(-DN"+S+J [(2J+1)(2J'+1) (2N+1)(2N'+1) jS J')(-l) /N' 2 N 2 J N j(-l)r /N' 2 N\ } VO 0 0/ It is straight-forward to obtain explicit expressions except for the dipolar term cd^-^.S), where the 9-j symbol is not listed in standard tabulations (22), and the algebraic expressions given by -122-Mizushima (23) contain yarious small but important errors. For complete ness we list the corrected forms of the relevant 9-j symbols in Appen dix IV. Our phase choice in eqs. (5.5)-(5.8) corresponds to that of Bowater, Brown and Carrington, where N is treated as a space-fixed operator (19), and the order of coupling the vectors in eq. (5.4) is also the same as theirs. The third-order spin-orbit contributions will be unfamiliar since 3 they only occur for S^, that is for electronic quartet states or worse. In this study we have had to use the third-order contribution to the spin-rotation interaction, ys, introduced by Brown and Milton (25), and the corresponding correction to the isotropic hyperfine interaction, which we call bs (26). The term in y<. has a complicated history. The original formulae for 4E states derived by Budo (27) and Kovacs (28) accounted nicely for Nevin's data for (29), but not for the data on GeH (30) and SiF (31). Hougen (32) attempted to find the source of the discrepancy, and extended the theory to include the second spin-4 rotation parameter required by group theory arguments for a I state in the general case. Later work by Martin and Merer (33) showed that the original SiF spectrum has been misassigned, and that there were no dis crepancies, but their theoretical treatment was still incomplete. Finally Brown and Wilton (25) gave a full discussion of the higher order spin dependence of the spin-rotation interaction in I states. Their conclusion was that the single spin-rotation term in y given in eq. (5.2) will usually suffice, except in cases where very high resolut ion is available, or where another nearby electronic state interacts -123-strongly by spin-orbit coupling; in this case a second spin-rotation terra will be needed. This second term results from a third order interaction where the spin-uncoupling operator -2B(J S +J S ) is taken x x y y with the spin-orbit operator za 1..s- twice: the matrix elements in i i case (a), with z and n taken as signed quantities, are <SZ,Jft|H^)|S^±'I ,Jfi±l> = -J5Yc[S(S+l)-5z(z±l)-2] x [jfj+D-nCnii)]15 [sis+D-zCz+l)]15 (5.9) The experimental parameter y$ isa complicated function of the spin-orbit matrix elements, the energy separation to the interacting states and the differences AB between the B values of the interacting states and the state of interest. Barrow (34) has reported that the C4z" and X4z" states of VO both require two spin-rotation parameters (presumably in Hougen's formalism). The present work confirms this conclusion, although we have used Brown and Milton's definitions for the parameters. These states of VO are the first 4Z states known where two y's are definitely needed. The high precision of our data has required that we consider the corresponding effect in the isotropic hyperfine Hamiltonian. The mechanism for its appearance is entirely analogous: instead of the spin-uncoupling operator -2B(J S +J S ) we take the isotropic hyperfine xx y y operator zb.I.s- with the spin-orbit operator twice (26). The result is as if there were an effective operator = £ TVM^TVtS), T2(S..S.)), T2(C)) (5.10) iU i>j ~ -124-acting within the manifold of the \ state of interest. As given in ref. (25), the matrix elements of eq. (5.10) in case (bgJ) coupling for A=0 are <N'SJ-IF|HW|NSJIF> 4(-DJ+I+F jF I J'J[t2J+l)(2J'+l).I(I+D(2I+l)] 1S° jl J I ) x(-l)NV 2 N\ [(2N+l)(2N'+l)]Js /N' N 2\ t5-11) \0 0 0/ S S 3] lJ- J 1 x[35(2S-2)(2S-l)2S(2S+l)(2S+2)(2S+3)(2S+4)/3]2b$ The experimental parameter TQ(C), which would result from eq. (5.10), has been put equal to -y- (14)3* b^, in order to define a parameter b<-which is as similar as possible to Brown and Milton's YS• Calculation of the explicit matrix elements from eq. (5.11) in case (bDl) coupling was a long process because the 9-j symbols give uncompromisingly intractable algebraic expressions. For reference they are included in Appendix IV. No doubt there are simpler forms for the 9-j symbols with N'=N, but we have not tried to search for them. The final matrix elements, on the other hand, are quite simple because of extensive cancelling. Another approach to the matrix elements of these third-order terms is to set up the matrices in case (aQ) coupling and transform them al-p gebraically to case (bgJ) using the eigenvectors of the case (a^) rotational matrices. This is no real advantage for the hyperfine term -125-because the case (a.) matrices are already quite complicated, having P elements of the type Aft=0,±l, AJ=0,±1; we did however transform the diagonal blocks in this way in order to check the calculation of the 9-j symbols. On the other hand the spin-rotation term has a simple matrix representation in case (a) coupling, consisting only of the elements given in eq. (5.9), but a fearsome form in case (b):-<N'SJIF|H^]n_rot|NSJIF> = {[(2N+1)(2N'+1).J(J+1)(2J+1)]% x[2( 2S-2) (2S-1) 2S( 2S+1) ( 2S+2) (2S+3) ( 2S+4)/2>fz Yc; b (5.12) x I (2x+l)/ 3 x 1\ (-1)N' /N' x N\ ' "• x=2'4 \-l 0 1 / \0 0 o) J J 1 Accordingly we converted eq. (5.9) to case (b) algebraically using a Wang transformation followed by the similarity transformation H(b) = s-lH(a)s (5.13) If the element corresponds to |n|=3/2 and the H^) element to F3 or F^, the eigenvector matrix £, given by c -s" s c (5.14) has elements for a £~ state as follows:-•126-e levels (F, and F,): c = h{3{J-h, s = -^[(J+lj/Jj* (5.15) f levels (F2 and F4): %[(0-1)/(J+1)^ -^[3( J+|)/( J+DJ5 The two third-order terms have quite similar effects on the energy level structure. Table 5.1 gives the algebraic forms of the matrix ele ments we have used. It can be seen that when similar powers of N are cancelled, the spin-rotation and isotropic hyperfine energies follow the same type of expression: F, and F4 (J=N±f): E$r + E^ * ±§N(Y-*S) ±§C(bJ,bs)/2N - Q q (5.16) F2 and F3 (J=N±^): E$r + E.SQ = ±32N(y+|Ys) ±C(b+|bs)/2N where C = F(F+1) - J(J+1) - I(I+D (5.17) The result is that the F-| and F4 levels have different effective y and b parameters from the F2 and F^ levels. The energy level calculations based on the matrix elements of Table 5.1 require that two 16 x 16 matrices be set up and diagonalized for each F value. One of these matrices has basis functions extending from N=F+5 to N=F-5 in steps of 2, and the other has basis functions from N=F+4 to N=F-4, also in steps of 2. Unfortunately the structures of the two matrices are not the same, and there are problems with missing levels at low F values: the complete 16x16 matrices first appear for F=5. We have reduced the general forms of the matrix elements -127-Table 5.1 Matrix elements of the spin and hyperfine Hamiltonian for a Z state in a case (b ) basis. Diagonal elenents Fl(>,»3) : + [-^KCSfD^XK + c,* + jctb + ) - ^gj^^ ] /(2,+3) F2(J=N+|): ir(N-3) + [-|YSN(3N+5) + 2X(N+3) ]/(2N+3) + [ClC-;2K(K+l) + (N-3)} + -|<:{b(2M-9) + 3bg(3N+2+ ) + c( JL^- + 7)} . e2QSX(^3)(2N-3) ]/[(2N+1) (2Nf3) ] 21(21-1) .NQN+3) F3(J=S-|): + [-|ys(N+l)(3N-2) + 2A(N-2) ]/(2N-1) + [c C{2N(N+1)-(N+A) } - -|c{b(2N-7) + 3bg(3N+l+ ) + c( - 7)} . e% X(N-2)(2N+S) ]/[(2s_1)(2Wfa)] 21(21-1) .(N+1)(2N-1) F4(>K-|): - yy(N+l) + [yr£N(N+l) - 2X(N+1) + CjCCN+l) - |c{b c+b (N+1) 2 t 2K-1 2r e Qq>:CN--^> ]/(2s-i) 21(21-1) .N(2N-1) X - TC(C+1)-1(1+1) J( J+1) , C - ?(F+1)-J(J+1)-I(I+1), and b « a -c/3 A u 2 : The rotational Hamiltonian has only diagonal elements, equal to BN(N+1)-DN (N+1)' <N-2 J F|H!NJF>- j [3(J-i)(J+|)]>5/(2N-l)} [2X + { 5( J+|) (J-N+l)-2} Yg + icC + 3e^2s_X + ic {2(5N+3) ( J-N+l)-| > } /( 2 J( J+1) ) ] \ 1(21-1) (2J-1M2J+3) 2S 2 J where X and C have been defined in the diagonal elements -128-Table 5.1 Continued. <K J-l F|HiKJT>- [(F+J+I+D (J-F+I) (F+J-I) (F-J+I+l) 3^ [R(«/AJ] [b-Cj + bgQ(N) (n(n+1HJ2_19/M I 2 {F(F41)-I(I+1)-(J-1)(^1)1 I 3 + (2N-l)(2N+3) " 41(21-1) (J-D (J+D / where R(N) and Q(K) are given by J=N+§: R(K)- [3N/(N+1)]N Q(»- -C^)/(4N+6) is • [(2N-l)(2N+3)/(N(.+ l)}]ii [12K(»» -3]/[2( 2K-1) (2M3) 1 J=N+2 = 1- [3(N+1)/N]li -(4N+l)/(4N-2) <K-2 J+1 F IH j K JF>" -|[(F+N+I+|)(N-F+I-|)(F+N-4)(F-N+1+|)/{N(N-1))]H <>^2 J-l F ,K jNJF>= 1(21-1) (2N-3M2N+1) [(F+J+I+D (J-F+I) (F+J-D (F-J+I41)] ^ [w(K)/{8J(2N-l)}] « r ,2. JF(F+1)-I(I+1)-(J-D(J+1)^ _b A d I ] lc + 3e Q<5 41(21-1) (J-D (J+D s 2 S J>N_2 where 6 1 is the Kronecker delta, and W(N) is given by J»«— J«N+i: W(N)« [3(K-1) (2N-l)(2N+3)/N]!s >N-|: 4[(W-l)(N-2)]'1 J-N-|: [3N(2N-l)(2N-5)/(N-l)]li -129-to algebraic expressions, rather than programming the computer to call subroutines for the Wigner angular momentum coupling coefficients, in order to save computing time; immense amounts of cancelling occur in the calculation of the algebraic expressions from the Wigner coeffi cients, and even without this it takes 20 seconds of CPU time on the University's Amdahl 470 V/8 computer for one least squares iteration to N=25 using the algebraic expressions. D. Analysis of the spectrum (i) General description of the band 4 - 4 -The (0,0) band of the C I - XT system of V0 is quite strongly red-degraded, and has two well-marked heads at 17426.4 cm-1 (R-j and R^) and 17424.2 cm-1 (R2 and R3). The spectrum is very crowded down to about 17400 cm-1, with typical line densities of the order of 50 per wavenumber. To the red of this the band opens out and the rotational lines become well separated; the eight hyperfine components of each 51 line resulting from the 1=7/2 spin of the V nucleus are clearly re solved, and the lines can readily be assigned to their respective elec tron spin components by their distinctive hyperfine patterns. Some typical hyperfine patterns are illustrated in Fig. 5.3. The hyperfine patterns of the F-j and F^ branches are three times as wide as those of the ?2 anci F3 branches, as can be understood from the diagonal matrix elements of the hyperfine Hamiltonian given in Table 5.1. There is no problem with the rotational assignments in the tail of the band because the constants given by Barrow (34) usually reproduce -130-the line positions to within 0.2 cm"1. We have recorded the band out to 17288 cm"1 (the P(41) group), where the branches have nearly died out. All four electron spin components suffer from rotational perturba tions by other electronic states, which mainly appear as discontinuities in the branch structure; we have found extra lines in only two of the perturbations, though Lagerqvist and Selin (8) have identified extra lines in various higher N perturbations in their arc spectra. The assignment of the hyperfine F quantum numbers is easy for the F-| and F^ electron spin components because the hyperfine structure follows the Lands interval-type pattern described by eq. (5.17) and the diagbnal elements of Table 5.1: the hyperfine patterns open out at the high F side, and the higher F lines have greater intensity. For the F2 and F3 branches the internal hyperfine perturbations cause the patterns to be irregular over the complete range of N values we have studied, even though the maxima in the internal perturbations are at N'=5 and 15. It can be seen in Fig. 5.3 how the most intense hyperfine lines (which have the highest F values) are clustered together, in con trast to the patterns for the F-j and F^ branches. The reason is that the F2 and F3 spin components of the ground state are only 0.5 cm-1 apart at N=30, and the hyperfine matrix elements between them, which are F-dependent and of the order of 0.1 cm"1, are able to reverse the Lande-patterns. (ii) Internal hyperfine perturbations As the F2 and F3 branches are followed to lower N values extra lines induced by the hyperfine perturbations start to appear at N=21. -131-Fig. 5.3 Hyperfine structures of lines from the four electron spin components of the VO cV - xV (0,0) band. -132-These extra lines, though, not resolved into individual hyperfine compo nents, had been observed by Richards and Barrow (9) and Hocking, Merer and Milton (.12). The patterns of hyperfine lines become very complica ted because the F order of the hyperfine components inyerts at an inter nal hyperfine perturbation, producing a kind of band-head in the hyper fine structure for both the main lines and the extra lines. The pertur bation-induced extra lines can be seen for about five N values on each side of the maximum, so that the effects of the upper and lower state internal perturbations run into each other and produce extra lines over the complete range N=4-21. These lines have proved to be very valuable in determining the spin and hyperfine constants, as will be described in Section F. The internal hyperfine perturbations are best understood from plots of the energy levels against N. Fig. 5.4 shows the quartet electron spin structure of the ground state with the hyperfine effects omitted. The F-| and F^ levels cross near N=10, but their J values differ by 3 units, so that they do not interact. The F2 and F3 levels cross in zero order near N=15, but must avoid each other because of the AF=AN=0, AJ=±1 matrix elements of the hyperfine Hamiltonian. The avoided crossing is shown magnified in Fig. 5.5 with the hyperfine structure drawn in. Only seven of the eight hyperfine components of each level actually avoid each other. The reason is that the range of F values is different in the two levels; F2(J=N+32) has F=N-3 to N+4 while F3(J=N-*i) has F=N-4 to N+3. Hyperfine components with F=N-3 to N+3 occur in both electron spin levels, and therefore perturb each other, but the F=N+4 component of F2 and the F=N-4 component of F3 pass through the avoided VO X4I spin splitting pin fine structure of the VO X*E v = 0 level plotted as a tational and hyperfine structures are not shown. N-4 N + 3 CO I Fig. 5.5 Calculated hyperfine energy level patterns for the F2 and F3 electron spin components of the X^Z~ v = 0 state of VO in the range N" = 9 - 22. The calculations are from the iinal least squares fit to the ground state hyperfine structrues, and levels with the same values of F"-N" are connected. -135-crossing region unaffected. They give rise to the isolated strong lines in the centres of some of the hyperfine patterns which are very characteristic, as can be seen in Fig. 5.6. Figure 5.6 shows the P3 lines for N"= 15-18. This is almost the only region where the perturbed lines are not overlapped by other branch structure. The P3O8) line has most of its intensity in the short wave length components (left hand side), which are the zero order P^ transi tions. The intensity transferred to the induced lines (on the right) depends on two factors, the separation of the zero-order hyperfine com ponents, and the value of F. For P3(18) the two factors approximately balance for F=15-18, but the higher F components, which are starting to form a hyperfine 'head', are much weaker. The central unperturbed F=14 line (F=N-4) is very distinct. With decreasing N the intensity of the P3 lines is progressively transferred to the long wavelength components, and the 'heads' in the hyperfine structure become very pronounced. As might be expected from Fig. 5.5 the effects pass through a maximum at N"=15. An interesting effect of the reversal of the hyperfine energy order at the perturbation is that the different hyperfine 'branches' (with the same value of F-N) have their minimum separation and most nearly equal intensities at different N values: for instance the F=N+3 hyperfine components have minimum separation at N=14, but the F=N-3 components have minimum sepa ration at N=17. This caused us some difficulty in the early stages of the least squares fitting, because we needed to establish the exact parentage of a hyperfine component in order to match it with an eigen value from the diagonalization. -136-hternal hyperfine perturbations in VO, X4Z O 01 0-2 cm-' Fig. 5.6 The P3 branch lines of the C4I~ - X4Z" (0,0) band in the region n" = 15-18, showing the hyperfine patterns near the ground state internal hyperfine perturbation. The FM quantum numbers for the hyperfine components are marked. -137-Below N"=15 the patterns are unfortunately blended because the upper state spin splittings are smaller than the ground state pertur bation doublings; also other branches interfere. Figure 5.7 shows the ?2 and P3 lines for N"=ll-14. At this stage the N values are low enough for the energy 'spread' of the hyperfine structure of the F2"(N) and F3"(N) levels to be noticeably different, as can be seen in Fig. 5.5. This difference governs many of the features of the low N hyperfine patterns, and results from the factors (2N+9) and (2N-7), respectively, in the diagonal elements of bl.S for the F2 and F3 components. The result in the spectrum is that the F2 branches are very open while the F3 branches begin to collapse into sharp spikes where the hyperfine structure is often not fully resolved. The factor (2N-7) for the F3 levels in fact causes the hyperfine energy order to invert between N=3 and 4. The upper state has a similar hyperfine perturbation centred near N'=5. The energy level pattern is shown in Fig. 5.8. Parts of this pattern are anomalous because the N values are so low that the full complement of eight hyperfine components is not present. Also the in version of the hyperfine energy order for the lower set of interacting components does not occur: the reason is that the inversion in the F3 components between N=3 and 4 cancels the inversion caused by the fact that they turn into F2 levels at the perturbation. The only seeming irregularity in the lower set is that the F=N+3 components lie above the F=N+2 for N'=4 and 5. The P2 and P3 branches in the region N"=5-8 are shown in Fig. 5.9. The P2 lines are more than 0.5 cm"1 to the blue of the P3 lines with the -138- 0.1 cm-' Fig. 5.7 The P2 and P3 branch lines of the V0 C*T, - X*Z (0,0) band in the range N"= 11-14. Numbers above the spectra arc F" values for the hyperfine components of the P2 and P^ lines; other lines belonging to overlapping branches are indicated below the spectra. -139-Fig. 5.8 Calculated hyperfine energy level patterns for the anc* ^3 4 -electron spin components of the C £ v=0 state of VO in the region of the internal hyperfine perturbation (N' = 2-13). Levels with the same values of F'-N1 are connected. P,(8) P3(8) R4(29) R»(7) .::~»..-:.:.?.B:»- D(7) 29 28 27 ~ <™-\2t a 30 rlv'' 2D 263 24 23 22 3 _ ,oc-i 25 24 23 22 RJilO) R4(27) ^ 4 23 5 24 23 22 21-^26) R,f24) P4(6) o i Fig. 5.9 The P2 and P3 branch lines of the VO cV - xV (0,0) band in the region N" = 5-8 ; the F" quantum numbers of the hyperfine components are marked above the spectra. Overlapping high-N R lines and low-N P, and P, lines are indicated below the spectra. All four tracings are to the same scale. 1 4 -141 -same W" value, reflecting the large spin splitting in the ground state, and their appearance is altogether different because of the different overall hyperfine energy 'spread' described above. The P3 lines are only partially resolved even at our resolution of 100 MHz, and it is fortunate that the induced lines in the P2 branches are so well resolved, otherwise it would not be possible to follow the upper state hyperfine energy pattern. The hyperfine assignments are very difficult to make in this region, because the line positions depend critically on the spin and hyperfine constants of both states; this was in fact the last region of the band to be assigned. ( iii) The band centre The centre of the band contains R lines with N"=15-20 (correspond ing to the ground state internal hyperfine perturbation) together with the very low N lines. The R lines confirm the hyperfine patterns given by the P lines, but blending limits their usefulness. The low N lines on the other hand are very interesting because they carry most of the information about the dipolar I,S interaction and the quadrupole con stants. Often they are quite difficult to assign because the hyperfine patterns are fragmentary, and detailed calculations of the energy levels are needed. Typical patterns are shown in Fig. 5.10. The upper tracing, which covers the region just to the blue of the band origin, shows the P2(l), R3(2) and Qefik) lines, superimposed on the perturbed R2(17) and R3(17) lines. The line strengths in our spectra are such that hyperfine compo nents with F"-S2 are usually not seen. However in the range F"=3-7 we 20 19 V P,(4) R,(22) P2(3) 1 t i Fig. 5.10 Two regions of the VO C^E ~ - X^E- (0,0) band near the band origin. Low-N lines are marked in roman type with hyperfine quantum numbers indicated as F'-F"; high-N lines are marked in italic type with only the F" quantum numbers of the hyperfine components indicated. Cross-over signals (centre dips) are marked 'cd' . The two tracings are at the same scale. -143-frequently observe lines with AF/AJ, and, where lines with a common lower level lie within the same Doppler profile, we observe centre dips. These two effects are well-known in sub-Doppler spectroscopy, particu larly for 12(35), and require no further explanation. The advantage of the additional lines that arise is that they give direct hyperfine com bination differences, which break the correlation between the upper and lower state hyperfine constants resulting from the parallel selection rules of the electronic transition. For example, in the line Qe^(%) p (or Q-|2(0), to give it its case (b) designation) we observe all four of the possible hyperfine components, and therefore obtain directly the separations of the F=3 and F=4 components of the two combining J=% levels. An energy level diagram illustrating this is given in Fig. 5.11. The lower tracing of Fig. 5.10 shows the P-j (4) and ?2(3) lines, against the background of R3(21) and R.j(22). The P-|(4) line has par ticularly clear AF=AJ hyperfine components, and also centre dips between them and the case (b) allowed AF=AJ=-1 components. An interesting centre dip involves the strong F'=7-F"=8 component and the unobserved F'=8-F"=8 component; this centre dip is quite weak, because the strength of a centre dip is proportional to the square root of the product of the strengths of the two contributing transitions (35). Altogether about forty AF^AJ hyperfine components haye been identi fied in the low N lines. We had not anticipated them in our original least squares programme for fitting the observed transitions, and had to include them as special cases. Similarly we had not anticipated that the Q branches would be so comparatively strong. Twelve hyperfine com ponents belonging to four Q lines have been assigned; the observed Q -144-F,(-1) F2(0) F 3 o O CD CO CD O CP d CD O LO LO CO LO LO 00 CO LO CT) ro • O LO LO cr> ro E exile /cm_1 - 17419.6774 17419.6165 -0.8193 -0.9341 Fig. 5.11 Energy level diagram indicating the assignment of the four hyperfine components of the line Qef(^)--145-lines are Qe^(%) and Q^eC%)> which form the A-doubling components of 4 - 4 -the Q{J=h) line of the EX - ET sub-band in a case (a) description, 4 - 4 -and the corresponding first Q lines of the E^2 ~ ^3/2 su':)-l:)and, Qef(|) arid Qfe(|). At these low J values the case (b) description of the levels and transitions breaks down, and some apparently impossible lines arise. p The Q(%) lines are good examples: Qef(}s) and Qfe(h) become Q-]2(0) an<^ Q2-|(-l), respectively. We have kept the case (b) notation for the main branches since they show no discontinuities when followed down from high N. This 4 breakdown in notation for a E state in fact only happens when the Q=J component in case (a) corresponds to the F-| and F2 levels, that is when the spin-spin parameter X is greater than the rotational constant B(33); both the C4E" and X4z" states of VO have X>B. The inner band head formed by the R2 and R3 branches is very complex because it contains many extra lines caused by the internal hyperfine perturbations, together with overlapping R-| and R^ lines. Nevertheless all the features have been assigned with the aid of com puter calculations. The assigned lines of the band are listed in Appendix VI , Table I. 4 _ E. Electronic perturbations in the C E state 4 _ Nine electronic perturbations have been found in the C z v=0 level. Seven of these are shown in Fig. 5.12, which is a plot of the energy levels against J(J+1); two others, at F2(^4) and F3(85), which 3 b i 17800 17600 r 17400 0 H,2 = 0 50 cm-' 1000 >V 0.069 cm-' 2000 J(J+1) 3000 i Fig. 5.12 Rotational energy levels of the C4£ v = 0 state of V0 (with scaling as indicated) plotted against J(J+1). Dots indicate rotational perturbations, and the perturbation matrix elements, 2 H12, are given where they can be determined. The dashed line is probably a component of a II state (see text) with Beff= 0.482 cm -1 -147-were discovered by Lagerqvist and Selin (8) lie beyong the range of the figure. The figure also includes the perturbation matrix elements where they can be determined from the induced extra lines. The only regularity we can recognize is that the perturbations at F^(26) and F3(45) are caused by the two A-components of an orbitally-degenerate state, which could possibly be n, as we now show. (i) The F^C26) perturbation The perturbation at F4(26) is particularly annoying because the 4 -perturbing state has almost the same B value as C I ; its effects there fore do not die away rapidly to zero on either side of the maximum of the avoided crossing. It can be shown that the perturbation shift at the origin of C4E~ is still 0.006 cm"1, so that since we can determine the line positions to better than 0.0005 cm"1 we must allow for the effects of the perturbation over the complete range of F4 levels. The details of how this was done are given in Section F. We can assign the perturbations at F^(26) and F3(45) to the same perturbing state because it is possible to interpret the F^(26) pertur bation in detail from the laser spectra. Ignoring the hyperfine struc ture initially, we fitted the upper state term values with N'=24-28 (including the single observed 'extra' level) to the eigenvalues of the Hamiltonian matrix H = Tc + BCN(N+1) '12 112 T ^ + B .N(N+1) pert pert (5.18) •148-It was necessary to fix (for the C4z" state) at an effective yalue for this N range calculated from preliminary least squares work, and to assume that the perturbation matrix element can be treated as constant over such a small N range. The results are given in Table 5.2.The fit is good, and the parameter TQ comes out to within 0.1 cm-1 of the C state origin; also the perturbation matrix element H-^ is given to an accuracy of ±0.005^ cm-1. Eq. (5.18) assumes that the rotational energy of the perturbing state is proportional to N(N+1); however the N range of the fit is so small that we can convert the results, without loss of accuracy, to the case where the energy of the perturbing state is linear in J(0+1). When this is done it is found that Bpgrt agrees to within 0.0007 cm"1 with what is obtained if the ^(26) and F3(45) perturbations are assumed to be caused by the same perturbing state. This is excellent agreement, and unambiguously proves a connection between the two perturbations. Because the F^ and F3 levels have different e/f symmetries the perturb ing state must have rotational levels with double parity (i.e. it must be orbitally-degenerate); the two perturbations are therefore caused by different A-components of a perturbing orbitally-degenerate state. 4 4 Extrapolation of the vibrational structures of the B n and A n states (34) rules them out as candidates for the perturbing state. Of course a third 4n state could be responsible, though we see no evidence for such a state in our Fourier transform spectra, which extend down to 6000 cm"1: the emission transition to X4£~ would be spin and orbi-tally allowed. On the other hand the hyperfine structure suggests that -149-Table 5.2 Analysis of the C Z , FA(26) perturbation. Upper state energy levels _Nj with F = N-1 (cm-1) Obs-calc (cm"1) 24 17716.011 -0.003 25 17740.641 0.003 26 17766.039 17767.084 0.000 -0.000 27 17793.169 0.002 28 17820.725 -0.001 Least squares results: (lo) T = 17420.055 ± 0.018 cm"1 c B = 0.49336 (fixed) c T = 17447.313 + 0.020 pert B = 0.4550 ± 0.0003 pert H = 0.496 ± 0.0055 -150-the perturbing state has only moderate spin-orbit coupling, so that a possible candidate would be a n state from the same electron configura tion as A4n (probably 4sa13dS14p-iT1). The sum of hyperfine energies of the doubled F^(26) levels is found to be linear in F(F+1), so that the perturbing state must follow case (a„) or (b ',) coupling; also the spacing of its hyperfine levels P 3d is found to be almost exactly the same as that of C^i", F^(26). If we write the hyperfine energy expression for a rotational level (N,J) as Ehfs = To + kF(p+1) (5J9) where k is a function of N, J and the hyperfine constants, the deper-turbed values are k(C4z", N=26, J=24i) = 0.000234 cm"1, \ , (5.20) k(perturbing, J=242) = 0.000205 cm Now case (bnl) states have wider hyperfine spacings at high J than case 3d (a ) states for the same hyperfine parameters, as can be seen from the 3 diagonal elements of bl .£: the case (bgJ) expression <NASJIF IbKSJNASJIF> = -b[N( N+1)-S(S+1)- J( J+1)]£F( F+l) -1( 1+1)-J( J+1)] 4J(J+1) (5.21) has essentially an extra factor of J compared to the case (a^) expression (11,36) -151-<JftSEAlF|bI .S| JfiSEAlF> = bnzlF(F+l)-I(I+l)-J(J+lU — 2JtW) (5'22) Therefore the comparatively large value of k for the orbitally-degene rate perturbing state indicates a considerable tendency to case (b^j) coupling, or in other words that it has comparatively small spin-orbit coupling. This is what we expect for a n state from the same confi guration as A4n (where A = 30 cm-1), but not what we expect for a 2n 4 -1 state from the same configuration as B n (where A^70 cm ). If the 2 perturbing state is indeed a component of the n state corresponding to A4n, the positive sign of the hyperfine parameter k suggests that it is the F-j component, though its magnitude is only a quarter of what we calculate for case (b coupling, indicating that the spin-uncoupling is only quite partial1. 1 4 The argument runs as follows. The hyperfine parameter b for A n is 4 - 1 known to be virtually identical to that of X E , namely 0.0273 cm . 2The n state from the same configuration as A n should, in first approximation, have a b-value three times as large, because the isotropic hyperfine operator is strictly E b. I.s. rather than bi .S. Therefore "' . , l ~ ~i ~ ~ I electrons from eq. (5.21) we calculate, for case (b^j) coupling, k(2n,F-,, J=24%) = 0.00084 cm"1 K(2JT,F2, J=24*s) =-0.00080 cm"1 Further evidence that the spin-uncoupling has not progressed very far comes from the spin-orbit matrix elements given by Kovacs (37): in pure case (b) coupling 0 » so that no perturbation would have been observed. -1 52-(ii) The F-j(37) perturbation The very small perturbation at F-j(37) has been reported already (.38) It forms an instance where an avoided crossing occurs within the hyper fine structure of a single rotational level, and the analysis can be carried out by treating the hyperfine structure as a fragment of rota tional branch structure. Two regions of the spectrum are shown in Fig. 5.13. The lower tracing is the P-j(27) line near 17353 cm"1, which is unperturbed and shows the Lande-type pattern; the lower state F values are given underneath. The upper tracing shows two lines, the P-j(38) and P3(38) lines. The P-j(38) line consists of 13 components, rather than eight, and the intensity pattern is anomalous. The P3(38) line has been included to give the intensity scale; nevertheless its hyperfine pattern is found to be irregular as well, as a result of the internal hyperfine perturbation in the ground state described earlier. Intensity considerations allow the F" quantum numbers to be assigned to the components of P^(38), as given in Fig. 5.3. It is evident there has to be a perturbation within the hyperfine structure of the F-j(37) rotational level of the upper state. The lower state rotational-hyperfine energies can be calculated from the rotational constants got by fitting the ground state combination differences The upper state term values can then be obtained by combining these with the line positions, as in Table 5.3. When the upper state energy levels are plotted against F(F+1), the classic pattern of an avoided crossing (41) emerges (see Fig. 5.14): there are two sets of energy levels which have minimum separation where the intensities of the corresponding lines are equal, and the averaged energy levels are P, (38) F=4140 38 36 34 36 37 38 39 40L 36 37 38 39 40 41 S P, (27) uuuuu 42 43 F= 25 26 27 28 29 30 31 32 on 00 I Fig. 5.13 Two regions of the intermodulated fluorescence spectrum of the C4£ - X4S (0,0) band of VO. Upper tracxn ing: the P_(38) and perturbed P^S) lines. Lower tracing: the unperturbed P1(27) line. 18111.55 h 1200 1400 1600 1800 F(F*1) Fig-5.14 Upper-state term values (cm"1) of the hyperfine levels of the perturbed F^) rotational level of the cV , v=0 state of V0 plotted" against F(F+1) . -155-Table 5.3 Analysis of the C E , ¥^37) perturbation. Pl (3B) lines 1^(36) lines Tl '(37) levels 104(0-C) F' lower upper F^'OB) energv lower upper Fj-oe) energv lower upper lover upper 35 17 304, ,0117 4.1577 807.3564 17385.7700* 5.9101 725.6015 18111. ,3674 1.5129 1 2 36 3. ,9807 4.1195 7.3933 5. 7339 5.8716 5.63S3 1. .3731 1.5114 0 0 37 3. ,9^51 4.0824 7.4314 5. ,6954 5.6360 5.6764 1, .3757 1.5131 -1 -3 38 3, .9044 4.0479 7.470E 5. .6587 5.8034 5.7158 1 .3749 1.5190 -1 0 39 3. .8601 4.0173 7.5115 5. .6131 5.7565 1 .3706 1.5231 _2 1 40 3 .6116 7.5534 5 .5654 5.7986 1 .3645 6 41 3 .7585 7.5968 5 .5116 5.6420 1 .3545 -3 42 3 .7031 7.6414 5 .4568 5.8E69 1 .3441 1 Values in cn"1; * means blended line. Allowance has beer, made in the averaging for an absolute calibration shift of 0.0007 CE"1 between the FjOS) and K^C36) lines. Least squares results: 18111.6888 i 0.0020 crT1 (lo) -0.000178 t 0.0000012 18111.1122 t 0.0032 0.000241 * 0.000002 0.0685 * 0.0001 Standard deviation • 0.00028 etc"1 CV : T(1) o kl pert : T(2) o K2 H12 -156-linear in F(F+1). The perturbation matrix element is very small, but analysis is nonetheless possible because the k parameters are very dif ferent for the two interacting levels. We have now disentangled the R-|(36) lines from the strong over lapping P2(17) lines and can refine the parameters reported previously (38). The effect of averaging the (36) amd P-j(38) data has been to improve the standard deviation of the least squares, fit considerably. The results are given in Table 5.3. The model used was a simple 2x2 ' matrix for each F value, akin to eq. (5.18): H = T^ + k1F(F+l) H12 H12 if} + k2F(F+l) (5.23) Nothing was held fixed in the least squares treatment, and the accuracy of the model and the fit must be assessed by comparing the observed and calculated k values for C4z" , F-j(37) : k(cV, N=37, J=38^)ca1c = -0.000185 cm"1 (5.24) k(C,V, N=37, J=3835)obs = -0.000178±0.00004 cm_1(3a) 4 -The perturbing state has a lower B value than C z , which means 4 -that further perturbations in the other spin components of C z might be expected at lower N values. We have not identified any such per turbations, and can unfortunately say nothing about the nature of the perturbing state except that its hyperfine splitting is large. An -157-Tnteresting effect of the perturbation is that the hyperfine structures of the levels within about four units of N on either side of the avoided crossing are noticeably irregular; this reflects the fact that the un perturbed level separations show a strong dependence on F. A second small perturbation occurs in the F-| component at N=36. The ?2 component is also perturbed at this position. Unfortunately extra lines do not occur and we can say nothing about these perturbations except that they appear to be unrelated to each other or to the other perturbations described. F. Least squares fitting of the line positions Ht has been a formidable problem achieving a least squares fit to the observed data that reflects their precision adequately. We dis covered at an early stage that a full matrix treatment of the hyperfine structure was required, and the only approximation we have made has been to omit the AJ=±2 elements of the electric quadrupole interaction; these are the only elements which do not add to matrix elements of the magnetic interactions, and in any case are calculated to be very small. We also quickly found that the absolute calibration of the isolated lines in the tail of the band was less precise than that of the crowded lines in the head of the band where the overlapping of VO lines between successive 1 cm-1 scans of the laser permits several scans to be cali brated at once (see Section B). 4 -However, the main difficulty has been the fact that the C E F^ levels are shifted by electronic perturbations throughout the N range -158-4 -of our spectra; also we cannot trust the other C E spin components not to have been shifted similarly after about N'= 25. When we attempted to fit the raw data we were unable to obtain a satisfactory converged fit unless we restricted ourselves to N<20, and even then the upper state centrifugal distortion parameter D was unrealistically low (>6.2xl0"^ cm"1, compared to the Kratzer relation value of 6.62 x 10"^ cm"1). After some experimenting with higher order terms we realised that it would be necessary to allow for the effects of the state crossing the levels at N=26, and not to attempt to fit the upper state beyond N"=25; the ground state h^F" combination differences, although less precisely determined because of calibration problems, could be included for the full range of our data, to N=40. 4 -(i) Deperturbation of the C z level positions The 'deperturbation' procedure obviously depends critically on the nature of the perturbing state (hence the detailed discussion in the previous Section). To summarize, we are certain that the perturbing state is orbitally-degenerate, has an effective rotational energy ex pression Epert = 0,482 J(J+1) 011-1 (5.25) and that the interaction matrix element <Jpert=24 |H|C4z~, N=26, J=24 > is 0.496±0.006 cm"1. It seems unlikely that A states, or states with 2 4 2S+1>4 will be important, so that we are restricted to n and n states. Provisionally we favour n because we might expect to see a perturbing -159-4 4 -n state directly in emission to the ground z state, and particularly because of arguments based on the hyperfine structure. Fortunately, as can be seen from the tables of Kovcics (37), the matrix elements between F4 and any component of ^n(a) or ^n(a) are essentially independent of J, inasfar as matrix elements of BL can be neglected compared to those of Ea^K; the single exception isAl-^' ^n1s 1S ru^ec' out ^y hyperfine arguments since the hyperfine splittings of the perturbing state are either much too big or of the wrong sign. We can therefore take the interaction matrix element as being in dependent of J, and can calculate the downward shifts caused in the \~ F^ levels according to eq. (5.18), with the zero order perturbing state energy written as in eq. (5.25). The shifts are given in Table5.4, where they are seen to rise from 0.0060 cm-1 at N=3 to 0.0300 cm-1 at N=22. For the final least squares work we raised the F4 levels by the amount fromTable5.4 in excess of 0.0060 cm-1; the quantity 0.0060 cm"1 4 -is thus incorporated into the effective A parameter for C z . Similar corrections should be needed in the F3 levels (as Fig, 5.12 shows), but it is easy to prove that they are an order of magnitude smaller. We have not included them specifically, so that they are taken up in the effective spin and centrifugal distortion parameters. The principal justification of this deperturbation procedure is that it works - it removes the systematic trends in the least squares residuals for the rotational structure, and it makes the centrifugal distortion parameters more realistic: for instance the upper state D value, determined from levels up to N'=24 only, rises to 6.44 x 10"7 -160-Table 5.4 Calculated perturbation shifts in the VO C E v=0 F4 level N Av/cm 3 0.0060 4 0.0062 5 0.0065 6 0.0067 7 0.0071 8 0.0074 9 0.0078 N Av/cm 10 0.0082 11 0.0087 12 0.0093 13 0.0099 14 0.0107 15 0.0116 16 0.0126 N Av/cm 17 0.0139 18 0.0156 19 0.0177 20 0.0204 21 0.0243 22 0.0300 23 0.0393 -161--1 2 -7 -1 cm , compared to the Kratzer relation value of 6.62 .x 10 cm We emphasize that the ground state rotational constants are unaffected: they are determined principally from the A2F" measurements up to N=40. Similarly the hyperfine constants of both states are unaffected; a least squares fit of the uncorrected data up to N=6 gives essentially the same hyperfine constants as the deperturbed data up to N=24, though the precision of the latter is greater. (ii) Least squares results The least squares fitting was carried out in two steps. In a first step all the lines to N""=23 and the high N A2F"'S up to N=40 were fitted simultaneously. This gives a good determination of the spin and rotational constants, though the lower accuracy of the high N A2F"''S affects the statistics for the hyperfine constants. In a second step the ground state rotational constants were held fixed, and only the more accurately calibrated data up to N"=23 were fitted. The spin and hyperfine constants do not change, but their standard errors improve consi derably. Anomalous D yalues in apparently unperturbed excited states of transi tion metal oxides are not uncommon. For instance the A6z+ v=l state of MnO (24) has a D value three times higher than expected, and many of the upper levels of the 'orange system" of FeO have unusually large D values (A.S-C. Cheung, A.M. Lyyra and A.J. Merer, work in progress). -162-The model used was the full matrix of Table 5.1 in each case. The only constraints applied were that Ap and b^ were set to zero for the ground state, and that the nuclear spin-rotation parameters Cj were fixed. It was found that convergence was very slow with the cT parame ters floating, but that the difference ACT=CJ'-Ct" was well determined and remained constant in successive iterations. Accordingly, since the apparent nuclear spin-rotation interaction, cTI.N^ arises principally from second-order spin-orbit effects (18), as does the electron spin-rotation interaction yS^, we made the arbitrary choice cT(cV) _ Y(cV) {5 26) Cj(xV) YuV) Effectively this portions out the contributions to ACj from the two states in the ratio of the two Y'S, on the assumption that the spin-orbit terms are similar. The results are given in Table 5.5. The error limits for the spin and rotational constants are 3a values, taken from the first fit, in cluding the A2F''"s, where the overall standard deviation, normalized to unit weight, is 0.00092 cm"1 (28 MHz). The error limits for the hyper fine constants are from the second fit, using only the 1363 lines up to N'"=23, where the normalized standard deviation is 0.00076 cm"1 (23 MHz). G. Hyperfine parameters The electron spin resonance spectrum of V0, xV, in an argon matrix at 4 K has been measured by Kasai (10). He derived values for the iso-Table 5.5 Rotational, spin 4 and hyperfine constants for the C 4 -Z and X Z states of V0. cV '. v=0 X< 'z"r v=0 T 0 17420.10257 ± o.oooi7 0.0 B 0.4937896 + 0.0000033 0.5463833 + 0.0000029 107 D 6.44 ± 0.03 6.509 ± o.oiA Y -0.018444 ± 0.000069 0.022516 ± 0.000066 X 0.7469? ± 0.0003 2.0308? ± 0.0002^ lO7 YD 5.43 ± 0.50 0.56 ± 0.32 io6 xD -4.3 ± 0.5 0.0 fixed lO5 Tg -23.1 ± 1.4 -1.0 ± 1.5 b -0.00881 + 0.00003 0.02731 + 0.00004 c -0.00114 ± 0.00009 -0.00413 ± 0.00008 2n e Qq 0.00139 + 0.00023 0.00091 ± 0.00088 106 cT -3.9 fixed 4.7 fixed i-io5 bs 4.5 + 1.8 0.0 fixed Values in cm"1; error limits are three standard deviations; o = 0.00076 -1 cm The bond lengths (rQ) are: cV 1.6747 A, XAZ 1.5920 A. •164-tropic and dipolar interactions which are closely similar to our gas phase values. With the conversions b-ac4c=Ai=A,so-Adip; c-A,, -Ax=3Adip (5.27) the values are b = 0.02731 0.00004 cm"1 (gas) = 0.02792 0.00002 cm"1 (matrix) and c =-0.00413 0.00008 cm"1 (gas) =-0.00408 0.00003 cm"1 (matrix) (5.28) (5.29) There is excellent agreement for the dipolar constant c, but there is a small though definite difference between the gas and matrix values of the isotropic parameter. As pointed out by Kasai (10), these parameters provide strong evidence for the ground state electron configuration 4S01 3d6 : the sign of b for transition metal d electron radicals is negative because of spin polarization effects unless s electrons are also present (39), and c will be negative also. The parameter c is a sum over the valence electrons of the terms 3 12 = 3gJjBgIjJN<n ] r" .^(3cos e-l)|n> (5.30) where r is the distance between an electron carrying spin angular momen tum and the vanadium nucleus. If we make the approximation that the states n closely resemble V atomic orbitals, the sum becomes -165-c=3x(.2/3)guBgIjjN<3d6 |r"3.J5(3cos2e-l) 13d<S> (.5.31) where the factor 2/3 arises because only the two 3d6 electrons, out of the three valence electrons, give non-vanishing average values of 3cos 6-1. For atomic-like orbitals the average value expression is (.18,20) -3 2 2 -3 <n£m|r .3j(3cos e-l)|njyn> = ^m^cos e-1|an><n£|r |n£> = _[3m2-ii(£+l)]<r-3>no (5'32) (2£-l)(2£+3) which, for the ground state of V0, gives c in cm-1 units as c=-7WiyN<r"3Vhc (5-33) The observed value c = -0.00413 cm"1 therefore gives <r"3>,. = 3.0 x 1024 cm"3 (5.34) which is 85% of the value given by the Hartree-Fock calculations of Freeman and Watson (40) for the free V atom. It is interesting that this simple model also accounts for the value of the ground state electric quadrupole parameter e Qq. Assuming that only the two 3d6 electrons are responsible for the quadrupole parameter we find e2Qq=-(4.803x10'10esu)2x0.27x10"24cm2x2x-y<r"3>3d/hc (.5.35) -166-2 -1 from which the experimental value e Qq = 0.00091 cm gives <r"3>3d = 2.5 x 1024 cm"3 (5.35) However the error limits on e2Qq are so very large that the agreement >3d in the values of <i""3>od is probably mainly fortuitous. It is not so easy to apply these agreements to the C4E~ excited state because the 4sa electron has been replaced by a 4pa electron. As expected the isotropic parameter b is negative because of spin polariza tion, and the dipolar parameter c is much smaller than in the ground state. The decrease of c on electronic excitation can be understood from the model given in eq. (5.30). For the C4i" state the expression becomes 3 2 3 2 1 3 2 It is not easy to obtain independent estimates of <r >^^, but with the -3 -3 very crude approximation that <r >3C|=:<r >4p> we obtain ,4-3 ,2-3 » c=gPBgIyN(-7<r >3d+3-<r >4p)/hc =-0.171gpBgIuN<r"3>3d/hc = -0.00126 cm"1 Somewhat surprisingly, this number agrees, almost to within the experi mental error, with our observed values. The upper state quadrupole parameter does not fit this model . As for the other two hyperfine parameters, bs and cT, we have diffi culty determining the parameters separately for the two electronic states, -167-though the difference between the parameters on electronic excitation is well determined. This is a consequence of the selection rule AF=AN, which applies except at the lowest N values; obviously, direct measure ments of the hyperfine level separations will be needed to break the correlation. We estimate that the measured difference ACj=Cj'-Cj" is accurate to about 10%, or, in figures AcI=cI,-cI"=(-8.6±0.9)xlO-6 cm'1 (5.39) The standard deviation in the least squares fit was improved by about 2% when we included the Cj terms, though as explained above we had to fix the Cj parameters in the ratio of the y parameters in the final least squares fitting. The symptom showing that the Cj terms were needed was that the least squares residuals for the hyperfine structures of all  four electron spin components showed a systematic trend from positive to negative with increasing F; the effect is only about 0.002 cm"1, but it disappeared at once on inclusion of the Cj terms. 4 -The third-order cross term bg is well determined for the C E state provided we set bs" equal to zero. This is not an unreasonable approxi mation because the value of bs reflects the positions of nearby electro nic states coupled through spin-orbit interaction, and we know that the C4E" state suffers many local perturbations. Also the related parameter Ys is essentially undetermined for the ground state but well-determined for the C4z~ state. It is unfortunately not simple to interpret the parameter b<., so that all we can do is point out the need to include -168-t>s in precise work on excited electronic states of quartet and higher multiplicity. H. Discussion 4 - 4 -, This analysis of the C E -X E (.0,0) band of VO at sub-Doppler resolution is the most detailed account so far of an electronic band involving quartet states. The most interesting aspects are the internal hyperfine perturbations in the two electronic states, which we can fit 4 in detail using the complete E spin and hyperfine Hamiltonian. 4 _ Electronic perturbations are unfortunately widespread in the C E state, which means that we cannot obtain a fit to the higher N data that does justice to their precision. In fact we have had to apply corrections to all the F4 levels of the C4E~ state to allow for the comparatively large perturbation at N = 26. Without these corrections the least squares results show systematic residuals even at low N, and the centri fugal distortion parameters are unrealistic. The final fit was performed with 'deperturbed' data only up to N' =24, plus ground state A2F" com bination differences to N = 40. The upper state term values beyond N = 24 are calculated from this fit to within about 0.1 cm-1, as can be seen from Fig. 5.15. This figure shows the residuals for the R branch lines up to N1 = 42, calculated using the final constants of Table 5.5. The chaotic courses of all four spin components are readily appreciated. 4 -The C E state also suffers from 'global' perturbations affecting all levels, besides the local rotational perturbations. These require the introduction of a second spin-rotation parameter ys» following the formalism of Brown and Milton (25). This parameter is a third-order -169-spin-orbit effect, and gives evidence for a close-lying electronic state that interacts strongly through the spin-orbit operator. We have also had to introduce the corresponding isotropic hyperfine parameter b^ (26), which arises in similar fashion as a cross-term between the I .S hyperfine operator and the spin-orbit interaction. The parameters y$ and b<. only occur for states of quartet and higher multiplicity. The magnetic hyperfine constants for the two states have been accurately determined. For the ground state there is good agreement with the e.s.r. values of Kasai (10), inasfar as the gas phase and matrix values can be compared. The magnetic constants for the upper state show 2 1 clearly that its electron configuration is 3d6 4pa . The electric qua drupole parameters are unfortunately not well determined, though there appears to be consistency between the ground state electric quadrupole and magnetic dipole parameters, which involve roughly the same averages over electron coordinates. The electron spin-spin parameters A for the two states have been accurately determined, despite the parallel selection rules, because of the observation of Q branches at low N values. The Q lines essentially 4 permit an exact determination, according to the E relation 4A-2Y=F2(N)+F3(N)-F1(N)-F4(N) (5.40) (which follows from Table 5.1) because they can be combined with the main branch R and P lines to provide the separations F2(N)-F-j(N) and F4(N)-F3(N). Less precise values could still have been obtained without the Q branch measurements because the low N line positions are quite sensi--170-Fig- 5.15 Residuals (obs-calc) for the R branch lines of the VO 4 - 4 -C £ -XI (0,0) band, as compared to the positions predicted from the constants of Table 5.5, plotted against N'.'Raw' data have been used, so that the R^ lines from N' = 9-22,.which were deperturbed for the least squares treatment (thick lines) have i-zero residuals. Vertical bars indicate the spread of the non-hyper fine structure. -171-ti ve to the exact values of the two A parameters. 4 -The low N energy levels of VO .X E , v=0 will be of interest to astronomers concerned with the detection of VO in interstellar space and to spectroscopists concerned with microwave and far infra-red stu dies of the ground state. We therefore list the ground state rotational and hyperfine energy levels (as calculated from the constants of Table 5.5)up to N" = 5 in Table 5.6. The experimental ground state hyperfine combination differences, between levels with the same F value in the F2 and F3 electron spin components over the range N" = 8-20, are listed in Table 5.7. 4 - 4 -To summarize, the VO C E - X E electronic transition provides 'textbook' examples of the effects of electron spin and hyperfine structure in a quartet state in case (bgJ) coupling; it is frustrating that electronic perturbations prevent the higher N lines from being fitted to an accuracy that matches their precision. .4 -Table 5.6 Rotational and hyperfine energy levels of the X E -1 from the constants of Table 5.5. Values in cm . N J F^.T-7/2 P=J-5/2 P-J-3/2 F=J-l/2 F-J+l/2 v=0 state of VO for N >5, calculated F=J+3/2 F=J+5/2 F-J+7/2 0 1 1.5 -2.8982 -2.8391 -2.7595 -2.6576 2.5 -1.6426 -1. 6160 -1.5759 -1.5220 -1.4541 -1.3716 1.5 1.2592 1.2536 1.2434 1.2264 0.5a -3.1255 -2.9492 3.5 0.6665 0.6766 0.6969 0. 7273 0.7681 0.8194 0.8814 0.9545 2.5 4.0231 4. 0275 4.0340 4.0425 4.0527 4.0642 1.5 5.9068 5.9379 5.9782 6.0269 0.5a -0.8199 -0.9343 4.5 4.0522 4.0685 4.0930 4. ,1258 4.1670 4.2167 4.2751 4.3424 3.5 7.6362 7.6389 7.6442 7 . ,6525 7 .6638 7.6785 7.6967 7.7190 2.5 9.0674 9. .0741 9 .0835 9 .0948 9.1070 9.1187 1.5 6.2261 6.2351 6.2490 6.2694 5.5 8.5121 8.5326 8.5599 8, .5943 8.6357 8.6842 8.7401 8.8035 4.5 12.2171 12.2219 12.2294 12 .2396 12.2531 12.2703 12.2918 12.3183 3.5 13.3492 13.3505 13.3531 13 .3565 13.3601 13.3632 13.3647 13.3636 2.5 10.0328 10 .0224 10.0072 9.9875 9.9638 9.9369 6.5 14.0522 14.0756 14.1050 14 .1405 14.1820 14.2297 14.2838 14.3443 5.5 17.8219 17.8279 17.8364 17 .8476 17 .8621 17 .8803 17.9032 17.9318 4.5 18.7437 18.7449 18.7463 18 .7475 18.7479 18.7468 18 .7430 18.7354 3.5 15.1594 15.1529 15.1400 15 .1208 15.0955 15.0643 15.0275 14.9855 J - h and F^(2), J = h must be treated as F^( -1) and F2(0), respectively, since \ > Table 5.7 Gronud state hyperfine combination differences, F2(N)-F3(N), -1 , in cm , tor the X4E", v=0 state of VO in the range N=8-20. N F=N+3 F=N+2 F=N+1 F=N F=N-1 F=N-2 F=N-3 8 -0.4193 -0.4606 -0.4896 -0.5101 9 -0.3325 -0.3770 -0.4066 -0.4272 -0.4411 -0.4510 -0.4559 10 -0.2638 -0.3103 -0.3411 -0.3629 -0.3735 -0.3804 11 -0.2091 -0.2581 -0.2871 -0.3057 -0.3163 12 -0.1699 -0.2190 -0.2461 -0.2615 -0.2692 -0.2700 -0.2677 13 -0.1484 -0.1947 -0.2184 -0.2300 -0.2330 -0.2287 -0.2213 14 0.1455 -0.1829 -0.2021 -0.2080 -0.2056 -0.1958 -0.1795 15 0.1589 0.1854 0.1977 -0.1993 -0.1909 -0.1750 -0.1501 16 0.1813 0.1992 0.2045 0.2001 -0.1859 -0.1628 -0.1281 17 0.2101 0.2197 0.2195 0.2104 0.1913 0.1631 -0.1202 18 0.2413 0.2400 0.2248 0.2040 0.1723 0.1252 19 0.2619 0.2466 0.2222 0.1888 0.1406 20 0.2694 0.2438 0.2095 0.1618 The tabulated values are observed quantities corresponding to the differences betw hyperfine levels with the same F value shown in Fig 5.3. -174-Chapter 6 Laser-Induced Fluorescence and Discharge Emission spectra of FeO; Evidence for a 5 A.j Ground State -175-A. Introduction Ferrous oxide, FeO, is probably the most important of the diatomic oxide molecules whose spectra have so far defied detailed interpretat ion. It is of interest in astrophysics, as well as molecular spectros copy, because of the high cosmic abundances of both iron and oxygen. The difficulties with FeO have been its low dissociation energy (which means that it is quite difficult to prepare in discharge systems), its involatility, and the tremendous complexity of its spectrum. Considerable argument has surrounded the nature of the ground state of FeO. Quite recently Engelking and Lineberger(1) have inter-5 preted the photoelectron spectrum of FeO in terms of a A ground state for FeO, with a vibrational frequency of 970 ± 60 cm"1, and De Vore and Gallaher (2) identified a band at 943.4 ± 2.0 cm"1 in infra red emission experiments on FeO, but it is now clear, from the matrix isolation work of Green et al. (3), that FeO has a ground-state vibra tional frequency of about 875 cm"1. This number is also found for the lower state of the well-known electronic band system in the orange region (4-6), which must therefore involve the ground state. The orange band system is unusually complex (7,8), but has been found to contain a few surprisingly simple parallel bands consisting of single P and R branches (4-6), apparently of type 1z- 1i. Harris and Barrow (6) recognize that the bands must be more complex than this, since theoretical predictions give, variously, 5A(9,10) and 5z+ (11) for the ground state. -176-The purpose of this chapter is to report new emission and laser-induced fluorescence spectra of FeO which proye that the simple bands in the orange system are Q' = 4 - Q" = 4 bands. This provides strong evidence that the ground state is A.., since n = 4 components do not 5+5 7 + arise in the other possible candidate states, E , n, and E . This identification agrees also with Weltner's report (12) that matrix isolated FeO gives no ESR spectrum under conditions where orbitally nondegenerate species normally give strong signals. -177-B. Experimental Details Emission spectra of FeO were excited by a 2450-MHz electrodeless discharge in a mixture of flowing argon, oxygen, and ferrocene (dicy-clopentadienyl iron) at low pressure. The discharge is unfortunately not very stable, and gives mainly CO spectra if the buildup of solid rust-like products becomes excessive, because these interfere with the transmission of the microwave power. As a result, photographic expo sures longer than 1 hr were often unsatisfactory, but this time was sufficient to give good spectra in the region 5500 - 6300 K using Kodak Ila-D plates in a 7-m Ebert-mounted plane grating spectrograph. The temperature of the emitting molecules, as estimated from the de velopment of rotational branch structure, is about 500°C. Laser-induced fluorescence of FeO was produced using a Coherent Inc. CR-599-21 tunable dye laser operating with rhodamine 6G, and pumped by an argon ion laser. The optical arrangement for this experi ment was the same as in chapter 5. The laser beams were sent through the end of the flame of the microwave discharge system described above, and observed at right angle to the stream of molecules. Broadband and single frequency laser excitation spectra, and Sub-Doppler intermodulated fluorescence spectra (.13) of certain regions of the 5819^ band were recorded, as well as resolved fluorescence spectra. The photographic "survey" spectra were measured on a Grant auto matic comparator, and reduced to vacuum wavenumbers using a four-term polynomial. Calibration spectra were provided by an iron-neon hollow cathode lamp, for which the wavelengths have been listed by Crosswhite (14). The laser spectra were calibrated by means of the iodine spec--178-trum atlas of Gerstenkorn and Luc (H), with the correction of 0.0056 cm"1 to give absolute wavenumbers applied. -179-C. Results Harris and Barrow (6) have identified three bands involving the level v" = 0 of the state which appears also in the matrix isolation experiments of Green et al. (_3); they occur at 5583, 5819 and 5911 K. The 5583- and 5911 A bands lie in crowded spectral regions where blending is severe, but the 5819-A band is in a comparatively clear region and we have selected it for study. Its head is illustrated in Fig. 6.1. The band is a very strongly red-degraded parallel band, and the assignments of the P and R lines are those of Harris and Barrow (6). A feature of this band is that where small rotational perturbations occur they usually appear as two lines of equal intensity. Because this implies that there are several exact coincidences of perturbed and per turbing levels if the states involved have Q = 0 (as postulated by Harris and Barrow (6) we suspected that there could be A-doubling pre sent which is not resolved in the grating spectra. Sub-Doppler inter-modulated fluorescence spectra of the unperturbed line R(15) at a resolution of about 75 MHz showed at once that this is true (see Fig. 6.2); in this figure the line is seen to consist of two equally intense closely spaced components separated by about 120 MHz. Given that ft' = ft" f 0, we searched for possible Q lines in the grating spectra, because the J numbering of the first Q line would give the ft value. Quite a number of lines occur in the expected region, but a branch could be picked out. Its numbering was established by plotting the line positions against n(n +1), where n is an arbitrary running number, and choosing the best straight line. The result gave ft = 4. •180-The Q(4) line of the 5819-A band runs: into the R(..10) line in Fig. 6.1 because of the strong exposure; the lines are seen resolved in the sub-Doppler spectrum shown in Fig. 6.2b. In view of the fact that there are many weak background lines underlying the 5819-A band which could easily be mistaken for Q lines (see Fig. 6.1) confirmation was sought from rotationally resolved laser-induced fluorescence experiments. These turned out to be absolu tely conclusive and leave no doubt about the Q branch and its numbering. Some of the patterns observed are illustrated in Fig. 6.3. As expected, excitation at the wavelength of the first line, Q(4), gives only Q(4) and P(5) emission, while higher J lines of the Q branch give R-, Q-, and P-branch emission. Excitation of the R-branch lines gives consis tent patterns, and somewhat surprisingly, weak Q-branch emission can even be seen when the line R(16) is excited. In the end it was possible to follow the Q branch on the grating spectra from its first line, Q(4), up to Q(26), where it becomes lost in the background of weak lines. The assigned lines of the 5819-A band and the other bands we have studied are listed in Appendix VI Table II. o Figure 6.1 also shows the head region of the 6180-A band. This band was shown by Harris and Barrow (6) to have the same u " value as the 5819-$ band, but with v" = 2 rather than v" = 0. Quite a strong Q branch can be seen, which can be numbered unambiguously by means of the A-|F" combination differences for the level -v" = 2. Again the first line is J = 4, which is consistent with the laser-induced fluorescence experiments described aboye. Similar Q branches have been identified in the 5583- and 5866-A bands (v" = 0 and 1). .181. 4 5 10 i iiiiniiiiiMiiiuiiiiiii R. 4 5 10 5819 A 6180 A 10 4 10 12 Q4 16174.03 Cfn-1 j I I I I I | I | I 16152.15 cm-1 • iiiiiiiiiiiiHiMiipm ILLS ' » " , i i* I I I I 4°. tot Rz Fig 6.1 Head of the 5819-A band of FeO. Lower print: head of the 6180-A band of FeO. (a) (b) Q(4) R(10) 17167.6050 cm-' I I I I I 1 I 1 I I 300 MHz morkers Iodine fluorescence spectrum 17176.1197 cm-' Fig 6.2 Two regions of the intennodulated fluorescence spectrum ofs FeO:(a) The twoA components of the R(15) line of the 5819-A band, (b) The Q(4) and R(10) lines of the 5819-A band; the A doubling is not resolved for these lines. -182-5823 5829 5821 5830 Q(6) R06) excitation 5823 5829 5832 5846 Wavelength markers 6.3 Resolved fkorsecence spectra of FeO produced by excitation of various lines of the 5819-Aband: excitation of Q(4),Q(5}, Q(6) and R(16). The intensity of the excited line is anomalousl high as a result of scattered laser light. -183-D. Discussion The work presented here proves, that the comparatively simple bands analyzed by Barrow and his co-workers (5,6) have Q" = 4. Some of these bands (including the 6180-$ band illustrated in Fig. 6.1) form a lower-state progression which gives vibrational constants (.6). that are almost identical to those obtained from the infrared spectrum of matrix-isolated FeO by Green et al. (3), viz., Gas: CJ0 = 880.61 cm""1, WQX0 = 4.64 cm-1; 6 i 1 (6J) Matrix: we = 880.2 cm we*e = 3.47 cm . Therefore the ground state of FeO contains an n = 4 spin-orbit compo nent. The ground electron configurations of the transition oxides immediately before FeO are known (15,16) to be (Appendix V) TiO (4sa)1(3d6)1 3Ar VO (4sa)1(3d6)2 V (6.2) CrO (4sa)1(3d6)2(3d^)1 5nr MnO (4soV(3d6)2(3d^)2 6z+ where the energy order of 4sa and 3d6 is not certain. In FeO the extra electron could go into the next unoccupied m.o., 3dCT , giving a 7 + 5+5 E ground state, or into the 4sa or 3d6 m.o.'s, giving z or A as the ground state. Theoretical computations (9-11) are divided between 5Z+ and 5A, through the CI calculations of Bagus and Preston (9) con-184-5.+ elude that the ground state is not bz . The fact that there is an 5 n = 4 component in the ground state is only consistent with where 3 + 7 + the"fi values run from 0. to 4; the highest n values in z and z states are 2 and 3, respectively. 5 It is interesting that the A state under discussion, which comes 13 2 from the configuration (4sa) (3d6) (3C1TT) , must be inverted, with its ft = 4 component as the lowest in energy. This is probably the reason why the matrix-isolation vibrational constants (3) agree so exactly with the gas-phase constants (5,6) because at the low temperature of the matrix only the Q, = 4 component is likely to be appreciably popu lated, assuming spin-orbit intervals of about 100 cm"1 (by analogy with TiO (17), where there is also an unpaired 3dS electron). The other 5 ii components of the A state will have slightly different effective vibrational frequencies because of the variation of the spin-orbit coupling with vibration. The subbands so far analyzed carry no direct information about the spin-orbit coupling of the ground state. For a start the subbands are all parallel-polarized (ft1 = ft"), and as yet only one spin component has been identified. However, these ft = 4 subbands, though prominent in the spectrum, account for only a small fraction of the total emission intensity, and subbands involving the other spin components must also be present. The prominence of the ft = 4 subbands probably results from the fact that the A-doubling is unresolved, so that their lines have apparently twice the strength of other lines belonging to subbands in the same region with resolved A-doubling; this effect is 5 5 also pronounced for the ft = 3 subbands of the A n-X n system of CrO (18) •185-In the CrO spectrum the five subband heads form a regular series, which is obvious in low-dispersion spectra, but the same is not true in the FeO spectrum. It appears, that there are extensive interactions between two or more excited electronic states in FeO which produce an almost random distribution of Q, substates, as if the spin coupling were case (c). Because the lvalue of each band has to be determined individually in the FeO spectrum, it will be a lengthy process assembling data for all five spin-orbit components of the ground state. At present even the bond length is not accurately given by the available B value for 5 because of the spin-uncoupling. In principle, it would be possible to use the difference between the apparent centrifugal distortion con-stant for the component and the value given by the Kratzer relation to estimate the spin-orbit separations and then correct the B value for spin-uncoupling. In practice we find that the error limits on ^apparent^ A4^ are to° ^ar9e for tni's aPProacn to succeed. A least-squares fitting of the A2F"'s we have measured gives B .(5i.) = 0.51089 ± 0.00003r cm"1 (la) ((- ,A apparent 4' 5 15.3) D +(5A„) = 6.6n x 10"7 ± 0.2, x 10'7 cm"1 apparent^ 4' 0 3 (in close agreement with the results of Barrow et al . (5,6)) . A comparison of the molecules FeO and FeF (19) is instructive. FeF is known to have a AI ground state which arises from the electron 1 3 2 1 configuration (4sc) (3dS) (3dir) (3da) ; in other words the extra electron in FeF goes into the 3da m.o. rather than into 3d6 . This -186-shows the close analogy between FeO and FeF, because the ligand field effect of an F atom is not as great as that of an 0 atom so that the splitting of the iron 3d manifold Is smaller. In FeF Hund's rules apply to the three 3d orbitals and the 4s orbital as a group, producing a high-spin situation, while in FeO Hund's rules apply only to 3d6, 3dir, and 4sa. In further laser-induced fluorescence experiments which are not reported here, we have analyzed various sub-bands with ft" = 0, 1, 2 and 3 in the orange system of FeO. The regularity of the B" values leaves no doubt that the lower levels form the other spin-orbit compo-5 ° nents of the X A., state. The ground state bond length is 1.619 A, and the spin-orbit intervals are about 190 cm"1. -187-Chapter 7 Predissociated Rotational Structure in the 2490-A Band of 15N02 -188-A. Introduction Slightly predissociated jmolecular band systems where there is a sizable isotope effect offer the possibility of selective dissociation of one member of a mixture of isotopes. The experimental requirements are a suitable source of radiation which can be tuned to an appropriate wavelength and a scavenging system (chemical or other) which can collect the dissociated products without interference from the undissociated compound. Various experiments of this type have been successfully carried out using narrow-line lasers, for example, on s-tetrazine by Karl and Innes (1) and by Hochstrasser and King (2), and on ICfc by Liu et al. (3). The 2490-A absorption transition of N02(22B2 - X2A-,) is slightly predissociated (4) and therefore allows the possibility of laser-induced isotope enrichment. For this reason we have studied the 15 corresponding band of N02 with a view to identifying those wavelengths where irradiation would selectively dissociate one isotope and permit 14 15 the separation of N and N. o 15 Spectroscopically, the 2490-A band of N02 is very similar to that of 14N02, though its origin is shifted 14.5 cm"1 to higher energy. An analysis of the quartic centrifugal distortion constants of the upper state has been carried out. B. Experimental Details The absorption spectrum of 15N02 in the region 2480 - 2520 K was photographed in the 23rd order of a 7-m plane grating spectrograph, using Kodak SA-1 plates. The absorption path was 1.85 m, and photo--189-graphs were taken with the cell at room temperature and yarious tempe ratures up to 200°C. Doppler broadening of the lines becomes appreci able at the higher temperatures, and the highest temperature where useful spectra could be obtained was 120°C. The background continuum was supplied by a 1000-W xenon arc, and exposure times, with the spec trograph slit set to 25 -jim, were about 1 hr. Calibration lines were supplied by an iron-neon hollow cathode lamp, the reference wavelengths for which have been given by Crosswhite (5). The plates were measured on a Grant automatic comparator, and reduced to vacuum wave numbers with a four-term polynomial. C. Analysis of the 2490fl Band of 1 5N0Q The theory of the energy levels of asymmetric top molecules in multiplet electronic states is now fairly well understood (6-9). For reference we give the matrix elements required for the NC^ spec trum in the absence of hyperfine effects: < NK |H | NK >= ]-(B + C)N(N + 1) + [A - h(B + C)]K2 - %[0(J + 1) - N(N + 1) - 3/4][ao + a{3K2/(N(N + 1))- 1} - nK4/(N(N + 1))] - AKK4 - ANKN(N + 1)K2 - ANN2(N + l)2 + + HKiK6 + HKNK4N(N +1) < NK ± 2|H|NK >= {^(B - C) -h b[J(J + 1) - N( N + 1) - 3/4]/[N(N +1)] - 6NN(N + 1) - %6K.[K2 + (K ± 2)2]} (7.1) x [N(N + 1) - K(K ± 1)J1'2[N(N + 1) - (K ± 1)(K ± 2)f 2; <N - 1K|H|NK> = ( |a - 1 nK2)K[N2 - £ Zf2/H-<N - IK ± 2|H[NK> = ± |b[N(N + 1) - K(K ± I)]5* x [N + K - 1)(N + K - 2)]%/N. -190-The basis set for Eq.(7.1)is the type Ir representation case (b) basis |NJSK>, where the quantum numbers J and S have been suppressed because the elements are diagonal with respect to them. The coupling scheme (8) N + S = J (7.2) has been used, and the phases, are appropriate for the definition of the rotational angular momentum vector N as a space-fixed operator rather than a molecule-fixed operator. The quartic centrifugal distortion terms are complete, but only the largest sextic terms have been included. The spin-rotation constants in Cartesian form (10) are related to those of Eq. (7.1) by ao = " T (eaa + ebb + £cc); a = " 1 (2eaa ' ebb " ecc); b = " J(ebb ' ecc> <7-3> and n is the leading centrifugal distortion correction to the spin-ro tation interaction parameter e (called n„ = , by Dixon and Duxbury (11), aa aaaa nK by Brown and Sears (12) and A^ by Cook et al. (13)). The ground state of ^N02 has not been studied as comprehensively 14 as that of N02. The microwave spectrum of N02 is very sparse because of the large A rotational constant, and although it has been carefully measured by Bird et al. (14) and Lees et al. (15) the available lines do not carry enough information to determine all the centrifugal distortion parameters required to describe the energy levels to the precision of the optical spectrum. The missing data have been supplied for 14N09 by the high-resolution infrared spectrum (16), but as yet the -191-15 infrared data for NC^ 07-19) are less complete. Even so, the best 15 centrifugal distortion constants for N02 are in fact those from the microwave spectrum (14), though, as recognized by Lees et al. (15), they are not particularly good because so few lines are available. In analyzing the 2490-A band of N02 we proceeded as follows. First we converted the ground-state centrifugal distortion constants (14) from Kivelson and Wilson's T'S to Watson's formalism according to the recipe of Yamada and Winnewisser (20). Next the ground-state energy levels were calculated employing the second-order approximation formula of Cabana et al. (21) for the spin corrections to the rotational energy levels. With the spin parameters written in spherical tensor notation this formula is ESp1n(J = N ± 1/2) = ±l/2{(a - aQ ± l/2b61 K)N(N ± 1) - 3aK2 + nK4 - 9a2K2[l - K2/(N + 1/2 ± 1/2) 2]/4B" }/(N + 1/2 ± 1/2). (7.4) where the term ±lb6i „ refers to the two asymmetry components of 2 1 >^ K =1, not to the two J components of an (N,K) level. This approximat-ion works very well, since N02 is so close to being a prolate symmetric top, and breaks down (22) only where, by accident, a near-degeneracy oc curs between levels (N,K) and (N - 1, K + 2). Its advantage is that it permits the rotational energy levels to be calculated using the forma lism for a singlet electronic state, where the matrices are half the size of those required for an exact treatment of a doublet state. At this stage the lines could be assigned by standard combination difference techniques, since there are no perturbations. The band is a -192-normal asymmetric top type A Cparallel), band, with the N and K structures 15 ? both strongly degraded to the red. The head.of the N02 249.0-A band is 14 illustrated in Fig. 7.1 with the corresponding band of N02 printed alongside in its correct relative position. The line assignments 15 14 refer to the lower print, which is the N02 spectrum; the N02 assignments are not repeated, having been given in Ref. (4). It can be seen that the bands of the two isotopes are quite similar, particularly in the head region, but that they are sufficiently different that it 15 is not possible to assign the N02 lines by simply comparing the two spectra. The two spin components of the ^P^(2) line are clearly resolved, 14 and their relative intensities are found to be exactly as in the N02 spectrum; this confirms the assignment of the ordering of the F-j( J = N + 1/2) and F2 (J = N - 1/2) spin component lines made in Ref. (4). The argument goes as follows. The case (b) selection rule AN = AJ forbids any satellite branch transitions of the type that can be used in case (a) coupling situations to identify which spin component is which. At very low N, however, the relative intensities of the two spin components of a given line will be noticeably unequal, because the intensities are governed essentially by the number of components in the combining states, or in other words the J values. For the qP1(2) line the two com ponents are J* = ~-J" = 1-1 (F£ - F2) and J' = ll - J" = ll (F] - F^ , so that the F-j component is expected to be stronger. It can be seen in Fig. 7.1 that the stronger line is the long wavelength component. A very weak j" = ]~ - J" = ITJ- satellite transition is predicted to occur; this should fall between the F-j and F-, main branch lines, but has not been observed. 7.1 Head of the 2490-A bands ( 2 B, - X A. ) of N02 (above) and NC>2 (below). The line assignments refer to the ^N0„ spectrum. -194-Figure 7.2 shows the K = 7 and 8 subbands, printed from a plate taken with the gas at 120oC, where the branches run to N values rather higher than in Fig. 7.1. The spectrum is seen to be still quite com plex, despite the wide separation of the subbands,' but the characteristic large spin doublings of the branches are clear in the right-hand part of the figure. The K dependence of the spin splitting for constant N can be seen when the splittings for K = 7 and 8 are compared. The upper-state rotational constants were determined by adding the energies of the unblended lines to the lower-state energies, to obtain the upper-state term values, and fitting these by least squares. The Hamiltonian used for the upper state was the same as that for the lower state. The results are given in Table 7.1. The quartic centri fugal distortion constants of the upper state are very much what would be expected by analogy with 14N02 (4), but the sextic constants are completely different. The reason is that the ground-state Hamiltonian includes no sextic centrifugal distortion, since the microwave data do not allow these constants to be determined. We therefore set all the sextic constants to zero for the ground state, which means that the upper-state sextic constants in Table 7.1 are strictly the differences between the upper- and lower-state sextic distortion constants. When this is appreciated the constants are found to be very much as expected. For instance, in 14N02, Hal 1 in and Merer (4) could only determine the constants HVK and H1^, all the others being too small to measure, and they found H'K - H"K = (-3.26 ± 0.46) * 10"6 cm"1, H'KN - H"KN = (-1.85 ± 1.30) x 10"8 cm'1. (7.5) Fig. 7.1 K = 7 and 8 subbands of the 2490-A band of N0? (in the region 2502 - 2508 A). -196-Allowing for the mass difference these quantities compare favorably with those in Table 7.1. We did not attempt to refine the ground-state quartic centrifugal distortion constants in this work, even though" they clearly need impro vement. The reason is that the most poorly determined microwave con stants are precisely those which the parallel selection rules of the elec tronic transition prevent us from improving. Specifically, the constant is probably the least well determined because the microwave results go only to K = 3. The electronic spectrum also contains information on the K-stack separations up to K = 3 in the form of SR branches, but naturally it is far less precise; for K > 4 there are no transitions observed in the electronic spectrum except = 0 branches. Also the constant is difficult to obtain from the electronic spectrum because it is determined from differences between the asymmetry components of the low K stacks, where the lines involved lie in the most crowded region of the spectrum and blending is most severe. A second reason is that because of the predissociation in the upper state the lines are wider than the Doppler width alone. Because of the difficulties with the ground state we have not been 15 able to obtain as good a fit to the data for N02 as Hall in and Merer (4) could for 14N02, eyen though the data themselves are of comparable -1 15 quality. The overall standard deviation is 0.032 cm for N02, which -1 14 1 should be compared to 0.025 cm for N02-The assigned rotational lines have been collected in Appendix VI , ^In other words we could perhaps have lowered the standard deviation for 15N02 by refining the ground-state centrifugal distortion but because of the problems described above the improvement would almost certainly have been arti fi ci al. -197-TABLE 7.1 Rotational Constants for the 2490-$ Band of 15N0,(cm"1) 2 B9, 000 (upper state) 000 (ground state) To 40140. .339 ± 24 0.00 A 3, .9266 19 7.63047 B 0, .403300 94 0.433735 C 0, .363946 102 0.409440 106 AK 460 51 2297 io6 ANK 13 .17 158 -17.69 io6 AN 0 .439 31 0.2817 106 AK 8 .8 86 2.704 105 AN 0, .098 34 0.03189 106 HK -1 .93 42 -io8 HKN -1 .9 20 -£ aa -0 .1724 40 0.1718 ebb 0 0.000256 ecc 0 -0.003163 n. - -0.000103 a 0 .032 -Notes: Ground state rotational constants from ref. (14), with the quar-tic centrifugal distortion converted from the T values, which are Taaaa= (-9.12 ±0.13) x 10'3 TLk,.= (-1.382 ± 0.005) x 10"5 cm"1 aaaa j. bbbD , Taabb= C5.87±0.06) x 10"b xabab= (8.16±0.03) x 10 Ground state spin constants from ref.(15) except n which was calculated as Taaaa- eaa/2A (.11). Uncertainties are three standard deviations, in units of the last significant figure quoted. -198-D. Ccncl us ion The 24904 bands of both 14N02 (.4) and 15N02 are found to be slightly predissociated, and to have essentially the same linewidths. The predissociation lifetime was determined in Ref. (4) to be 42 ± 5psec. 14 The possibility of selective dissociation of N02 in the presence of 15 N02, and vice versa, has been explored in this work. The two spectra are compared in Fig. 7.1 It is seen that there would be no difficulty 15 14 dissociating N02 in the presence of N02 because the isotope shift of 14.40 cm"1 for the qRQ heads means that many strong ^N02 lines lie in a region where only weak sparse SR lines of 14N02 fall. On the 14 15 other hand, dissociating N02 in the presence of N02 would require narrow-line lasers specifically tuned to certain wavelengths. The most promising regions appear to be in the heads of the qR-| and qR„ branches 14 14 of N09, where many close-lying strong N02 lines fall in gaps between the 15N02, qP.| , qR2, and qP2 lines. The exact wavelengths can be cal-15 culated from the tables of assignments given in Table III for N02 and the Appendix to Ref. (4) for 14N02-c 15 Spectroscopically, the 2490-A band of N02 confirms the analysis 14 of the corresponding band of N02 in detail, and poses no questions. The need for a more detailed examination of the infrared spectrum of 15 NO <2 is pointed out. It is likely that the resulting changes in the ground-state constants can be transferred, directly to the upper-state constants reported in this work, since we have essentially determined the differences between the upper- and lower-state constants in this work. However, it is not impossible that the upper-state constants may-need to be reworked if the ground-state changes are considerable, since •199-the relationship between the two s.ets is not one-to-one because of the large changes in the rotational constants on electronic excitation. -200-Chapter 8 Fourier Transform Spectroscopy of VO; 4 4 -Rotational Structure in the A JI-X E System near 10500 & -201-A. Introduction Vanadium monoxide, V0, is present in considerable amounts in the atmospheres of cool stars, to the extent that its two electronic band systems in the near infra-red are used for the spectral classification of stars of types M7-M9 .(1). Both of these systems, A-X near 10500 A and B4n-X4E_ near 7900 ft, were in fact first found in stellar spectra ('2,3) before laboratory work, respectively by Lagerqvist and Selin (4,) and Keenan and Schroeder ( 5), proved that V0 is the carrier. The purpose of this chapter is to report rotational analyses of the (0,0) and (0,1) bands of the A-X system from high dispersion Fourier transform emission 4 4-spectra; the A-X system is shown to be another n- E transition. The A4n state of V0 is found to have quite small spin-orbit coupling, so that the rotational and hyperfine structure follows case (a^) coupling at low rotational quantum numbers, but is almost totally uncoupled to case (bQ1) coupling at the highest observed quantum numbers. The hyper-51 fine structure caused by the V nucleus (I = 7/2) is not resolved in the spectra reported here, but an interesting result is that the hyperfine parameter b for the A4n state can be estimated from the line shapes at high N values and is found to be essentially the same as in the ground X4E" state. The conclusion is that the A4n state comes from an electron configuration containing an unpaired 4so electron, as does the ground state. 4 In contrast to the other excited states of V0 the A n,v = 0 level is unperturbed rotationally; it therefore provides one of the very few examples known where the energy formulae for 4n states can be checked directly against observation. -202-B. Experimental details The near infra-red electronic transitions of V0 in the region 6000-14000 cm"1 were recorded in emission using the 1 meter Fourier Transform spectrometer constructed by Dr. J.W. Brault for the McMath Solar Telescope at Kitt Peak National Observatory, Tucson, U.S.A. The source was a microwave discharge through flowing VOCil-j and helium at low pressures, which was focused directly into the aperture of the spectro meter. An indium antimonide detector cooled by liquid nitrogen was used, and the resolving power of the spectrometer was set to approximate ly 800,000. Forty-two interferograms, each taking six minutes to record, were co-added for the final transform. The resulting spectrum, consist ing of tables of emission intensity against wave number for every 0.013608 cm"1, was processed by a third degree polynomial fitting pro gramme to extract the positions of the line peaks. C. Appearance of the spectrum The spectrum of V0 in the near infra-red down to 6000 cm"1 consists 4 4 - 4 4 -of the two electronic transitions B Ji-X z and A n-X z . The B-X system is very much stronger than the A-X system under our discharge conditions, so that the B-X progressions and sequences mask most of the A-X system except for the (0,0) and (0,1) bands.. Even the (0,0) band of the A-X system (which is by far the strongest band) is not free from overlapping B-X structure, which causes some difficulty in the analysis. The main heads of the A-X (0,0) band are illustrated in Fig. 8.1; each of the S R four sub-bands produces one strong head ( R43, R3, Q21 and R-j), and there VO A4n-X4E\(0,0) 4n 5/2 4n V2 4n 1/2 Fl. 8.1 Fourier transfer, spectrum of VO In the region 9410-9570 cm'1 shoving the heads A4H - X4E (0,0) band of VO. -204-4 4 -is also a less prominent Q-j head in the Ji - z sub-hand. Two other heads, belonging to the B-X (.1,4) band, appear in the region of the 4 4 -n5/2" E sub-band; they have not been identified in the Figure, though their branch, structure is readily picked out at higher dispersion. The A-X (0,1) band is qualitatively similar, though since it is weaker the background of B-X lines is more troublesome. The A-X (1,0) band is so heavily overlapped by B-X structure that we have not been S _i able to analyse it; the R^ head appears to be at 10503.3 cm"' but even this is not definite. 4 4 D. Energy levels of JI and z states 4 Energy levels for n electronic states have been considered by a number of authors (6-11"). The most detailed treatment is that of Femenias (.9), who has given a full explanation of how to calculate the matrix elements for the higher order centrifugal distortion terms. Detailed analyses of 4n states, against which to test the formulae, are less common; the best examples come from the spectra of 0^+ (10) and NO (12). 4z states, on the other hand, are much more numerous, and have been extensively treated (6,7,9,10,13-17)r It will therefore only be necessa ry to sketch the Hamiltonian and its derivation, and to give the matrices we have used. Following van yleck (l8)we take the rotational Hamiltonian, the first and second-order spin-orbit interactions and the spin-rotation interaction, respectively, as -205-H = B(r)(J-L-S)2 + A(r)L.S + |A( r) (3S2-S2). + Y(r)(.J-S) .S (.8.1). The expansion of the parameters A, B, A and y» which are functions of the internuclear distance r, in terms of the normal coordinate, produces centrifugal distortion terms, which are conveniently written in operator form as Hr d = -D(J-L-S)4 + W\DI(J-L-S)2, LzSz]+ 'cd, + \ h^3SU2)' + *DICO.-L-:U2.{&^]+ (8-2) where [x,y]+ means the anti-commutator xy + yx, which is necessary to 4 preserve Hermitian form for the matrices. The A-doubling of the n state 4 was calculated by setting up the 12x12 matrix for a n state interacting with a single 4z~ state according to the first two terms of eq. (8.1,), applying a Wang transformation to convert to a parity basis and treating the elements off-diagonal in A by second order perturbation theory. The effect is as if there were an operator HLD = ^o+p+q)(S2+S2M(p+2q)(j+S++J_S_)+ ^(jJ+J2) (8.3) acting only within the manifold of the 4n state (l1,19) - The A-doubling parameters (o+p+q), (p+2q) and q are related to matrix elements of the spin-orbit operator, as given in ref. (ll). The centrifugal distortion corrections to eq. (8.3) are obtained in the same way that eq. (8.2) is 1 2 2 constructed from eq. (8.1). The spin-spin operators -g a(.r)(S++S_) and 4 A^(r)(3S2-S2) are incorporated into the terms in (o+p+q) and A, respectively. Table 8.1 Matrix elements of the rotational Hamiltonian for a \ state in case (a) coupling. |-S> T.L + (B-^A0+2Ad) (z+1) -D(z +5z+l) i 3(J+'-i)D o+p+q -/3l[B-Vf-AD-2D(z+2)] */3[(o+p+q) + U+2)D0+p+q  +'5(2z-DDp+2q] \ + (B-'5A0-2v0)(z+3) -D(z2 + 13z + 5)'(J+',)[(p+2q) +30 + D o (z +3)+D (z-1 )] o+p+q p+2q q -/Slz^T} [ZD(J+b) «i(p*Zq)» DQ + p+q ^(z+DDp+2q * 4(z-2)Dq] -2/ri[B-'-5Y-2AD-20(z+2) l^(J+g<qiDp+2q +D (z+2))] q Symmetric T3/y+(B*'sAD-2»D)(z+l) -D(z2+9z-15) •(z-l)(J+»i)Dn */fz-l)(z-4) [!,q 3 q 3°P+2q + ,-i0"(Z-2)] /3(z-1)(z-4) [-20 -+4D (J+4)] -/3(z-4) [B-'-iY+AD -20(z-2)] Tr,/:,+(B+^AD+2A0) (z-5) -0(z2-7z+l3) z - (J+'-s)2. Upper and lower signs refer to e and f rotational levels respectively. The basis functions | Jsi • have been abbreviated to |si> -207-Table 8.2 Matrix elements for spin and rotation in a Z in case (a) coupling . I!> 2 2x + Bx - D(x +3x) -/3X[B-12Y-Ys-12YD(X+7=I-{2J+1}) - \i - 3Y[)x -2D(x+2)*{J+**})] -2\ + B(x+4)-D[(x+4)2+7x+4] symmetri c -7y-Yd(7X+16) * 2[B-i5Y-!sYD(x+n)+|Ys - 2D(x+4)](J+is) x = (j+4)2-l. Upper and lower signs give the e( F-j and F3) and f(F2 and F4) levels respectively. The basis functions |JZ> have been written |z> -208-The resulting Hamiltonian matrices which we have used are gi.yen in Tables 8.1 and 8.2 (for 4n and 4i states respectively). The X4E~,v=0 parameters were not varied in this work since they have been determined 4 - 4 -with great precision from the C z -J z transition using sub-Doppler techniques (17). The parameter y$ in the Z matrix represents the third-order spin-orbit contribution to the spin-rotation interaction (16,17); neither y^ nor the centrifugal distortion correction y^ appears in the 4 H matrix because they are not needed. Hyperfine effects have not been considered in Tables 8.1 and 8.2 be cause the hyperfine structure is not resolved. However, with the large 51 spin and nuclear magnetic moment of V (I = 7/2), the hyperfine structure is important in determining the details of the branch structure, as will be shown below. E. Analysis of the branch structure ft n _ Rather surprisingly, the analysis of the h'Ti-Vz bands of VO proved to be remarkably difficult because of unresolved hyperfine structure effects and overlapping sequence bands from the B-X transition. The pro blem with the hyperfine structure is that only when the hyperfine 'widths' of the combining levels making up a rotational line are the same does the spectrum consist of sharp rotational lines (where the eight hyperfine transitions lie on top of each other). Since the four electron spin components of the ground state have hyperfine widths that differ from one to the next by about 0.2 cm"1, rotational lines with the same upper state which go to different electron spin components of the ground state -209-have noticeably different line-widths. The broader the line-widths the more the intensity is spread out, and the more the line tends to get lost in the background of overlapping B-X structure. Therefore al-though a n- E transition should have 48 branches, most of them are broadened beyond recognition by the hyperfine structure in this case. There are only two regions of clear branch structure in the (0,0) band. One of these, shown in Fig. 8.2, lies between the two shortest wavelength heads. The obvious branch, later identified as Q43, could be assigned at once to the F3 spin component of the ground state because it contains the characteristic internal hyperfine perturbation pattern at N" = 15 discovered by Richards and Barrow (20) in the B--X and C-X systems. This internal hyperfine perturbation is a remarkable occurren ce, where the F2 and F3 electron spin components (N = J-^ and N = J+h respectively) would cross at N = 15, because of the particular values of the rotational and spin parameters, were it not for the fact that they differ by one unit in J, and therefore interact through matrix elements of the hyperfine Hamiltonian cf the type AiN = AF = 0, AJ = ±1. Extra lines are induced, and, since the detailed course of the ground state levels is known (17), their positions tell whether a branch containing them has F2" or F3", and also give its N-numbering. Given the numbering of the obvious F3" branch, the other three F4' branches marked in Fig. 8.2 could be numbered easily using ground' state spin and rotational combination differences, The R^ and Q4 branches are hyperfine-broadened, and even though they are intrinsically strong they are by no means obvious in the spectrum. At this stage the lower states of the branches were known, but the nature of the upper state was still -211-unclear. The other region of obvious branch structure is the tail of the band, part of which is illustrated in Fig. 8.3. There are at least ten sharp branches in this region, but only eight of them actually belong to the A-X (0,0) band. A further complication is that there are no ground state combination differences connecting any of these eight. The analy sis was performed by comparing the (0,0) and(0,1) bands, since the separations between corresponding (N,J) levels of the )(4E~ v=0 and 1 levels are known from the analysis of the C-X system (2l). This method gives at most two possible N-numberings for the branches, but it is less easy to determine the ground state spin component since the intervals are very nearly the same for the four spin components. Eventually all eight of these branches were identified, and assigned to their respec tive ground state spin components. The resulting pattern can be inter preted as the Q and P main branches of a 4n-4z transition where the 4n state is close to case (b) coupling at these high N values, and all four components show A-doubling. The analysis is confirmed by the identifica tion of the four R branches, and various weak hyperfine-broadened spin satellite branches. The branch is interesting because it is a sharp branch at the high N values of Fig. 8.3, but hyperfine-broadened at the lower N values of Fig. 8.2. It is possible to follow the -branch over the complete range of N values, and to see how it changes from broad to narrow fairly quickly in the region N=40-50. The reason for the sudden disappearance R R of the Q43 branch near N=35 (see Fig. 8.2) is then clear - the Q43 1 cm-' -213-branch is prominent at low N because the hyperfine structure of the 4 4 -Jl F4 level is initially the same as that of the J. z level, but with increasing N spin-uncoupling changes the 4n hyperfine level pattern until 4 - "R at high N it becomes the same as X E F^; as a result the branch TO becomes broadened. In addition the intensity of Q^, which is a spin satellite branch that becomes forbidden in a 4n(b)-4E(b) transition, must diminish as spin-uncoupling sets in. 4 4 What emerges finally is a 'text-book' example of a n - E transition 4 where the n state has quite small spin-orbit coupling so that it changes fairly quickly from case (a) to case (b) coupling. The 4n state is shown to be regular (with a positive spin-orbit coupling constant) because there is no detectable A-doubling in the F^ component (4n^2) before about N=45, whereas the other three spin components show A-doubling effects almost from their first levels. The A-doubling and spin-uncoupl ing patterns are shown qualitatively in Fig. 8.4, where the upper state energy levels, suitably scaled, are plotted against J(J+1). The curvature in the plots of Fig. 8.4 is a consequence of the spin-uncoupling. The assigned lines of the (0,C) and (0,1) bands of the A-X system are given in the Appendix; only the sharp lines are listed, because they are suffi cient to determine the upper state constants, and in any case it is often quite difficult to obtain the exact line centres for the hyperfine-broadened branches. F. Least squares fitting of the data One of the unexpected effects of the ground state internal hyperfine perturbation is that the F2" and F3" levels are appreciably shifted from 2000 4000 6000 8000 J(J+1) Flg 8.A Reduced energy leve!s o£ the A** state of VO plotted against JO+l). The ,ua„tity plo the upper state ter- value less (0.50865 + 0.00365U ) CJ*2 - 6.7 « 10 OH) « Table 8.3 Corrections applied to the observed F2 and F3 line positions to allow for the internal hyperfine perturbation shifts. N F2 F3 N F2 F3 N F2 F3 4 -0.030 -0.003 14 -0.079 +0.055 24 +0.029 -0.026 5 -0.031 +0.008 15 ±0. .080 25 0.027 -0.025 6 -0.031 0.012 16 +0.075 -0.086 26 0.026 -0.024 7 -0.033 0.017 17 0.065 -0.075 27 0.025 -0.022 8 -0.034 0.022 18 0.051 -0.060 28 0.023 -0.021 9 -0.036 0.025 19 0.047 -0.058 29 0.023 -0.020 10 -0.053 0.031 20 0.043 -0.043 30 0.022 -0.019 11 -0.060 0.033 21 0.038 -0.039 31 0.021 -0.018 12 -0.065 0.034 22 0.035 -0.031 32 0.021 -0.018 13 -0.070 0.043 23 0.032 -0.028 33 0.020 -0.017 I ro i The corrections were obtained by subtracting the rotational energy calculated in the absence of hyperfine effects from a weighted average of the rotational-hyperfine energies given by a full calculation of the hyperfine structure. -216-the positions that they would have in the absence of hyperfine structure. Therefore it is necessary to correct all the line positions in the branches involving F2 or F3 lower levels for this effect. It may seem surprising that a hyperfine effect can shift the positions of rotational levels, but the hyperfine matrix element acting between F2 and F3 levels with the same N value is about 0.08 cm"1, while the zero-order separation of the F2 and F3 levels (which depends cn the spin-rotation parameter y) remains less than 1 cm"1 even some distance from the N-value of the internal perturbation. The calculated shifts are given in Table 8.3. After applying these corrections to the F2" and F^" branches we fitted the lines directly to the appropriate differences between eigen-4 4 -values of the n and 1 matrices. No attempt was made to vary the 4 -X 1 , v=0 parameters in the present work since they have been determined with high precision by the sub-Doppler spectra of (17), where the reso lution is a factor of ten higher. Our procedure is therefore equivalent 4 to fitting the term values of the A n, v=0 state to the eigenvalues of Table 8.1. The (0,1) band was then fitted similarly, but with the A4n upper state parameters fixed at the values derived from the (0,0) band; the results give essentially the differences between the parameters for 4 -XI v=0 and v=l. The final parameters are assembled in Table 8.4. The overall stan dard deviations listed correspond to unit weighting of all the data; they are not as low as we had expected, but in view of the blending and the unusual line shapes produced by unresolved hyperfine structure effects in some of the branches we see no reason for concern. 4 4 -Table 8.4 Parmeters derived from rotational analysis of the A II - X S (0,0) and (1,0) bands of VO in cm -1 A4n, v = 0 \l2 T3/2 9555.500 ±0. 011 (3o) T 0 9512.432 ±0. 017 B 7 \l2 9477.830 ±0. 023 10 D T-./2 9449.710 + 0. 021 A B 0.516932 ±0. 000006 If 107D 6.782 ±0. .010 105vs q -0.000151 + 0, .000012 108YD p+2q -0.01349 ±0 .00027 o+p+q 2.107 ±0 .008 Y 0.00383 ±0 .00010 10?D Q 0.023 ±0 .022 10?V2q -2.32 iO .68 105°O+P+q -4.95 ±0 .42 XD 0.000050 + 0 .000004 xV, v 0 1001 . 812 ±0.011 (3<J) 0.5463833 0. .542864 ±0.000013 6.509 6. .54 ±0.03 2.03087 2, .028 ±0.002 0.022516 0 .0226 fixed -1 -1 fixed 5.6 5 .6 fixed f i xed I ro i Standard deviations (unit weight):- A%, v = 0: 0.024 cnf1 ; xV. v - 1: 0.024 cm' Bond lengths: A4n. r = 1.6368 A; XV, rQ - 1.5920 A, r 1 .5894 A , = 0.548143, ae = 0.00351g cm" ) -218-G. Discussion (i.) Spin-orbit coupling Constants and indeterminacies 4 Since n states are comparatively uncommon it is instructive to see what parameters can be determined in this case, and what happens to the problem of the indeterminacy of some of the parameters in the general case. Veseth (21) has pointed out how y and [the spin-rotation interac tion and the centrifugal distortion correction to the spin-orbit coupling) cannot be determined separately in a 2n state, and Brown et al (19) have proved this rigorously. Brown et al have also shown that an indetermina-cy exists between B, AD, AD and y for case (a) n states, essentially because there are only three effective B-values for the three spin-orbit components, but four parameters to be determined from them. The indeter-minacy can be avoided if the levels can be followed to high J values, where case (b) coupling applies, because there is additional information in the effective D-values of the three spin-orbit components. No such indeterminacy occurs for 4n states because there are now four effective B-values to determine the same four parameters; only if higher-order terms such as y$ (the third-order spin-orbit correction to the spin-rota tion interaction (16,17.)) are needed will further indeterminacies arise. ' It is very clear from our data that AD is effectively zero for the A4JI state of VO. If AN is floated the standard deviation increases marginally, and AN is giyen as (4 ±12) .x 10~6 cm"1. Nevertheless if it were not so small it would in principle have been determinable from the data. -219-Another indeterminacy may arise in the sub-state origins for the components of a multiplet 71 state. These origins can be expressed, in terms of the spin-orbit and spin-rotation parameters, as Tfi=T0+AAE+|Al3E2-S(S+l)J (8<4) + yIftZ-SCS+l)J + nAlz3-(;3S2+3S-l).E/5] where n is the third-order spin-orbit interaction (.22,23). From the previous discussion it is seen that for a n(a) state only effective yalues of T0, A and A can be determined, but that all five parameters can be determined for a n state, because y can be obtained from the rotatio nal struct ure. Because y has to be determined separately we have written the sub-state origins in Tables 8.1 and 8.4 in the form of T^ values. However, it would be entirely equivalent to use expressions derived from eq. (8.4) in the least squares work. Converting from the Tfi values given in Table 8.4 we have T = 9498.878 cm"1 ; A = 35.193 cm"1 (8.5) A = 1.867 cm"1 ; r, = 0.331 cm"1 It is interesting to see how comparatively large the second-order parameter A is compared to A. As is well-known (18) the second-order pa rameter A includes the diagonal spin-spin interaction, but since the latter cannot be estimated easily it is not possible to say how much of the observed A is caused by it. The obseryed A for the A^n state is -220-similar to that for the X4E~ state (see Table 8.4), so that its large size is not unexpected. To our knowledge an accurate value of the third-order parameter n has only previously been obtained for the level y = 4 of the 4ii^ state of 02+ (23), though estimates have been made for the A5n and X5n states of CrO (22). ( ii) A-doubling parameters In the approximation where a single V state causes the A-doubling in a 4n state the parameters o, p and q are given by O = -35<4n!AL+|4r>2/AE]iE p = -2<4n!AL+|4z"x4n|BL+|4z">/AE]iz (8.6) q = -2<4n|BL+| V>2/AEnz Two approximate relations between the A-doubling parameters follow at once: and (8.7) p/q = A/B 2 „ (8-8) p = 4oq Equation (8.7) should in fact be obeyed quite well no matter what the states causing the A-doubling are because it assumes only that the matrix elements of AL+ and BL+ are in the ratio of A to B; from Table 8.4 we find (p/q)/(A/B) = 1.26 -221-(8.9) which is not far from unity. Equation (8.8) on the other hand is not obeyed at all, and the experimental ratio p /4oq is -0.13. There are two possible reasons. One is that the off-diagonal spin-spin interaction parameter a (which should be subtracted from the expression for o in eq. (8.6)), is important; the other, which is rather more likely, is that there is a nearby strongly interacting electronic state of different multiplicity. Assuming that the spin-orbit operator is responsible, 4 such a state will have rotation-independent matrix elements with A n, so that it will contribute to the parameter o, but not to p or q. 4 As far as we can tell from our spectra the A U, v=0 level is unper-4 turbed rotationally, and the principal perturbations in B II are by 4 - < \ 2 another E state; however, there is evidence \\7) for a n state perturbing C4E", V=0 (at 17420 cm"1), which possibly comes from the same electron configuration as A^n and is a good candidate for causing the effects described. (iii) Hyperfine structure of the A4n state Section E described how the main branches (AN = AJ) in all four 4n-4E_ sub-bands become 'sharp' at high N values (where the spin coupling approximates case (bgJ) in both states) although they are often hyperfine-broadened at low N. It has been possible to obtain the approximate hyper fine widths of the four components of A4n from detailed measurements of the line shapes in the various branches, together with the known hyper fine structure of the ground state (17); the results are shown in -222-Fig. 8.5. This figure should be considered only as an "artist's impress-4 4 -ion" because the hyperfine structure is never resolved in the A n-X z transition, and the deconvolution of the Doppler and hyperfine profiles has not been attempted. The error bars given for the F2 and F^ components show that it is relatively futile to try to obtain values for any of the hyperfine parameters except b, but on the other hand the value of b can be obtained with reasonable accuracy. To understand why only the hyperfine parameter b is determinable we consider the magnetic hyperfine Hamiltonian [24] in detail: Hman hfc = a I-L + b I-S + c I7S7 + jd(e2l'*I S + e"2i*I+S.) (8.10) nmag.hfs ~~ z z 2 --  + In this equation the first term is the interaction between the electron orbital motion and the nuclear spin, the second term is a combination of the Fermi contact interaction and the dipolar interaction, and the last two terms are dipolar interactions, respectively diagonal and off-diagonal in A in a signed quantum number basis. The term in d gives rise to 4 different hyperfine structures in the two A-doubling components of n^2> and its effects can be seen in Fig. 8.5, where there is a definite diffe rence between the hyperfine widths of the F2e and F2f levels up to about J = 50. This difference can be measured fairly accurately because the line widths in the P2 and Q2 branches are quite obviously different, though the absolute values of the hyperfine widths are uncertain to the extent of the error bars in Fig. 8.5. In case (a^) coupling the diagonal matrix elements (25) of the first three terms of eq. (8.10) are ro r-o • Fig 8.5 Hyperfine widths, AEhfs= Ehfg(F=J+I) - Ehfs(F-J-I), of the four spin components of the A4n state of VO, plotted against J. Points are widths calculated from the ground state hyperfine structure and the observed line widths, without correction for the Doppler width. •224-<JfiAIF|Hhfs|JflAIF> = lF(F+l)-I(I+l)-j(J+l)MaA+(b+c)E]/[2j( J+1)] (8.11) while the d term contributes ± d(S+3s)(J+35)[F(F+1)-I(I+1)-J(J+1)]/[4J(J+1)] to the diagonal elements for = h when S is half-integral. The hyper fine widths (in other words the separations of the hyperfine components with F = J + I and F = J - I) for a 4n state where I = 7/2 are therefore AEh+-c = 7(J+J2Ma+(b+c)E]/[J(J+l)] htS 9 (8.12) ±7(J+^d6^/[J(J+l)] Equation (8.12) implies that the hyperfine widths should decrease as 1/J except that there is a J-independent contribution of ±7d in the two 4 A-components of n^. In case (b^j). on the other hand, the diagonal matrix elements of the magnetic hyperfine Hamiltonian are <NASJIF|H. - I :NASJIF> = [F( F+1)-I(I+1)-J( J+1)] ( aA2X(NJS)  ntS 4J(J+1) \ N(N+1) + bX(JSN) - c[3A2-N(N+l)][3X(SNJ)X(NJS) + 2X( JSN) N(N+1) ] 3N(N+l)(2N-l)(2N+3) ± d[3X(SNJ)X(NJS) + 2X(JSN)N(N+1)] 6,., -, I l* W 2(2N-l)(2N+3) lAl*- | J/ where X(xyz) = x(x+l) + y(y+l) - z(z+l). It is not so easy to see the J-dependence in these formulae, but order-of-magnitude considerations show that the coefficients of a and c decrease as 1/J, while the coeffi cients of b and d are almost independent of J. The hyperfine energy -225-expressionsfor 4n(b) states are roughly F^J = N+3/2) Ehfs = - | (b±hd) X(JIF)/(2N+3) F9(J = N+l/2) - i (b±>»d) X(JIF)(2N+9)/.[(2N+l)(2N+3)] 2 \ (8.14) F3(J - N-l/2) 2 (b±%d) X(JIF)(2N-7)/I(2N-l)(2N+l] F4(J = N-3/2) | (b±*jd) X(JIF)/(2N-1) where the terms in ±kd refer to the A-doubling components; for I = 7/2 the approximate hyperfine widths in the four spin components, in units of 7(b±^d)/2, are 3,1,-1 and -3, respectively. 4 Fig. 8.5 shows that the hyperfine patterns in the A n state of VO, over the range J = 10-80, correspond to a spin coupling intermediate between cases (a„) and (b„,)- As described above, the different hyperfine P pO widths in the F2e and F2f components represent the dipolar d term, but the observed difference is a complicated function of how far the spin-uncoupling has proceeded. The d term should show up again as a small difference between the Q and P branch widths for the high N F-j and F^ lines, but this is not observable at our resolution. The high N pattern corresponds to almost pure case (b^j) coupling, with the parameter b being very nearly the same as in the ground state (hence the 'sharp' main branch lines where the hyperfine components all fall on top of one another). The experimental value of b is b(A4n) = +0.026 ± 0.002 cm"1 (8.15) compared to the ground state value 0.02731 ± 0.00004 cm-1 (17). •226-We have not attempted to obtain values for a, c and d from Fig. 8.5, since the pattern is clearly dominated by the parameter b, with the exact details being governed by the extent of the spin-uncoupling. The fact that b(A4n) is closely similar to b(X4E~) indicates that the same 4sa electron responsible for the Fermi contact interaction in 4 the ground state is also present in the A n state. 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Schawlow "Microwave Spectroscopy" McGraw-Hill, New York (1955). 26. I.C. Bowater, J.M. Brown and A. Carrington, Proc. Roy. Soc. (London) A 333, 265-288 (1973). Appendices (Al) J.H. van Vleck, Rev. Mod. Phys., 23, 213-227 (1951). (A2) J.A.R. Coope, J. Math. Phys. Y\_, 1591-1612 (1970). (A3) A. Carrington, D.H. Levy and T.A. Miller, Advan. Chem. Phys. 1_8, 149-248 (1970). (A4) J. Jerphaguon, D. Chemla and R. Bonneville, Advan. in Phys., 27, 609-650 (1971). (A5) A.R. Edmonds, "Angular Momentum in Quantum Mechanics", Princeton University Press, Princeton (1974). (A6) B.L. Silver, "Irreducible Tensor Methods", Academic Press, London (1976). (A7) M.D. Brink and G.R. Satchler, "Angular Momentum", Clarendon Press, Oxford (1968). (A8) T.M. Dunn, in "Physical Chemistry, An Advanced Treatise, Vol. V", (H. Eyring, Ed.), Chapter 5, Academic Press, New York (1972). -239-(A9) C.J. Cheetham and R.F. Barrow, Adv. High Temp. Chem., 1, 7 -41(1967) (A10) R M Van Zee, CM. Brown, K.J. Zeringne, and W. Weltner, Jr., Accts. Chem. Res. 13, 237 -242(1980). -240-Appendix I Transformation between Cartesian Tensors and Spherical Tensors -241-. This appendix gives the tranformation between cartesian and spheri cal tensor components for operators of up to second rank, and outlines some problems that arise when angular momenta referred to different axis systems commute in different ways. In practice, it is not always necessary to formulate every, inter action in spherical form; for example, the electron spin-rotation inter action can be written conveniently in cartesian form (Al). However for uniformity all terms in the Hamiltonian will be written in spherical tensor form in this thesis. Tensors of rank 0 are scalars, and are the same in cartesian and spherical form. First rank tensors (i.e. vectors) can be expressed either in a cartesian basis |i> with i={x,Y,Z} or in a spherical basis |l,u> withp= {-1,0,1}. The unitary transformation <i|£,u> between the two bases is |1,1> |i,o> p,-l> |X> -(2)'h 0 (2)~h |Y> 0 (A.I.I) |Z> 0 1 0 i.e. for the components of a vector T. rl - •(2)"l5(Ty + i TY) (A.I.2) -242-It is implicit in (A.I.I) that cartesian components satisfy the commutation relations with the "normal" sign of i, tTX'TY] = TXTY " TYTX = +iTZ (A.1.3) The transformation of second-rank tensors from cartesian to spheri cal form can be performed in a two step process. Starting from a second-rank cartesian tensor (whose elements are T.- ^one first changes the V2 cartesian coordinates into spherical coordinates through a product of unitary transformations <i|s,,.y> . This results in a reducible spherical tensor, which is reduced by the well-known properties of the Clebsch-Gorda* coefficients to give the irreducible spherical tensors T^ (A 2,A 4). Tjj = <J m I 1 1 u i 3J2 > < 1' yl I h > < 1' y2 I 1-2 > Tin i"2 ^AJ-4^ where < J m | 1 1 ^ ^> ^ the usual Clebsch-Gordan coefficient, which is related to the Wigner 3-j symbols by < J m I 1 1 y, u2 > = (-l)m (2J+lf2 n 1 0 \ (A.1.5) VI v2 ~ m) Substituting numerical values for the 3-j symbols, and using (A.I.I), a second rank cartesian tensor can be decomposed into a sum of spherical tensors of ranks 0, 1 and 2, T1 XT1 = T° + T1 + T2 (A.1.6) -243-The explicit expressions are i = - <3>"* {TXX + TYY + V Tj - 1 (2)-* (Txy - TYX) Tl _ 1 ' + W {(TZX - TXZ) ± 1 ^TZY " TYZ)} (A.I .7;) T0 = (6)"' {2TZZ " TXX " TYY} il= * <?> «TXZ + TZX> ± 1 (TYZ + TZY'}  T±2=l {(TXX - TYY) 1 1 <TXY + TYZ)} In molecular spectroscopy we must distinguish between space-fixed and molecule-fixed operators. Tensor components defined in terms of axes mounted on the molecule, which are denoted by x, y and z, have the sign of i in the commutation relations reversed (A 1), i.e. [T , T ] = -i T (A.I.8) x y z Components of molecule-fixed tensor operators will transform diffe rently from cartesian to spherical form. For components of a molecule-fixed vector T (A 3): l] = - (2)_35(T - i T ) 1 ' x y Tj = Tz (A.I.9) = (2)"35(TX + i Ty) -244-For second rank tensors, only TJ and are different from (A.1.7 ) 0 XY Y (A.I.10) ' ^ - TXZ) ± 1 (TZY-TYZ)} -245-Appendix II The Derivation of the Nuclear spin -Electron spin Dipolar Interaction Matrix Elements in case (bgJ) coupling -246-This Appendix gives a derivation of the matrix elements of --i^9»B9NiH.T1lI>-T1<S.Ci!> tA-II'1) U ns-es in the case (bDl) basis (i.e. | n N K S J I F> ) given as eq. (2.144) in Chapter 2. Matrix elements can only be evaluated when the operator and the relevant parts of the wavefunctions are in the same reference frame, i.e. either both in the molecule-fixed axis system (q) or in the space fixed axis system (p). Our procedure will be to expand the Hamiltonian in the first instance as a space-fixed operator, so that irreducible tensor methods can be applied in their standard form-, and then those parts that are physically appropriate are referred to the molecule-fixed axis system by means of the rotation matrix eq. (2.110). The operator Hns_es is a scalar product of two commuting tensor operators because^ and £ are in different spin spaces. Therefore, from Edmonds' eq. (7.1.6) (A5) < ri N' K'S' r r r MF I HNS_ES | N N K S J I F MF > = - (lo)3* g Pb gN yN VF 6M/ M (1)I+J+F (F J" i l i J < \ II T](I) || I > < n N' S' || T^C2) || n N K S J > (A.II.2) R3 where the reduced matrix elements are with respect to the space-fixed axis system. <\ \ \ T^I) | | I > can now be evaluated directly, because, -247-in case (b^j), the nuclear spin I is quantized in the space-fixed axis system, as is the operator (I). Therefore by eq. (2.109) < V || T](0 || I > = 6ri [I(I+1)(2I+1)]35 (A.II.3) 1 7 The operator T (!S,C ), in the second reduced matrix element, is a compound tensor operator constructed from simpler commuting operators; by using Edmonds' eq. (7.1.5) (A5), < n N' K- S" || T^S.C2) II n N K'S 0 , - <3)W)<2J'+1>3' R3 <S' ! I T1CS> ||S XT, N- K' 1 | Ci 11 n N K - I N' H 2 FT j S S 1 j' 0 1 (A.II.4) Again for a case (bgj) coupling scheme the electron spin S is quantized in the space-fixed axis system, so that, for the matrix elements < S' || T^S) || S > = 6$-s [S(S+1)(2S+1)]15 (A.II.5) The second-rank spherical harmonic C is defined in the molecul e-fixed axis system, but so far its matrix elements have been reduced with respect to the space-fixed system. The transformation from space-fixed to molecu le-fixed axes is carried out by means of the rotation matrix D, (2)* C2 = I D„ U) C2 (A.II.6) .q Q q -248-where co represents the three Euler angles (agy). (.2)* D (co)is the rotation matrix with no reference to the space-fixed • q' components, and its reduced matrix element is defined by eq. (A.II.17). When this is substituted into the reduced matrix element we obtain < n N K n N K > = E<n|C, n > (2)* < rf K' || D (co) || N K > (A.II.7) where < n | Cn | n > is an experimentally determinable parameter, which 7 will be redefined as T2(C) = < n | Cjjj n > R3 (A.II.8) (2)* Lastly, we must calculate < N' K' | | D (u)| | N K > (2)* which is the reduced matrix element of < N K M | DnC[ (co) | N K M >. We will first evaluate < lT K' M' | D$* (CO) | N K M > and then apply the Wigner-Eckart Theorem (A5) to get its reduced matrix element. Since ^ N K M > = "2N+1 (A.II.9) and its complex conjugate is < N' K' M' I 2N +1 8TT2 J (A.II.10) -249-the relationship between D* and D is (A7) D<£* W - (-DP"q^p)-q M (A.II .n) Therefore, < N' K' M | P(K)* | N K M > = £(2N +1)(2N+1)]^ (A.II .12) Using the relationship in Silver (A6) p. 43, we Ijave < M | Z^V) | N K M > = (-1)P"q (-1)M~K [(2N^1)(2N+1)]^ pq V k N\ /V k N X -P -iy \K' -q -Kj (A.II.13) The pair of 3-j symbols in (A.II.13) will be non-zero only if they satisfy the conditions NT + (-pl + (-M) =0 and K' + (-q) + (-K) = 0 ; (A.II.14) Also with the use of symmetry properties of the 3-j symbol, we finally obtain -250-<N'K'lT \D']* (CO) I N K.M > 1 Pq = (-1)M^K^[(2N^1).(2N+1)]17 N' kN\/N' k N> V ' kN\ / N  k \ \-M' p M]\-K' q KJ Applying the Wigner-Eckart Theorem, we find < N' K' M' \DM* (CO) I N K M > = (-1) rH / N k N\ l-M' p M j < N' K.' M D{K)* (co) || N K Therefore, < IT r || || N K > = (-l)r-K' [(2N'+l)(2N+l)f2 with the special case of k=2 (2)* - -< IT K' | | D | | N K > = (-1)N "K £(2N +1) (2N+1 )jV l\T 2 \K q Combining eqs (A.II.2), (A.II.3), (A.II.4), (A.II.5), (A.II.7) and (A.II.18) we finally get -251-n K' S J" I F | H ns-es N K S J I F > •(30f2g uB gN^N (-DJ+I+F jU'F J I 1 II(I+l)(2I+l)S(S+l)(2S-rl)]35 [(2J'+l)(2J+l)(2N'+l)(2N+l)r N N 2 S S 1 j' J 1 I (-D q r-r /N'2N\ T2(c) \-K'qKJ q (A.II.19) which completes the derivation. -252-Appendix III Derivation of the matrix elements of the operator i T1{I). T1[T3|T1(S). T2(£i,sj)},C2]/r?. i>j in a Hund's case (bgj) basis i -253-We begin by applying eqs. (7.1.5) and (7.1.6) of Edmonds (A5) to reduce the general form of the matrix elements of eq. (3.25) in Chapter 3, E T^D-T^T^T^S), ASLSJ)}, C2]/r3 . i>j J set up in a Hund's case (bc1) basis: < n NAS J I F I E T1(I)-T1[T3{T1(S),T2(s1-,si)}, C2J/r?. | n'N' A' S j'l F> i>J J = M)I+J +F (F J I )< I || T](X) || I>x (3)h [(2J+l)(2J%l)f2 (1 I J'l N ff 2 j S S 3 ( E < S || T3 {T]fS)-T2(s.,s .)} || S> < n N A | | C2 | | nV h > (A.III.l) The scaled spherical harmonic C is defined in molecule-fixed axes, so we need to project it into molecule-fixed axes according to C2 = E D(2)* (co) C2 (A.III.2) q ,q q where cu stands for the Euler angles of the rotation. The reduced matrix 2 3 element of C /r^. then becomes N A I! C2/r3, || n' N' A > - I < N A || D\Z) (U.) II IT A > E < n N A | | t /r. j | | n n ^ -> - u - » * M 1>j x z < n 1 Cl/r3 | n' > i>j = E (-1)N-AI(2N+1)(2N%1)]3'Y N 2 N'\ T2(C) (A.Ill .3) \-A q A / -254-In our case A = A= 0, so that q = 0. There will be just one experimental parameter TQ(.C), which will be proportional to b<-. Next we break up the reduced matrix element of the electron spin tensor product in eq. (A.III.l) using Edmonds' eq. (7.1.1): I < S || TWsJ.T2^,*.)} II S > = (7)* (-D2S+3 jl 2 3} [S(S+1)(2S+1)]^ x E < S | | T2(s.,s.) || S > (A.Ill .4) i>j ~' ~J where we have eliminated the: sum over states with total spin S" because < S || T](S) || S' > = [S(S+1)(2S+1)]35 6SS' (A.Ill .5) The tensor product in eq. (A.III.4) appears in the matrix elements of the dipolar electron spin-spin interaction and the A-doubling parameter o ;. with Edmonds' eq. (7.1.1) again we find < S || T2(s,,s.) || S > = (5)15 E (-1)2S+2 (1 1 2|< S || T1^) || S' > * (S S S) < S' I I T\S,) | | S > (A.III.6) The triangle rules on the 6-j symbol limit s' to S or S±l, but S = S+l is physically impossible, so that the tensor product becomes -255-< S || T2(Si,s.) || S > = (2) (-1) 2sr jl 1 2 )< S || T^(s.) || S-l Is S S-li < S-l r T l(s.) || S > + jl 1 2) <S HT1^) ||S ~J s s s < s || r(s.) || s > (A.III.7) To evaluate the scheme reduced matrix elements of ^ and s^ we define the coupling S = I s. = s1 + s,2 + • • • £. = s T + 5 a (A.Ill .8) Then, by Edmonds1 eq. (7.1.7) < S- || T1^) || S>- (-DSl+Sa+S+1 [(2S+1)(2S js, S' Saj < ST || T1^) || ST > (AvIII .9) where < s, | | T1^) II si > = l^Ws^)V' 0/2)*, following eq. (A.III.5), since S] - \. Using the definition^ - S - s,, and substi tuting for the Wigner coefficient, we obtain < S || T1^) || S > = •2-[(S+l)(2S+l)/S]1"2 which is a general expression holding for all s,. and < S-l || T1^) || S>= -\ [(2S-l)(2S+l)/sf2 (A.III .10) = _ < s || r^) II s-i > (A.III.11) -256-For the off-diagonal reduced matrix elements of the other spins we need to extend the coupling scheme: S = -s,l + *2 +h> 0Th=^2+h (A.III.12) Then < S-l || T'ls,) II S > - (-1)WS+1 R2S+D(2S-l)]^Sa S-l s,l ~^ (S S_ 1 ) <S. II T1^) ||S,, (A->"-13> a S still represents a coupled basis as far as electron 2 is concerned, ~a so that Now < Sa || T](s2) || % > = (-l)Sa+Sb+sZ+l..(2Sa+l) js2 Sa Sbj |Sa s2 1 j <s2 || T](s2) || s2> (A.III.14) which gives the results < S-l || T](,s2) |1 S > = ^[(2S+1)/{S(2S-1)}]1'2 = - < S || T1^) | | S-l > (A.III. 15) We can now return to eq. (A.III.7), and substituting the new expressions just obtained, we get < S || T2(s.,s.) || S > = [(S+l)(2S+l)(2S+3)/.{24S(2S-l)}]% (A ni-16) -257-We need the sum, E < S j | T2(s. ,s,) | | S >, for eq. (A.III .4), which is equivalent to multiplying eq. (.A.III .16) by S(.2S-1) since each pair of electrons is counted once only; this gives E < S I I T2(s.,S.) II S>= [(2S-l)2S(2S+l)(2S+2)(2S+3y24]is = 1 < S || T2(S,S) || S > (A.III.17) Finally, substituting into eq. (A.III.4) we find E < S || T3 il\s), T^s.,^)) || S > = [(2S-2)(2S-l)2S(2S+l)(2S+2)(2S+3)(2S+4)/640]ls (A.III.18) so that I F| E ^m^a^S)-!2^^) ,C2]/r^|VN-*- S'J- I F> < n N A S J (jjf (-Di+j'+f iF J1 ([i(i+i)(2i+l^' \640J J! i j'\ [(2J+1)(2J^1)(2N+1)(2N'+1)]15 x E(-DN-V N 2 N\| q V-A q A'/ N-A / M 0 M'\ / N M " 2 S S 3| J J' 1 [(2S-2)(2S-l)(2S)(2S+l)(2S+2)(2S+3)(2S+4)]3s T2(C) (A.III.19) Where q, A and A' are zero this becomes equivalent to eq. (3,20) in chapter 3 with T2(C) - ^ (14)35 b, (A.III.20) 0 6 5 -258-The useful results from the derivation are the general forms for the electron spin reduced matrix elements, (A.III.10), (A.111.11), (A.III.15) and (A.Ill .17). -259-Appendix IV Wigner 9-j symbols needed for the US dipol interaction and the third-order isotropic hyperfine interaction 3X(NSJ)X(NJS) + 2X(SJN) N(N+1) — " % [30(2N-l)2N(2N+l)(2N+2)(2N+3) .S(S+1) ( 2S+1) .J(J+1) (2J+1) ] where X(abc) - a(a+l) + b(b+l) - c(c+l) [N(N+1) + 3S(S+1) - 3J )[ (N+J-S) (N+S-J+1)(J+S-N) (J+S+N+l)] - •—• — !7 (15(2N-l)2N(2N+l)(2N+2)(2N+3).S(S+l)(2S+1).(2J-1)2J(2J+1) ] [ (J+S+N+l)(N+S-J)(N+J-S)(J+S-N+l)(N+S-J-2)(N+S-J-l)(J+S-N+2)(J+S-N+3)] |10(2N-3) (2N-2) (2N-t)2N(2N+l) .S(S+1) (2S+1) . (2J+1) (2J+2)GJ+3) l** [(J+S+N+l)(N+S-J)(N+J-S)(J+S-N+l)(N+S+J)(N+J-S-l)(N+S-J-l)(J+S-N+2)] [20(2N-3)(2N-2)(2N-1)2N(2N+1).S(S+1)(2S+1).J(J+1)(2J+1)^ [(j+S+N+1) (N+S-J) (N+J-S) (J+S-N+l) (N+S+J) (N+J-S-l) (N+S+J-l) ( N+J-S-2) ] (10(2N-3)(2N-2)(2N-1)2N(2N+1).S(S+1)(2S+1).(2J-1)2J(2J+1)]5 12(2X(JSN)X(SNJ)[5N(N+l)-2] + 2X(NJS)S(S*1)[2X(SNJ)-l] - SX(NJS) X(SNJ) X(JSNj -4X(SNJ)J(J*1) - 4X(JSN)N(N+1)S(S*1) } [lOS(2N-l)2N(2N+l) (2N+2) (2N*3) .2J(2J* 1) (2J+ 2) . (2S-2) (2S-1) 2S(2S+1) (2S+2) (2S+3) (2S+4)]11 where X(abc) = a(a+l) + b(b*l) - c(c*l) 2 [6(N+S+J+1) (J+S-N) (N+J-S) (N+S-J+l) )** x [35(2N-l)2N(2N+l)(2N+2)(2N+3).(2J-1)2J(2J+1).(2S-2)(2S-1)2S(2S+1)(2S+2)(2S+3)(2S+4)J-x {[5N(N+1) + S(S+1) - 5(J-1)(J+1) - 13/2](N(N+l) + S(S+1) - (J-1)(J+1) - 3/2] - 4N(N+1)S(S+1) + 3(J-l/2)(J+l/2)} 2] (N+S+J) (N+S+J+l) (N+S+J-l) (N+J-S-2) (N+J-S-l) (N+J-S) (N+S-J) (J+S-N+l) J*5 (35(2N-3)(2N-2)(2N-l)2N(2N+l).(2J-1)2J(2J+1). (2S-2) (2S-1) 2S( 2S+1) (2S+2) (2S+3) (2S+4)l*5 x (5J(J+1) + 5N(N+1) - S(S+1) - 10N(J+1) + 2] M(N+S+J)(N+S+J+l)(N+S-J)(N+S-J-l)(N+J-S-l)(N+J-S)(J+S-N+t)(J+S-N+2) ——— —r « (70(2N-3)(2N-2)(2N-1)2N(2N+1) . 2J(2J+1) (2J+2) . (2S-2) ( 2S-1) 2S(2S+1) (2S+2) (2S+3) ( 2S+4) P x [5J(J+D - 5N(N+1) + S(S+l) + 2(5N-1)] 2[(J+S-N+3)(J+S-N+2)(J+S-N+l)(N+S-J)(N+J-S)(N+J+S+l)(N+S-J-2)(N+S-J-l)] 7^3^27^^ 1 x [5J(J+D + 5N(N+1) - S(S+1) + 10NJ + 2] -262-Appendix V Molecular Orbital Description of the First-row Transition Metal Monoxides -263-This appendix gives a molecular orbital description of the first-row transition metal oxides. In this approach the molecules are formally 2+ 2 represented as M 0 and the splitting of the degenerate d orbitals of 2+ 2-the N ion by the ligand field of the 0 ~ ion is considered (A8). The relative stabilities of high-spin or low spin states will depend on the size of this ligand field splitting (A9.). A typical molecular-orbital energy level diagram for a first-row transition metal monoxide is presented in Fig. A.V.I. The relative positions of the 3ds and 4so orbitals in these oxides vary as the atomic number increases, as a result of the screening effect created by the elec trons. Therefore Fig. A.V.I represents only one way of writing this mo lecular-orbital energy level sequence, as found in ScO and TiO, for example. The ground state electronic configurations and term symbols that have been determined for the transition metal oxides (A8, A10) are 1 2 + ScO (4so)' E TiO VO CrO MnO FeO CuO (45a)1 (3d&)] 3Ar (43a)1 (3d6)2 V (4sa)] (3d6)2 (3d*)1 5n (4sa)] (3d6)2 (3d^)2 (4sa)] (3d&)3 (3d7r)2 5Ai (4sa)2 (3dS)4 (3d7i)3 2H. r -264-M orbitals MO orbitals 0 orbitals 4p 3d 4s / ~7-/ / / ' ^Y i / // \ \ \\ w \ \ \ \ \ \ \ \ \ = \ \ \ x \ \ \ \ \_ \ \ \ 4po / 4pir 3do 3 d TT 3d6 4sc 2pn 2po 2so \ \ \ 2p 2s Fig A. V . 1 Relative orbital energies in a general transition-metal monoxide molecule, MO. -265-Appendix VI Tables of assigned li -266-This Appendix contains the rotational line assignments on which the results of this thesis are based. The abbreviations used have meanings as follows: * indicates a blended line } indicates multiple line assignments for a given rotational transition ( ) indicates the line assignment is uncertain - indicates a line is expected but has not been observed indicates the line is perturbed. TABLE I ROTATIONAL LINES ASSIGNED OF THE C42~ X42~ (0.0) BAND OF BRANCH J " F ' -F N = - 1 R1(- 1 ) 0. 5 1 0 RQ21(- 1 )FE 0. 5 1 0 - 1 N = 0 RK 0) 1 . 5 0 PK 0) 1 . 5 -1 0 4 R2( 0) 0. 5 1 1 P012( 0)EF 0. 5 1 0 -1 N = 1 RK 1 ) 2 . 5 1 PK 1 ) 2 . 5 -1 0 R2( 1) 1 . 5 1 P2( 1 ) 1 .5 -1 0 N = 2 RK 2) 3 . 5 1 0 PK 2 ) 3 . 5 -1 0 R2( 2) 2 . 5 1 P2( 2) 2 . 5 -1 R3( 2) 1 .5 1 0 R043( 2 )FE 1 . 5 0 -1 N = 3 RK 3) 4 . 5 1 PK 3) 4 . 5 -1 0 R2( 3) 3 .5 1 P2( 3) 3 . 5 -1 0 R3( 3) 2 . 5 1 0 P3( 3) 2 . 5 -1 R4( 3) 1 . 5 1 PQ34( 3 )EF 1 .5 0 F"=J"-7/2 F=J-5/2 F=d-3/2 F=J-1/2 F = J+ 1/2 F=J+3/2 17422 . 5758 17421.0956 17424.6441* 17424.6288* 17424.5994* 17424.5545* 17424.5020 4 19.7569 4 19.7263 4 19.7113 4 19.6699 4 19.6500 17425.1708 17425.1478' 17425. 1 122 4 18.2762 421 .6615 -17425.0679 4 18.2437 4 18.228R 4 16.2974 421 .6483* 42 1 .652 1 17425.0127 41B.1876 F=J+5/2 17422.6435* 424.7946 424.7657* 17423.2760 17422.4357* 422.5170 422.3751* 420.4331 420.4952* 17423.8571 421 .0096 422.6854 420.3814* 17424 .4336 424.4514 419.5987 419.5757* 420.8046 4 19. 1927 17424.9461 4 18.119 1 416 . 2622' 4 16.2441 17423.7646* 420.9413 420.9015* 422.6779 420. 39 14 420.4246* 17424.3508 419.51 1 1 4 19.48 15* 422.7955 417.9737 420.7608 420.7637* 4 19.2260 17424.8682 4 18 0386 423.0844* 4 16.2255 F=J+7/2 17422.3960 424 .6161 424.5847* 17422.2735 422.9294 420.5507 420.6116 17423.6589 420.8177* 422.6718 420.4420 17424.2540 4 19.4076 422.7605 4 17.9324 420.7084 419.2 124 17424.7795 417.9451 423.0314 4 16. 1796 421.6055* 421.5935* 4 14. 6489* 423.0193 423.0218* 4 14. 5950* 423.0218* 4 17.4442 I ro cn —i i TABLE I (CONTINUED) BRANCH J" F ' - F " F"=d" -7/2 F = J -5/2 N= 4 R1( 4) 5 . 5 1 17425 . 6 131 * 17425. 5847 PK 4) 5 . 5 - 1 4 16 . 7986* 4 16 . 7734* O - 4 16. 7628* R2( 4) 4 . 5 1 - -R032( 4) 4 . 5 1 - -P2( 4) 4 . 5 -1 - 414 . 6593* R3( 4) 3 . 5 1 - -P3( 4) 3 . 5 - 1 P023( 4) 3 . 5 - 1 R4( 4) 2 . 5 1 P4( 4) 2 5 - 1 N= 5 RK 5) 6 5 1 17425 9648 17425 9313* PK 5) 6 5 -1 4 15 1702 4 15 1395 R2( 5) 5 5 1 R032( 5) 5 5 1 P2( 5) 5 5 - 1 4 12 9848* 4 12 9776 P032( 5) 5 5 - 1 R3( 5) 4 5 1 RS23( 5) 4 5 1 P3( 5) 4 5 - 1 4 1 1 9873 P023( 5) 4 5 - 1 R4( 5) 3 5 1 425 0202 425 0323 P4( 5) 3 5 - 1 N= 6 R1( 6) 7 . 5 1 17426 . 2207 17426 1828* PK 6) 7 .5 - 1 4 13 . 4495 4 13 4 156 R2( 6) 6 .5 1 424 .0326 P2( 6) 6 .5 -1 41 1 .2497* 41 1 . 2454* P032( 6) 6 .5 - 1 41 1 .1994* R3( 6) 5 .5 1 -RS23( 6) 5 .5 1 P3( 6) 5 .5 -1 4 10 .4471 4 10 .4398 P023( G) 5 .5 - 1 4 10 .4812 R4( 6) 4 . 5 1 425 .6393* 425 .6558* P4( G) 4 .5 -1 - -F=J-3/2 17425.5474 4 16.7366 4 16.7245 414.6489* 4 13.404 2 413.3995 F=J-1/2 17425.5001 4 16.6905 4 16.6765* 423.4934* 414.6349* 413 .3946* 4 13. 3920 17425 4 15 8922 1004 412 .9689 41 1 .9820 425.0539* 17426 4 13 424 4 1 1 411 423 4 10 425 4 12 1427 . 3743* .0013 . 2402* . 1754 . 2582 . 4329 .6823* . 7606 F=d+1/2 17425.4452 4 16.6346 416.6176 423.4796 414.6167* 413.3898 4 13.3827 17425 4 15 8442 0531 412 .9577 412.9528 41 1 .9769* 425.0794 17426 4 13 423 4 1 1 4 1 1 0954 3260 9826 2268 1573 425 4 12 4 12 7173 7931 8040 F-J+3/2 17425.3802 4 16.5686 4 16.5487* 423.3872 423.4573 4 14.5889* 4 13.3807* 4 13. 5095 17425 4 14 7887 9970 4 12.9394 4 12 .9325* 41 1 .9707* 425. 4 14. 4 14. 17426. 413 423 411 4 1 1 423 1 178 1441 1574 0393 2702 962 1 2135* 1377 3383 410.4254* 410.4204* 425 4 12 F=d+5/2 17425.3057 4 1G . 4920 423.3652 423.4261 4 14.5551 * 413.3743* 4 13.4989 424.3073 415.2546* F=J+7/2 17425. 4 14 423 7255 9323 6966 4 12 .9 137 4 12.9002* 422 4 1 1 425 4 14 17425 4 13 423 8330* 9677* . 1623 . 1891 9771 2072 9400* .7599 . 8337* 17425 4 16 17425 4 14 423 423 4 12 6529* 8594 6723 . 7488* .8776 4 12.8102 423.3440 410.4204* 425.8088* 4 12.8845* 4 12.9002* 422 4 1 1 412 425 4 14 17425 4 13 423 41 1 41 1 4 10 425 412 412 8330* 9689* 0513 2 146 . 2424 9069 1364* 9126* 0930 . 1536 . 22 15 . 4053 423.3504* 414 422 4 13 .5059 . 3534 . 37 13 4 13.4850 413.4614 .424.3527 415.3016 17425 4 14 423 5738 778 1 6463 4 12.8190 422 422 41 1 412 425 4 14 9153 8383 9769* .0453 . 2754 . 3045 17425.8300 413.0574 423.8819* 411.0756 4240 8668 9427* 9608 410.4958* 4 10.4355 425.9313* 4 13.0090 I ro CD co I TABLE I (CONTINUED) BRANCH J" -F» F"=J"-7/2 F=J-5/2 F=J-3/2 F=J-1/2 F=J+1/2 F=J+3/2 F=J*5/2 F-J+7/2 N- 7 R1( 7) 8 . 5 P1( 7) 8 . 5 R2( 7) 7 . 5 P2( 7) 7 . 5 P032( 7) 7 5 R3( 7) 6 5 P3( 7) 6 5 P023( 7) 6 5 R4( 7) 5 5 P4( 7) 5 5 N = 8 R1( 8) 9 5 P1( 8) 9 5 R2( 8) 8 5 R032( 8) 8 . 5 P2( 8) 8 .5 P032( 8) 8 . 5 R3( 8) 7 . 5 P3( 8) 7 . 5 P023( 8) 7 . 5 R4( 8) 6 . 5 P4( 8) 6 .5 17426.3805 41 1 .6312 424 . 1946 409.4051• 409.44 18* 423.6267 408.7877 426.0828 17426.3446* 4 1 1 .5950 424.1800* 4093903* 409.4418* 423 .6324 408.8013 408.7660 426.1077 17426.4376 409.7154 424.2666 407.5051* 407.5605* 407.0122 426 . 3808* 409.5310* 17426.3985 409.6764 424.2540 407.4902 407.5605* 407.0188 426 . 4079* 409.5560 17426.2991 411.5529* 424. 1639 409.3735 409.4384 423.6377 408.8085 408.7554* 426.1358 41 1 .2497 17426.3540 409.6334 424.2389 407.4744 407.5549 407.026 1* 426.4428 409.5886 17426 41 1 424 409 409 423 408 408 426 4 1 1 2491 5032 1471* 3552 4292 .6422 .8 136 . 7488* . 1740 .2869 17426 4 1 1 424 409 409 423 408 408 426 4 1 1 1936 4475* 1253 3347 4 152 6480 8 189 . 7449* . 2203 . 3325 17426.1318 4 11.3850* 424.1018 409.3117 423.6552* 408.8249 408.7449* 426.2722 4 11.3850* 17426.3038 409.5827 424 . 2 192* 407.4560 407.5476 407.0301* 426.4842 409.6290* 17426.2489 409.5273 424.2010* 424.3428 407.4357 407.5349 423.8465 407.0375 426 .5311 409.6764* 17426. 409 . 424 . 424 . 407 . 407 . 423 . 407 . 406 . 426 409 1872 4651 1756 3251 4121 .5151 . 8556* .0453 .9432* .5842 . 7297 17426 4 1 1 424 409 409 423 408 408 426 4 1 1 17426. 409 . 424 . 424 . 407 . 407 . 423 . 407 . 406 . 426 409 FROM HERE ON ALL THE LINES HAVE DELTA F EQUAL TO DELTA N BRANCHES LABELLED RQ32. P032. RS23 AND P023 ARE INDUCED BY INTERNAL HYPERFINE PERTURBATIONS 0634 3 160 0743 2858 3636 .6645 .8312 . 7488* . 332 1 .4451* . 1 198 3989* 147 1 3005 . 3842 . 4869 . 8660* .0545 .9496* . 6445 . 7896 17425.9890 4 11.2402 424.0396 409.2571 423.6774 408.8386* 408.7618 426.3986* 41 1 .5120 17426.0471 409.3240 424.1102 407.3495 423.88 19 407.0677 406.9647 426.7098* 409.8563 I cn to i TABLE I (CONTINUED) BRANCH F"=J"-7/2 F=J-5/2 F=J-3/2 F=J- 1/2 F = J* 1/2 F = cH 3/2 F=J+5/2 F=J+7/2 N= 9 RK PK R2( P2( P032( R3( P3( P023( R4( P4( 9) 9) 9) 9) 9 ) 9 ) 9) 9) 9) 9 ) 10 10 9 9 9 8 8 8 7 7 17426.3957 17426.3542 407.6953 407.6570* 424.2448 424.2333 405.5092 405.4956 405.5876* 405.5876* 405.1270 426.5492 407.7252 405.1317 426.5802 407.7555 17426.3091 407.6107 424 . 2 192* 405.4809 405.5831 423.9205 405.1366 426 .6174 407.7913 N=10 R1(10) 1 1 . 5 17426 . 2491 17426 2083 17426. 1627 P1(10) 1 1 5 405 5765* 405 5349 405 4894 R2(10) 10 5 424 1253* 424 1 150 424 1018* R032(10) 10 5 3894 P2(10) 10 5 403 4 155 403 4035 403 P032(10) 10 5 403 5190 403 5 167 R3(10) 9 5 423 8959 P3(10) 9 5 403 1321* 403 1343 403 1386 P023(10) 9 5 R4(10) 8 5 426 5960 426 6301 426 6689 P4(10) 8 5 405 8008 405 834 1 405 87 19 N= 1 1 R1 ( 1 1 ) 12 5 17426 0025 17425 9602 17425 9 136 PK 1 1 ) 12 5 403 3551* 403 3 125 403 2651 * R2( 1 1 ) 1 1 5 423 9091 4 23 9002* 423 8878 R032( 1 1 ) 1 1 5 2017 P2( 1 1 ) 1 1 5 401 2233 401 2 132* 401 P032( 1 1 ) 1 1 5 401 3506 R3( 1 1 ) 10 5 423 7610* 423 7610* 423 7610* P3( 1 1 ) 10 5 401 0289* 401 .0289* 401 0313* P023( 1 1 ) 10 . 5 603 1 R4( 1 1 ) 9 . 5 426 .5266 426 . 5629 426 P4( 1 1 ) 9 .5 403 . 7575 403 . 7923 403 .8321 17426.2584 407.5G06 424.2010 405.4640 405.5763* 423.9249 405.1420 426.6603 407.8334 17426.1113 405.4 386 424.0869' 403.3747 403.5092 423.9002* 403.1432 426.7130 405.9161 17425.8609* 403.2 147 423.8725 401.1872 401.3455 423 . 7646* 401.0343 400.8881* 426.6488 403.8775 17426 407 424 405 405 423 405 405 426 407 . 2031 .5051 .18 13 .4436 .5639 .9314* .1491 .0368* . 7098* .8824 17426. 407 . 424 . 405 . 405 . 423 405 405 426 407 17426.0554 405.3828 424.0674* 403 403 423 403 403 426 405 . 3548* . 4994 . 9067 . 1506 .0118 .7631* . 9655 17425.8065 403.1583* 423.8556* 401.1695 401.3356 423.7705 401.0398 400.8825* 426.6989* 403.9279 1427 . 4432 . 1579 .4203 .5457 . 9400* . 1573 .0368* . 763 1 * . 9368 17425.9963 405.3216 424.0437 403.3317 403.4813 423.9149* 403.1583* 403.0165* 426.8168 406.0200 17426 407 424 405 405 423 405 405 426 407 17425 405 424 424 403 403 423 403 403 426 406 17425. 403 . 423. 424 401 401 423 401 400 426 403 747 1 .0993 . 8322* .0509 . 1467 . 3 192 . 7782 .0483* . 8825* . 7540 . 9827 .0762 . 3768 . 1272* . 3907 .5180* .9527 . 1687 .0424 .823 1 .9970 .9313 . 2566 .0135 .2 132 .3015 .4537 .9314* . 1707 .02 18 .8776* .0807 17425.6823 403.0344 423.8015 401 401 423 401 400 426 404 1 153 29 19 7928 0622* .8879* .8140 .0429 17426.0052* 407.3041 424.0869* 405.3516 423.97 14 405.1855* 405.0578* 426.8886 408.0636 17425.8609 405.1855 423.9678 403.2570 423.9493 403.1899 403.0373 426.9419 406.1457 17425.6131 402.9647 423.7488 401.0628* 423.8146 401.0825* 400.9062 426.8776* 404.1076 I o i TABLE I (CONTINUED) BRANCH N=12 R1(12) PK 12) R2( 12) R032(12) P2( 12) P032( 12) R3( 12 ) RS23(12) P3( 12 ) P023( 12 ) R4( 12 ) P4( 12 ) N=13 R1(13) PK 13) R2( 13 ) R032(13) P2( 13 ) P032(13) R3(13) RS23( 13) P3(13) P023(13) R4(13) P4( 13) N=14 R1(14) P1( 14 ) R2(14) R032( 14 ) P2( 14 ) P032(14) R3( 14 ) RS23( 14 ) P3( 14) PQ23( 14 ) R4( 14 ) P4( 14 ) 13 . 13 . 12 . 12 . 12 . 12 . 1 1 . 1 1 . 1 1 . 1 1 . 10 10 14 14 13 13 13 13 12 12 12 12 1 1 . 1 1 . 15 . 15 . 14 14 14 14 13 13 13 13 12 12 5 5 5 5 5 5 5 5 5 5 5 5 5 5 . 5 . 5 . 5 • 5. . 5 .5 .5 . 5 . 5 .5 5 5 5 5 5 5 5 5 5 5 5 5 F" = J" -7/2 F=J -5/2 F = J -3/2 F = J -1/2 17425. 6529 17425. 6089 17425. 56 18 17425. 5 103 401 . 0290 400. 9863 400. 9389 400. 8880* 423 . 5900* 423 . 5834 423 . 5742 423 . 56 18 398 . 9331* 398 . 9258 398 . 9163 398 . 9038 399 . 0869* 399 . 0839* 423 . 5195* 423 . 5195* 423 . 5195* 423 . 5195* 398 . 8174* 398 . 8174* 398 . 8 174* 398 . 8 174 * 398 . 6655* 398 . 6556 398 . 648 1 * 42G 3447* 426 . 3808* 426 . 4240 426 . 4699 401 .6059 401 6428 401 . 6840 40 1 . 729B 17425 . 1974 17425 . 1535 17425 . 1058 17425. 0539 398 . 6036 398 .5600 398 .5124 398 . 4607 423 .1612* 423 . 1525 423 . 3980 396 . 54 17 396 .5375* 396 .5316* 396 . 5205 396 . 7235* 396 . 7235* 422 .9492* 396 .4995* 396 . 4922* 396 .4922* 396 . 3220 396 .3088* 396 . 2992* 426 . 0492 426 .0900 426 .13 17 426 . 1794 399 . 34 10 399 . 3792 399 . 42 17 399 . 4687 F=J+1/2 F=J+3/2 F=J+5/2 F=d+7/2 17425 400 423 423 398 399 423 423 398 398 426 40 1 17424.6399 396.072 1 422.6502* 422 8875 394.0504* 394.2455 422.7179 394.0738 425.6465 396.9664 17424 396 422 422 394 394 422 422 394 393 425 397 . 5947 .0277 .6502* . 9006* .0504* . 2557 . 7089 . 4708 .0651 . 8695 . 686 1 .0059 17424 395 422 422 394 394 422 422 394 393 425 397 .5470 . 9798 . 6473 . 9055* .0465 . 26 13* . 7034 . 4544 .0590* . 8527 . 7 305 .0496 17424. 395 . 422 . 422 . 394 . 394 422 422 394 393 425 397 4950 . 9282 . 6401 .908 1 .0394 . 2637* . 6999* . 4420* .0558* .8404* .7780 .0964 . 4546 .8324 . 5459 . 7784* .8876 .0767 .5251* . 3000* .8225* .6426* . 5209 . 7809 17425. 400. 423 . 423 . 398 . 399 . 423 423 398 398 426 401 17424.9983 398.4051 423. 1373* 423 .3915* 396.5061 396.7178* 396.4941 * 396.2915* 426.2302 399.5199 17424 395 422 422 394 394 422 422 394 393 425 397 .4397 .87 17 .6294 . 9055* .0282 . 2613* . 6999* .4323 .0558* . 8308* .8300 . 1482 3953 . 7730 .5251 . 7646* .8652* .06 1 1 . 5327 . 3000* . 8304* .6426* .5761 .8358 17424. 398 . 423 . 423 . 396 . 396 . 423 . 422 . 396 . 396 426 399 9392 . 3457 , 1 169* . 3793 . 4879* . 7039 .1717* .9187 . 4995 . 2890* . 2851 .5750 17425. 400. 423 . 423 . 398 . 399 423 423 398 398 426 401 17424 398 423 423 396 396 423 422 396 396 426 399 17424. 395 . 422 . 422 . 394 . 394 . 422 422 394 393 425 397 3809 8127 .6130 .896 1 .0112 . 2509 . 7034 .4293* .0590* .8268* . 884 9 . 2029 33 12 . 7085 .4934 . 7386 .8345 .034 1 . 5459* . 3058 . 8426* .6481 .6354 .8955 .8757* . 2823 .0875 3559* 4576* 6798 1849 9236 5108 2937* 3447* .6345 17424. 395 . 422 . 422 . 393 . 394 . 422 . 422 394 393 425 397 3177 7498 . 4420 . 7309 . 8404* .0857 . 7 130 .4301 .0683 . 8279" . 9436 . 2622 17425.2630 400.6400 423.4297 398 .77 14 423.5684 423.3229 398.8652* 398.6655* 426.6989* 401.9594 17424.8090 398 . 2 145 423.0077 396.377 1* 423.2075 422.9390 396.5313* 396.3088* 426.4079* 399.6982 17424.2517 395.6832 422.4831 393.8815 422 422 394 393 426 397 .8765 . 5875 .231 1 .9859 .0052 .3252 TABLE I (CONTINUED) -BRANCH J" N=15 R1(15) 16.5 P1(15) 16.5 R2(15) 15.5 R032(15) 15.5 P2(15) 15.5 P032(15) 15.5 R3(15) 14.5 RS23(15) 14.5 P3(15) 14.5 PQ23(15) 14.5 R4(15) 13.5 P4(15) 13.5 N=16 R1(16) 17.5 P1(16) 17.5 R2(16) 16.5 RQ32(16) 16.5 P2(16) 16.5 P032(16) 16.5 R3(16) 15.5 RS23(16) 15 5 P3(16) 15.5 P023(16) 15.5 R4(16) 14.5 P4(16) 14.5 N= 17 F"=J" -7/2 F = J -5/2 F=J -3/2 F = J -1/2 17423. 9774 17423. 9314* 17423. 88 19* 17423. 83 18* 393 . 4373 393 . 3923 393 . 3440 393 . 2926 422 . 0307* 422 . 0346* 422 . 0346* 422 . 0307* 422 . 2928 422 . 3051 • 422 . 3 145* 422 . 3201 * 391 . 4571* 391 . 4613* 391 . 4613* 391 . 457 1 * 39 1 . 674 1 39 1 . 6886 391 . 6979* 39 1 . 7042* 422 . 1572 422 . 1409* 422 . 1304* 422 . 124 1 42 1 . 8808 42 1. 8601 42 1. 844 1 391 . 54 14 391 5251* 391 . 5144 39 1 . 5079 39 1 . . 3080 39 1 2872 39 1 27 10 425 . 1339 425 . 1745 425 . 2 190 4 25 . . 2670 394 . 4824 394 . 5229 394 .5672 394 .6 150 F=J+1/2 F=J+3/2 F=d+5/2 F=d+7/2 17423. 393 . 42 1 . 422 . 391 . 391 . 422 . 42 1 391 391 425 394 17423.2103 390.6994 421.3102 421.5926* 388.7643 389.0035 42 1.4901 388.9030 424.5127 391 .89 10 17423. 390. 42 1 . 42 1 . 388 . 389 42 1 42 1 388 388 424 391 RK 17) 18 . 5 17422 . 3389 17422 PK 17) 18 . 5 387 . 8556 387 R2(17) 17 . 5 420. 4920 420 R032(17) 17 . 5 420. 7930 420 P2( 17 ) 17 . 5 385 . 9744 385 P032( 17 ) 17 . . 5 386 . . 2348 386 R3( 17) 16 . 5 420 .7171 420 RS23( 17 ) 16 . 5 420 P3( 17) 16 . 5 386 . 1586 386 P023( 17 ) 16 . 5 385 R4(17) 15 . 5 423 .7831* 423 P4( 17) 15 . 5 389 . 1906 389 1650* .6537 . 3 187* .6101 . 7737* .0228 . 4625 .18 16 .8755 . 6365 .5545* .9326 . 2928* . 8094 .3419 .6537 .8251 .0954 .6730 . 37 12 . 1 145 . 8547 . 8253 . 2326 17423 390 421 421 388 389 421 42 1 388 388 424 391 17422 387 420 420 3B5 386 420 420 386 385 423 389 .1161* .6051 . 3225 .6234* .7771* .0359* . 4472 . 1567 .8601 .6111 . 5994 .9779 . 2434 . 76 10 . 3203 . 64 15 .8034 .0833* .8177* .5050 . 2584 .9877* .8725* . 2788 17423 390 42 1 42 1 388 388 42 1 42 1 388 388 424 392 .0636 .5520 .12 13 .432 1* . 5781 .8451 * .4375* . 1366 .8506* . 59 13 .6482 .0258 17422. 387 . 420. 420. 385 . 386 . 420. 420. 386 . 385 . 423 . 389 . 1916* 7082 3035 6333 7864 0756* 8332 5 112* 2742 9944* 9205* 3274 7782* 2367 8254* . 122 1* .2517* . 5082* . 122 1 * .8317* . 5062* . 2584* .3194 . 6662 17423. 390. 42 1 . 421 . 388 . 388 . 42 1 . 42 1 389 38B 424 392 0077* 4967 . 1 124 .4312* . 5674 .8451* .6324* . 3225 .0451* . 777 1 * . 7000 .0776 17423. 393 . 42 1 . 422 . 39 1 . 391 . 422 . 422 391 39 1 425 394 17422 390 42 1 421 388 388 42 1 42 1 389 388 424 392 17422. 387 . 420. 420. 385 . 386 . 420. 420 386 385 423 389 1360* 6532 2914 6302* . 7730* . 0717 * .8452* .5128* . 2860* . 9944* . 97 14 . 3794 7 178 . 1780 . 8254* . 1304* .2517* .5120* . 3201 * .0231 . 7042* . 4496 . 3746 .7214 . 9492 . 4379 . 1089 . 4334 . 5634 .8451* .6339* . 3 168 .0451 * .7720* . 7548 . 1327 17423 393 42 1 422 391 391 422 422 39 1 39 1 425 394 17422 390 421 421 388 388 421 42 1 389 388 424 392 17422. 387 . 420. 420. 385 . 386 . 420 420 386 385 424 389 0772 .5944 . 2844 . 6302* . 7672* .0717* .8495* .5112* . 2910* . 9944 * .0278 . 4343 .6552 . 1 154 .8317* . 1409* . 2584* . 5248* .3145* .0108 .6979* . 4363 . 4333 . 7802 .8875* . 3760 .1114 .4422 .5668* .8506* .6324* . 307 1 .045 1 * .7630* .8131 . 1914 17423.5900 393.0491 421 .8553 391.2819 422.3008 42 1 . 9899 391.6835 39 1 .4140* 425 . 4955 394.8423 17422.8222 390.3105 421. 1239 388.5786" 42 1 42 1 389 388 424 392 6234* 2929 .0361* . 7478* .8757* . 2528 17422.0158 387.5330 420.2835 420.6347* 385.7672* 386.0756* 420.8495* 386.2910* 424.0869* 389.4926 17421.9509 387 .4677 420.2900 385.7730* 420.8452* 386.2860* 424.1471 389.5539 I PO ro TABLE I (CONTINUED) BRANCH J" N=18 RK 18) P1(18) R2(18 ) R032( 18 ) P2( 18 ) P032( 18) R3(18 ) RS23(18) P3( 18 ) P023(18) R4( 18 ) P4(18 ) N=19 R1(19) PK 19) R2( 19) R032(19) P2(19) P032(19) R3(19) RS23(19) P3(19) P023(19) R4(19) P4(19) N = 20 RK20) 21 P1(20) 21 R2(20) 20 P2(20) 20 P032(20) 20 R3(20) 19 RS23(20) 19 P3(20) 19 P023(20) 19 R4(20) 18 P4(20) 18 F"=J"-7/2 F-J-5/2 F=d-3/2 F=d-1/2 F=d+1/2 F=J+3/2 F=J+5/2 F=J+7/2 19 . 5 1742 1 . 3628 17421 . 3 158 19 . 5 384 . 9 112 384 . 8647 18 . 5 4 19 . 4485 4 19. 4 176 18 . 5 4 19. 77 1 1 4 19 . 7501 18 5 382 . 9627 382 . 9313 18 . 5 383 . 2442 383 . 2230 17 . 5 4 19 . 8368 4 19. 8957* 17 . 5 4 19 . 5738 17 . 5 383 . 3101 383 . 3695 17 . 5 383 . 0879 1G . . 5 422 9454 422 . 9889 16 . 5 386. . 3846 386. 4274 20 . 5 17420 .2835* 17420 . 2359 20 . 5 38 1 .8741 38 1 .8273 19 . 5 4 18 . 4 159 4 18 . 3845 19 . 5 4 18 . 7595 4 18 . 7363 19 . 5 379 . 9597 379 .9278 19 . 5 380 . 26 16 380 .2397 18 . 5 4 18 .851 1 4 18 .8991 18 . 5 4 18 .5566 18 . 5 380 . 3537 380 . 4022 18 . 5 380 . 1004 17 . 5 42 1 . 9999 422 .0439 17 . 5 383 .4733* 383 .5173' 17421 384 4 19 4 19 382 383 419 4 19 383 383 423 386 17419.0980 378 417 376 377 417 7066* 2740* 8502 1723 .7591* 377 . 2922 420.9476 380.4530 17419.0506 378.6594* 4 17.2433 376.8 187 377.1508 4 17.BOO1 377.3337 377.0124 420.9920 380.4975 2668 .8157 . 394 1 . 7362 . 908 1 . 2089 . 92 19 . 5899 . 3952 . 104 1 .0350 .4739 17420. 38 1 . 4 18. 4 18. 379 . 380. 4 18. 418 . 380. 380 422 383 1861 778 1 3603 72 1 1 . 9038 . 2255 .9255 . 5725 .4287 . 1 166 .0907 . 5637< 17421 . 384 . 4 19. 4 19 . 382 . 383 . 4 19 . 4 19 . 383 . 383 423 386 174 19. 378 . 4 17. 376 . 377 . 417 . 417 377 377 42 1 380 0007 6094* .2 190 . 7945 . 1355 . 8265 .4531 . 3598 .0286 . 0382* . 5446 2143 7630 376 1 7263* 8900 . 1998 . 9397 .5987* .4133 . 1123* .0844 .5231 1742 1 . 384 . 4 19 . 4 19 . 382 . 383 . 4 19 . 4 19 . 383 . 383 . 423 . 386 17420.1336 38 1 . 7247 •1 18 . 34 12 379 380 4 18 4 18 380 380 422 383 8846 2 147 9442 5823 4469 1260* 1409* 6 133* 17418.9477 378 5567* 4 17.1993 376.7747 377 . 1246 417 .8458 417. 4626 377.3792 377.0384 421.0891 380.5942 1590 708 1 3623 7209 8760 1937* 9524 6014* 4246 1 159* 1373 5750 1742 1 384 4 19 4 19 382 383 4 19 4 19 383 383 423 386 17420.0784 381.6702 4 18.3250 379.8697 380.2086 4 18. 9579 418.5879 380.4613 380.1312* 422.1924* 383.6656* 174 18.8930 378.5020* 4 17. 1832 376.7585 377 . 1 175* 4 17.8603 1009 6495 3525* . 7 196 . 8664* . 1937* . 9611* .6014* . 4346* . 1 159* . 1920 .6299 17421.0382* 384.5880* 4 19.3508* 17420.0204 38 1 .612 1 418.3151 379.8588 380.204 1* 4 18.9674 380.4705 380 422 1312* 2477 383.7203* 174 18.8350 378 . 4442* 417 . 1710 376.7467 377 . 1 175* 4 17.8708 382 383 4 19 4 19 383 383 423 386 8664* 1937* 961 1* 5987* 4346* 1123* 2502* . 6885 17420.9757 384 . 5235 419.3508* 382.8664" 4 19 4 19 383 383 423 386 961 1* 5899* 4346* 104 1 * .3116* . 7494 17419.961 1* 381.5509 4 18.3083* 379.8520* 380.2041• 4 18.9732* 380.4759* 380.13 12* 422.3051 * 383.7785* 377.3940 377.4047 377.0453* 377.0453* 421.1412 421.1962 380.6465 380.7018 17419.8957* 381.4869 4 18.3083* 379.8520* 418.9732* 380.4759* 3B0.1312* 422 . 3663 383.8390* 17418.7745 17418.7114* 378.3833* 378.3195* 417.1627 417.1597 376.7382 376.7349 377. 1 175* 417.8776 417.8804 377 .4 1 19 377.0453* 42 1 . 2538 380.7596 377 .4139 377.0453* 421 .3140 380.8199 ro TABLE I (CONTINUED) BRANCH J- F-J-7/2 F-J-5/2 F=U-3/2 F-J-,/2 F=U+./2 F=U+3/2 F=U+5/2 F-J-7/2 N=21 RK21) P 1 ( 2 1 ) R2(2 1 ) P2(21 ) P032(2 1 ) R3(2 1 ) P3(2 1 ) P023(21) R4(21 ) P4(2 1 ) N = 22 N = 23 N = 24 RK22) PK22) R2(22 ) P2(22 ) R3(22) P3(22 ) R4(22) P4(22 ) RK23) PK23) R2(23 ) P2(23) R3(23) P3(23) R4(23) P4(23) RK24) P1(24) R2(24) P2(24 ) R3(24 ) P3(24 ) R4(24) P4(24 ) 22 . 5 17417 . 8084 17417. 7604 17417 . 7108 17417. 6585 22 . 5 375 . 4483 375 . 4005 375 . 3508 375 . 2980 2 1 . 5 4 16 .0261* 4 15. .9954 4 15 9707 4 15. 9508 21 . 5 373 .6304 373 .6007 373 5765 373 556 1 2 1 . . 5 373 . 9745* 373 .9547* 373 9386* 20 . 5 4 16 . 5598 4 16 . 5967 4 16 .6221 4 16 . 64 14 20 . 5 374 . 1254 374 . 1620 374 . 1864 374 . 2066 20 . 5 373 . 8357 373 .8445* 19 . 5 4 19 . 7864 4 19 .8315 4 19 .8790 4 19 . 9288 19 . 5 377 . 3269 377 . 37 16 377 .4188 377 . 4689 24 . 24 23 23 22 22 21 21 25 25 24 24 23 23 22 22 174 17. 6027 375.2429 4 15. 9332 373.5393 23 . 5 174 16. 4 13 1 174 16. 3655* 17416. 3149 23 . 5 372 . 0866 372 . 0390 37 1 9888 22 5 4 14 .6701 4 14. 6407 4 14 .6167 22 5 370 . 3084 370 . 2783 370 . 2540 21 . 5 415 . 2546* 4 15 . 2884 4 15 .3130 2 1 . 5 370 . 8570 370 .8907 370 .9158 20 . 5 4 18 .5155 4 18 .561 1 418 .6093 20 . 5 374 .0930 374 . 1376 374 . 1864* 174 14 368 4 13. 366 . 4 13. 367 . 4 17. 370. 17413. 365 4 11 363 4 12 363 4 15 367 9 127 6174 2101* 8756 8440 . 4704 .1301 . 7522 . 3091 . 0467 . 6444 . 3486 . 3285 9938 6 102 2973 174 14 .8649 368.5700 4 13. 1815 366.8473 413.8744 367.5018 4 17. 1758 370.8021 174 14.8 148 368 . 5 193 4 13.1576* 366.8227 4 13.8990 367 .5261 417.2239 370.8500 17413 364 4 1 1 363 4 12 364 4 15 367 2612 9974 6 158 32 1 1 3578 0237 6568 3437 174 13 364 4 1 1 363 4 12 364 4 15 367 . 2 101 * . 9473 . 5923 . 2967 . 38 1 1 .0484 . 7052 . 3915 2622 9363 5950 2336 3323 370.9349 4 18.6592 374.2356 174 16 37 1 4 14 370 4 15 174 14. 368 . 4 13. 366 . 4 13. 367 4 17 370 17413 364 4 1 1 363 4 12 364 4 15 367 76 17 4668 1364* 802 1 9178* 5453 2740 + 9004 1576* 894 1 . 57 18* . 2763 . 4005 .0667 . 7557 . 4423 174 17.5452 375.1850 4 15.9209* 373.5261 4 16 374 6572 22 17 419.98 16 377.5216 174 16 37 1 4 14 370 4 15 370 4 18 374 17414. 368 . 4 13. 366 . 4 13. 367 . 4 17. 370. 17413. 364 4 1 1 363 4 1 2 364 4 15 367 207 1 88 10 5787 2164 3480 9512 7114* . 2883 . 7067 .4112 . 1 193 . 7850 . 9343 . 5610 . 327 1 .9528 1029 8390 5529* 2590 4 165 0829 8084 4948 4 16 374 . 6687 . 2333 420.0367 377.5767 17416.1493 37 1 .823 1 4 14.5652 370.2029 4 15.3601 370.9633 4 1B.7669 374.3432 174 14.6490 368.3536 4 13.104 7 366.7702 4 13.947 1 367.5736 4 17. 3822 371.0078 17413.0448 364.78 17 411.5395 363.2439 412.4 300 364.0952 4 15.8633 367.5498 17417 375 4 15 373 4840 1239 .9117 .5170* 4 16.6766 374.2407 420.0941 377.6343 17417.4214 375.0609 4 15.9060 373.51 17* 4 16.6808 374.2449 420. 1538 377.6939 174 16 37 1 . 4 14. 370. 4 15. 370. 418. 374 . 17414. 368 413 366 413 367 4 17 37 1 17412 364 4 1 1 363 4 12 364 4 15 367 0890 7627 555 1 * . 192 1 . 3690 . 972 1 .8240 . 4008 . 5889 . 2936 .094 1 . 7594 .9570 .5847 . 4392 .0652 .9848 . 72 14 .5272 23 16 4401 1064 9209* 6070 174 16 37 1 . 4 14. 370. 4 15. 370. 418 . 374 . 17414 . 368 . 413. 366 . 4 13 367 417 37 1 17412 364 4 1 1 363 4 12 364 4 15 367 026 1 • 7001 5478 1B52 . 3747 .9768 .8834 . 4601 .5261 2307 0858 75 13 9632 5900 4986 1248 9225 6597 5181 2233 4486 1131 . 9793 .6665 I ro -4 4^ TABLE I (CONTINUED) N = 25 N = 26 N = 27 BRANCH R1 (25) P1(25) R2(25) P2(25 ) R3(25) P3(25) R4(25) P4(25) R1(26) P 1 ( 26 ) R2(26) P2(26) R3(26) P3(26 ) R4(26) P4(26) R1(27) PK27) R2(27 ) P2(27 ) R3(27) P3(27) R4(27) P4(27) F"=J"-7/2 F=J-5/2 F=J-3/2 F=J-1/2 F-J+1/2 F=J+3/2 F=J+5/2 F=J+7/2 N = 28 26 . 26 . 25 . 25 . 24 . 24 . 23 23 27 27 26 26 25 25 24 24 R1(28) P1(28) R2(28) P2(28) R3(28) P3(28) R4(28) P4( 28 ) 5 5 5 5 5 5 5 5 5 5 . 5 .5 . 5 . 5 . 5 . 5 174 1 1 36 1 409 359 4 10 360 4 13 363 6003 3685 9726 .7140 . 7053 . 4086 . 77 16 . 7398 174 1 1 361 409 359 4 10 360 4 13 363 5496* 3203 9457 687 1 7350 4364 .8183 . 7864 17409.7867 357.5880 408. 1940 355.9693 408.9786 356.7132 412.5748 360.0464 17409.7376 357.5393 408 . 1678 355.9432 409.0060 356.7397 4 12.6189 360.0932 174 1 1 36 1 409 359 4 10 360 4 13 363 28 . 5 17407. 8654 17407. 8 178 28 . 5 353 . 7048 353 . 6554 27 . 5 406 . 3 112 406 . 2847 27 . 5 352 . 1217 352 . 0955 26 . 5 407 . 1429* 407 . 1683 26 . 5 352 . 9143 352 . 9409 25 . 5 4 10. 7 125 4 10. 7613 25 . 5 356 . 0296) 356 . .0755) 357 . .0774) 357 . 1235) 29 .5 17405 . B4 12 17405 . 7925 29 .5 349 . 7 179 349 .6673 28 . 5 404 . 3237 404 . 2968 28 .5 348 . 1768 348 .1511 27 .5 405 . 2016 405 . 2276 27 .5 349 .0140 349 .0402 26 .5 408 .7915 408 . 8388* 26 . 5 352 .6560 352 . 7039 5013 2696 92 16 663 1 7582 4601 . 8664 .8355 17409.6870 357.4889 408.1439 355.9192 409.0278 356.7622 4 12.6679 360.1422 17407 353 406 352 407 352 4 10 356 357 7659 6044 26 14 0720 1904 .9635 . 8086 . 1244 .17 18 17411.4475* 361.2169 409.9010 359.6412 410.7770 3GO.4788 413.9178* 363.8855 17405.74 18 349.6184 404.2743 348 . 1279 405.2497 349.0619 408.8880 352.7523 174 1 1 361 409 359 4 10 360 4 13 363 17409 357 408 355 409 356 4 12 360 6334* 4363 1222 8979 0470* 78 18 .7181 .19 16 17407.7129 353 . 55 10 406.2405 352.0521 407.2097 352 .9824 410.8598 356.1751) 357.2203) 17405.6887 349.5651 404.2538 348.1069 405.2680 349.0813 408.9392 352.8039 3930 1617 8834 624 1 7933 4946 9675 . 9385 17409.5793 357.3808 408.1048 355.8802 409.0633 356.7974 4 12 . 77 10 360.2455 17407.6570* 353.4960 406.222 1 352.0333 407.2261 352.9985 410.9125 356.2278 ) 357.2722 ) 17405.6336 349.5105 404.2345 348.0893 405.2848 349.0974 408.9922 352.8564 174 1 1. 3342* 36 1 . 1043 409.8681 359.6089 410.8069 360.5083 4 14.0231 363.9938 17409.5219 357.3247 408.0894 355.8654 409.0773 356.8120 4 12. 8256 360.3009 17407.6010 353.4387 406.2068 352.0184 407 2399 353.0122 410.9671 356.2819) 357.3251 ) * 174 1 1 36 1 409 359 410 360 4 14 364 2744 0444 8563* 5962 8 168 .5187 .0790 .0504 1741 1 360 409 359 4 10 360 4 14 364 2 135* 9832 8462 5868 8255 .5260 . 1377 . 1097 17405 349 404 348 405 349 409 352 5763 4534 2 193 0727 . 2991 .1117 .0470* .9114 17409.4623 357.2632 408.0758 355.8519 409.0887 356.8232 4 12.8823 360.3567 17407.5417 353.3795 406.1934 352.004 1 407.2529 353.0246 4 11.0236 356.3368 ) 357.3808) 17405.5180 349.3943 404.2050 348.0584 405.3113 349.1235 409.1033 352.9668 17409.3989* 357.2025 408.0658 355 . 84 14 409.0975 356 . 83 10 4 12.9407 360.4159 17407.4796 353.3181 406.1826 351.9934 407.2611 353.0325 411.0822* 356.3975 ) 357.4365)* 17405.4560 349.3326 404.1940 348.0478 405.3216* 349.1332 409 . 1619 353.0246 ro —I tn i TABLE I (CONTINUED) BRANCH d- F-J--7/2 F=d-5/2 F=d-3/2 F=J-1/2 F=d+1/2 F=d+3/2 F=d+5/2 F=d+7/2 N = 29 N = 30 N = 31 R1(29 P1 ( 29 R2(29 P2( 29 R3(29 P3( 29 R4( 29 P4(29 R1(30 P1(30 R2( 30 P2( 30 R3( 30 P3( 30 R4( 30 P4(30 RK31 P1(31 R2(31 P2(31 R3( 31 P3( 31 R4(31 P4( 31 R1(32 R2( 32 R3( 32 R4(32 P4( 32 N=33 R1(33 P1(33 R2( 33 P2( 33 R3( 33 P3( 33 R4( 33 N=32 30 30 29 29 28 28 27 27 31 31 30 30 29 29 28 28 32 . 32 . 31 . 31 . 30. 30. 29 . 29 5 5 5 5 5 5 5 , 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 5 5 5 5 5 5 5 5 33 . 5 32 . 5 31.5 30.5 30. 5 34 34 33 33 32 32 3 1 17403. 345 . 402 . 344 . 403 345 406 348 17401 341 7 129 6249 2290 1 137 1559* 0052* 7729 6240 478 1 4290 17403.6637 345.5754 402.2034 34 4.0888 403.1806 345.0289* 406.8206 348.6707 400.0305 339.9561 401 340 404 344 0027 .8908 .6556 .5278 17399.14 19 337 . 1277 397.7282 335.6900 398.7457 336.67 17 402.4313 340.3376 17401 34 1 . 4296 . 3784 400.0048 339.9308 401.0274* 340.9 153 404.7029 344.5751 17399.0922 337.0787 397.7029 335.6655 398 . 7714* 336.6967 402.4791 340.3844 17396.7000* 17396.6500 395.3172 395.2915 396.3814* 396.4037 400.1058 400.1537 336.0468 336.0944 17403.6126 345.5240 402.1802 344.0653 403.2021 345.0510* 406.8698 348.7200 17401.3777 341.3275 399.9825 339.9074 401.0481* 340.9360 404.7523 344.6252 17399.0407 337.0272 397.6802 335.6424 398.7905 336.7 176 402.5290 340.4346 17396.5992 395.2689 396.4246 400.2033 336.1466 17403 345 402 344 403 345 406 348 17394.1558 17394.1064 17394.0570 328.2 193 392.8034 326 . 8487 393.91 14 327.9225 397.6763 328.1706 392.7786 326.8245 393.9343 327.9461 397.7242 328.1188 392.7560 326.8019 393.9553 327.9665 397.7739 5595 47 17 1599 0443 2208 0705* 92 13 .77 15 17403.5050 345 . 4164 402.1420 344.0265 403.2379 345.0869* 406.9743 348.8234 17401.3240 34 1 . 2746 399.96 14 339.8869 401.0630* 340.9555 404.8040 344.6753 17398 .9876 336.9743 397.6598 335.62 14 398.8101 * 336.7347 402.5801 340.4861 17396.5453* 395.2488 396.4442 400.2546 336.1955 17 394.0020 328.0660 392.7352 326.7813 393.9740 327.9857 397.8249 17403.4471 345.3592 402.1259 344.0088 403.2526* 345. 101 1 * 407.0301 348.8790 17401 34 1 399 339 401 340 404 344 2694 2 198* 9432 8687 .0825* . 97 17 .8570 . 729 1 17401 34 1 399 339 401 340 404 344 2133* 1630 9268 8522 0976 9856 9 113 . 7839 17398.9329* 336.9197 397.6416 335.6034 398.8253* 336 . 75 19 402.6340 340.5395 17398 8760 336.8630 397.6249 335.5869 398.8401 * 336.7668 402.6876 340.5945 17396.4879* 17396.4334 395.2300 395.2 136 396.4611* 396.4763* 400.3080 400.3630 336.2494 336.3046 17393.9468 17393.8899 32B.0101 392.7169 326.7628 393.9907 328.0012 397 .8779 327.9533 392.6994 326.7462 394.0056 328.0167 397.9326 17403.3888* 345.3002 402.1115 343.9958 403.2652* 345. 1 125 407.0850 348.9348 17401.1535 34 1 . 1043 399.9129 339.8382 401.1108 340.9991 404.9683 344.8401 17398.8180* 336.8029 397.6106 335 . 57 19 398.8533 336.7796 402.7440 340.6499 17396 . 3765* 395.1984 396.4879* 400.4192 336.3605 17393.8278 327.8947 392.6848 326.7311 394.0190 328.0298i 397.9885 17403.3270 345.2397 402.1001 343.9847 403.2745 345. 1233 407.1429* 348.9929 17401.0926 341.0430* 399.9011 339.8263 401. 1213 341.0087 405.0259 344.8988 17398.7564 336.7424 397.5985 335.5595 398.8652* 336.7899 402.8017 340.7087 17396.3140* 395.1856 396.4995 400.4771 336.4186 17393 . 77 12 327.8338 392.6720 326.7193 394.0303* 328.0414 398.0466 PO (Tl I TABLE I (CONTINUED) N = 34 N=35 BRANCH RK34) P1(34) R2(34) P2(34) R3(34) P3(34) R4(34 ) P4(34) R1(35) P1(35) R2(35) P2(35) R3(35) P3(35) R4(35) P4(35) F"=d"-7/2 F=d-5/2 F=d~3/2 F = d- 1/2 F=d+1/2 F=d+3/2 F=d+5/2 F=d+7/2 N = 37 35 . 35 . 34 . 34 . 33 . 33 . 32 . 32 . 36 . 36 . 35 . 35 . 34 . 34 33 33 R1(37) P1(37) R2(37) P2(37) R3(37) P3(37) R4(37) P4(37) N = 36 R1(36) 37.5 PK 36) 37.5 R2(36) 36.5 P2(36) 36.5 R3(36) 35.5 P3(36) 35.5 R4(36) 34.5 P4(36) 34.5 17391.5145* 323.6086 390.1826 322.2716 391 .3347 323.3887 395 .1372 327 . 1620 17391.4678* 323.5598 390. 1584 322 . 2475 391 .3574 323.4127 395 . 1856 327.2106 17388. 318. 387 . 3 17 388 318 392 322 17385 385 314 384 312 385 314 389 317 695 1 8962 4764 . 5879 .651 1 . 7509 . 4938 .561 1 .7 700) .9101) . 1003 .6143 .8120 . 86 16 .0165 .7428 .8555* 17391 .4167* 323.5072 390. 1358 322.2252 391 .378 1 323.4318 395.2345 327.2604 17388 318 387 317 388 318 392 322 647 1 8456* 4518 5639 .6738 . 773 1 .5423 . 6098 17388 318 387 317 388 3 18 392 322 17385.7339) 385.8716) 3 14.0457 384 . 5880 312.7874 385 . 8840 314.0376 389.7914 317.9042* 38 . 5 17382. 9005 17382. 8523* 38 5 309 . 1084 309 . 0605 37 . 5 381 . .6934 38 1 .6702 37 5 307 .9339* 307 . 9089 36 . 5 382 . 9668 382 . 9890 36 . 5 309 . 1624 309 . 1845 35 . 5 386 . 8887 386 . 9366 35 .5 313 .0558 313 . 1056 5970 7943* .4288 . 54 10 .6951 . 7943* . 5924 .6605 17385.6984 ) 385.8360) 313 .9988 384.5679 3 12.7647 385.9040 314.0505 389 . 84 19 317.9539* 17382.7993 309.0095 381 .6476 307.8856 383.0093 309.2034 386.987 1 313.1552 17391 323 390 322 39 1 323 395 327 17388 3 18. 387 . 317. 388 . 3 18 392 322 17385 385 313 384 3 12 385 3 14 389 318 365 1 4522* 1 145* . 2044 . 3967 .4522 . 2862 .31 19 . 5449 . 7424 . 4074 . 52 1 1 . 7 128 .8121 . 644 1 .7121 6587 ) 8034 ) 9479 5476 7453 9226 0778 8933 .0065* 17391.3107 323.3998 390.0962 322 . 1866 391.4 140 323.4676 395.3397 327 . 3652 17388.49 14 3 18.6866 387.3870 317.5016 388.7297 3 18.8290 392.6973 322.7645 17382.7461 308.957 1 381.6274 307.8646 383.0279 309.2228 387.0387 3 13.2070 17391.2544 323.3428 390.0795 322.1694 39 1 .4286 323.4824 395.3941 327.4199 17388 3 18 387 317 388 3 18 392 322 4361 6314 3695 4851 . 7459 . 8456* .7519 . 8 193 1739 1 . 1970 323.2840 390.0641 322 . 1547 39 1 . 4419 323.4961 395.4503 327.4760 17388.3793 318.5723 387 . 3535 3 17. 4690 388.7583 3 18.8589 392.8075 322.8759 1373 2235 390.0511 322.14 12 4537 5072 395.508 1 327.5336 17391 323 39 1 323 17388.3202 318.5120 387.3388 317.457 1 388.7643* 3 18.8694 392 . 8655 322 . 9325 17385.6131 17385.5654 17385.5116 17385.4568 3 13 384 3 12 385 314 389 318 8926 5286 727 1 9395 .094 1 .9462 .0594" 3 13 384 312 385 314 390 318 . 8370 .5116 .7109 .9547 . 1088 .0008 . 1140* 17382.6914 308.904 6 381.6088 307.8450 383.0445 309.2391 387.0922 313.2598 17382.6349 308.8491 381 .59 17 307.8264 383.0601 309.2547 387. 1466 313.3151 313.7801 384.4966 312 .6943 385.9687 314. 1225 390.0572 318. 1709* 17382.5766 308.7927 381.5758 307.8100 383.0740 309.2690 387.2024 313.3697 313.7207 384.4825 312.6808 385.9803 314.1350 390.1145 318 .2275* 17382.5167 308.7342 381.5623 307.7972 383.0879 309.2810 387.2598 313.4288* I ro i TABLE I (CONTINUED) BRANCH d" F-d-7/2 F=d-5/2 F=d~3/2 F = d-1/2 F=d+1/2 F=d+3/2 F=d+5/2 F=d+7/2 N=38 R1(38 P1(38 R2( 38 P2( 38 R3( 38 P3( 38 R4(38 P4(38 N=39 R1(39 P 1 ( 39 R2( 39 P2( 39 R3( 39 P3(39 R4( 39 P4(39 N = 40 N=4 1 R1(40 P 1 (40 R2(40 P2(40 R3(40 P3(40 R4(40 P4(40 RK41 P1(41 R2(41 R3(41 P3(41 R4(4 1 P4(41 39 . 5 39 . 5 38 38 . 37 . 37 . 36 . 36 . 40 40 39 39 . 38 . 38 . 37 . 37 . 4 1 . 4 1 . 40. 40. 39 . 39 . 38 . 38 . 42 . 42 . 4 1 . 40. 40. 39 . 39 5 5 5 . 5 5 . 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 .5 5 . 5 .5 5 5 5 17379.8342 304.0095 ) 304.1553) 378.6378 302.906 1 379.9630* 304.2041 383.9286 308. 1335 17376.6630 298 . 9842 375.4894 297.8052 376.8533 299. 1469 380.8613 303.1179* 17373.3860 293.7550 372.2370 292.6033 293.9824 377.6885* 297.9912 17379.7841 303.9787) 304.1168) 378 302 379 304 383 308 6140* .8835 .9852 . 2260 . 9778 . 1829 17376.6130 298 . 9340 375.4660 297.7808 376.8749 299. 1683 380.9104 303.1653* 17373.3362 293.7053 372.2132 292.5783 373.6518 294.0026 377.7369* 298.0398 17379.7323 303.948 1 ) 304.0800) 378.5923 302.8599 380.0054 304.2461 384.0281 308.2344 17376.5610 298 . 8820 375 . 4439 297.7597 376.8943 299.1887 380.9610 303.2166* 17373.2838 293.6565 372. 19 18 292.5572 373.6712 294.0237 377.7869* 298.0914 17379.6785 303.9030) 304.0458) 378 . 57 19 302.8397 30O.O239 304.2647 384.0806 308.2841 17376.5077 298.8289 375.4234 297.7394 376.9136 299.2070 381.0125 303.2680* 17373.2300 293.5998 372.1706 292.5368 373.6902 294.04 12 377.8395* 298.1428 17370.0064 17369.9561 17369.9046 288.4225 368.8754 370.2947 288.7093 374.4071 292.7646 288.3720 368 . 8523 370.3162 288.7305 374.4601 * 292.8127 288.3207 368.8294 370.3367 288.7504 37 4.5069 292.8636 17379 303 304 378 302 380 304 384 308 624 1 8586 ) 0142) 5526* 82 16 0405 28 1 1 1333 . 3370 17376 . 4533 298.7735 297.7206 376.9302 299.224 1 381.0656 303.32 17 * 17373 293 372 292 373 294 377 298 1754 5456 1524 5 176 7074 0581 8932" 196 1 17369.8508 17369.7962 288.2666 288.2120 368.8092 368.7898 370.3548 370.3720 288.7689 288.7862 374.5585 374.6130 292.9153 292.9686 17379.567 1 303.8098 378.5356 302.8037 380.056 1 304.2967 384.1880 308.3920 17376.3962 298.7172 375.3880* 297.7040 376.9452 299.2398 38 1 . 1202 303.3765* 17373.1196 293.4895 372.1348 292.5000 373.7225 294.0735 377.9483* 298.2505 17369.7406 288.1553 368.7737 370.3880 288.8017 374.6674 293.0240 17379.5087 303.7575 378.5207 302.7895 380.0701 304.3105 384.2434 308.4480 17376.3384 298.6588 375.3725 297.6886 376.9596 299.2540 381 . 1758 303.4321 * 17373.0610 293.4307 372.1194 292.4846 373.7367 294.0877 378.0039* 298.3058 17369.6815 288.0974 368.7579 370.4027 288.8156 374.7226 293.0794 17379.4491 303.7020 378.5059* 302.7741 380.0826 304.3233 384.301 1 308.5054 17376.2783 298.5996 375.3577* 297.6739 376.9724 299.2660 381.2334 303.4894* 17373.0014 293.37 13 372 . 1054 292 . 47 16 373.7496 294.0998 298 . 3638 17369.6225 288.0362 368.7435 370.4150 288.8283 374.7800 293.1374 I PO ^1 oo i TABLE I (CONTINUED) BRANCH J" F "=J" -7/2 F = J -5/2 N = 42 R2(42) 42 . 5 R3(42) 4 1 . 5 0761 R4(42) 40. 5 1737 1 . 027 1 1737 1 . N= 43 R2(43) 43 . 5 17361 . 8357* 1736 1 . 8 137* R4(43) 4 1 . 5 367 . 5270* N = 44 R4(44) 42 . 5 17363. 8522* 17363. 8988* N= = 45 R3(45) 44 . 5 17356. 2819* 17356. 2946 R4(45) 43 . 5 360 2333* 360. 2842* N = = 46 R3(46) 45 . 5 17352 .3349* 17352 3554* R4(46) 44 .5 356 .3162* 356 .37 12* N-= 47 R3(47) 46 .5 17348 .3506* 17348 .3727* R4(47) 45 . 5 352 .5050* 352 .5545* N-= 48 R4(48) 46 .5 17348 . 4856 17348 . 5354 F=J-3/2 F=J-1/2 F = J+ 1/2 17365 37 1 17361 . 367 . 3637 1248* 7913* 6300* 17365. 37 1 . 17361. 367 . 3429 1781 7699* 6825' 17365. 371 . 1736 1 . 367 . 3243 2315 7512* 7360* F=J+3/2 17365.3067 366.9103 37 1 . 2873 F=d+5/2 F=J+7/2 17365.2915 17365.2772 366.9263* 366.9405* 37 1.3423 37 1.3995 17363.9533* 17364.0137* 17364.0829' 17361.7341* 17361.7173* 17361.7020* 17364.1654* 17364.2492* 17356 360 17352 356 17348 352 3135 3365 + 3746* 4214* . 3906* .6060* 17356 360 17352 356 17348 352 . 3368* . 3859* . 3941 * .4712* . 4097* . 6598* 17356 360 17352 356 17348 352 . 3516 . 4369* . 4104* . 5224* .4257* .7089* 17356 360 17352. 356 . 17348 352 3697 17356.3860 17356.4005* 4945* 360.5497* 360.6047* 4266* 17352.4420* 17352.4566* 5745* 356.6271* 356.6818* 4426* 17348.4567* 17348.4712* .7649* 352.8203* 352.8764* 17348.5862 17348.6383 17348.6909 17348.7462 17348.8022 17348.8585 I ro —I TABLE II ROATIONAL LINES ASSIGNED IN THE SI =4il"=4 BANDS OF FEO 5819 X BAND (V»=0) (CONTINUED) J " 4 17 180.200 17176. 120 5 79.990 75 . 091 17171 . 01 1 6 79.582 73 . 859 68 . 962 7 78.982 72 . 435 66 . 7 10 8 78 . 184 70. 808 64 . 26 1 9 77.201 68 . 994 61 . 6 18 10 76.035 66 . 989 58 . 781 1 1 74.686 64 . 798 55 . 756 12 73.163 62 . 432 52 . 545 13 7 1 . 472 59 . 888 49 . 156 14 69.598 57 16 1 45 . 593 15 67 .579 54 .284 4 1 . 848 16 65.402 51 . 243 37 . 949 17 63.074 48 .045 33 . 889 18 60.591 44 . 701 29 . 670 19 57.979 41 . 196 25 . 307 20 55.227 37 . 565 20. . 783 21 52 . 373 33 .800 16 . 135 22 49.350 29 . 926 1 1 . 347 23 46 . 2 15 25 .881 06 . 446 24 42.957 2 1 . 731 01 . 397 25 39.560 17 . 446 17096 . 228 26 36.161 13 .045 90 . 936 35.998 35.872 27 32.458 85 .511 32.401 28 28.647 80 .07 1 97 .9 12 79 . 791 29 34.282 74 . 342 33.530 74 . 287 24.630 30 27.118 68 .503 26.551 20.309 20.198 18.924 18.424 31 21.284 7 1 . 985 20.604 7 1 . 356 14.160 62 . 454 13.915 32 33 37 17 116.479 15.763 07 .427 12.542 11.271 10.355 34 09.723 06.897 06.504 04.860 35 06.067 03.175 02 . 46 1 00.992 36 17098.96 1 98. 1 14 96.782 94.656 93.698 92.564 3R 90.083 89.206 88.2 19 39 85. 148 84.5 16 83.566 40 82.393 8 1.173 79.536 41 77.289 76.289 74.4 16 42 72.432 7 1 . 677 68.655 17062.915 62 . 349 56.106 55.996 54.7 19 55.051 55.051 54 . 373 47 . 928 47.685 48.223 47.503 39. 16 1 42.261 40.983 40.072 37 . 4 1 1 34.588 34. 194 32.548 31 .747* 28.842 28.131 26.653 22.603 21.759 16.272 15.331 14.187 09.690 08.809 07.824 02.737* 02.105 01.158 16997.965 96.744 95. 1 1 1 I ro oo o TABLE II (CONTINUED) 5583 I BAND (V"=0) G2SO A BAND (V"=2) J" 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 17906. 06. 06 . 07 . 07 . 07 07 07 07 07 07 06 06 05 05 04 03 02 02 01 00 17898 97 96 063 544 977 278 543* 676* 676* 676* 543* . 358 .089 . 728 . 285 . 740 . 150 . 706 .827 . 958 .059 .07 1 .009 . 834 .9 13 . 446' 92 . 225 90.699 89.094 87.395 85.661 83 . 746 81 .787 79 . 664 77 .623 75 . 44 1 • 73 . 197* 17901 00 00 17899 99 98 97 96 406 943 437 798 138 310 .408 . 446* 17893 9 1 89 . 88 . 86 . 84 . 82 . 79 . 77 . 75 . 73 . 70. 68 . 65 . 62 . 59 . 57 54 . 5 1 . 47 . 44 4 1 38 34 3 1 28 24 20 17 13 09 05 272 639 897 137* 188 203 143 992 748 44 1 055* . 545 .075 . 336 . 864 . 946 055 .098 .074 .991 . 795* .624 . 325 .957 .527 .022 . 469 .838 .114 . 353 . 405 . 424 17797.218 93. 009* 88.670* R 0 P 16168. 326 173 . 531* 67 . 954* 74 . 032 67 . 484 16161. 879 74 . 463* 66 . 991 • 60. 470 74 . 795* 66 . 4 10 58 . 922 75 . 079* 65 . 748 57 . 348 75 . 288 + 64 . 010 55 . 686 75 . 460* 53 . 922* 75 . 460* 52 . 15 1 75 . 460* 50. 265 75 . 460* 48 . 310 75 . 288* 46 . 309 75 . 079 + 44 . 234 74 . 795* 42 . 082 74 . 463* 39 . 872 74 . 316 P 37 . 582 73 . 729 35 . 242 73 . 192 33 . 079 P 72 . 608 30. 4R4 7 1 970 27 948 7 1 . 264 25 . . 360 70 . 464 22 .7 17 69 . 735 20 .009 68 . 874 17 201 67 . 954* 14 . 473 66 .991* 1 1 . 6 10 65 . 97 1 08 .691 64 897 05 . 732 63 .773 02 . 701 62 .576 16099 . 647 61 . 343 96 .521 59 . 962 93 . 329 58 .538 P 90 . 100 56 .969 86 . 725 55 . 497 83 .292 P 53 . 922* 79 . 764* 52 . 286 76 . 307 50 . 582 72 . 708 48 . 789 69 .098 47 .052 65 . 406 44 .077 61 . 639 ro 00 TABLE III ROTATIONAL LINES ASSIGNED IN 2491 A BAND OF 15N02 K=0 K=1 K = 2 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 QR1 1+QR22 OP 1 1+QP22 40142.352 143.300* 144.025* 144.360* 144.360* 144.025 143.232* 142.035 140.450* 138.422* 135.978 133.135* 129.774* 126 . 164* 121 .976 1 17.471 122.630* 107.513* 101.851 095.915* 089.466* 40136.441* 134.013* 131 .431 128.403 124 .992* 121.293 117.081 * 112.630* 107.733* 120.428* 096.610* 090.466* 083.891 076.86 1 069.474* 061.624* OR 1 1+QR22 OP 1 1+0P22 40139. 139 . ( 139 . 140. 140. 14 1. 140. 14 1. 140. 141 . 139. 14 1. 138 . 140. 137 . 139 . 135 . 138 . 133 . 136 . 130. 134 127 131 124 128 120 125 1 17 12 1 1 12 1 17 107 1 12 102 106 097 100 09 1 094 084 088 202* 857* 857* ) 649 280* 131* 280* 456* 037* 366* . 394* .131* .422* . 450* .079 . 394* . 336 .079 . 153* . 346* . 737 .161* . 869* .696* . 566* . 754* .999* . 275* .081* . 378 . 630* .081* . 733* .211* . 608* 915* 083* 992* 035* 723 878* 045 (a (40133. 132 . 131 . 130. ( 129 . ( 127 . 126 . 124 . 122 . 12 1. 1 19 . 118. 115. 114. 111. 1 10. 106 . 106 . 101 . 101 . 095 . 096 . 090. 090 083 084 077 (076 070 07 1 063 064 055 056 (047 048 039 039 030 030 020 ) ) 798 880* 62 1 * 4 79 + 055* ) 869* ) 164* 992 + 864* 884* 294 336 327 569 012* 401 * 251* 096* 324* 324* 9 15* 128* 145 743* .891* . 878* . 372* .861*) .521* . 702* .2 17* .458* . 460* . 733* .583*) . 529* .02 1 . 857* .114* . 673* . 993* OR 1 1+QR22 0Q1 1 0022 40125.275* 124.992* 124 . 682* 124 .429* 40129. 130. 129 . 129 . 129. 129 . ( 128 . 129 . 127 . 128 . 126 . 127 . 124 . 126 . 122 . 124 199 122 1 17 120 1 13 1 18 1 10 1 15 106 1 12 102 109 097 105 092 101 086 074 8 19* 029 + 8 19* 942 + 363* 639 673* ) . 146* .641* . 430* . 164 * . 463* . 566* .164* . 376 .682 + .914 .864 + .081* .999 + .901* .600 .401* . 93 1 . 4 10* . 894* .025 . 357* . 279* .614* . 179* . 324* . 597* . 2 19* 40125.804* 125.392* 124.992' 124.682* 123.951 123.526 I ro co ro i (a) 40135.000 (F2) : 40134.790 (F1) TABLE III (CONTINUED) K = 2 K = 3 N QP11 OP22 QR11 OR22 3 4 5 40120.G19 40120.999* 6 119.294 * 119.621* 7 118.117 118.336* 8 116. 777* 9 115. 327* 10 113. 901 * 40111. 937 ' 1 1 112. 2 11* (40111. 547* ) 111. 868* 12 1 10. 585* 13 108 . 733* 111. 216* 111. 547* 14 107 . 208* 111. 012* 111 2 16* 15 104 . 988* 1 10. . 585* 1 10 . 800* 16 103 . 495 1 10 243* 1 10 . 4 10* 17 ( 100. 992* ) 109 600* 109 . 77 1 18 099 . 601 * 109 . 176* 109 . 357* 19 096 . 465* 108 .381* 108 . 540* 20 095 . 433* 107 . 733* 107 . 978* 2 1 09 1 . 7 14 ( 106 .727* ) 106 .9 15* 22 090. 996* 106 . 25 1 * 106 . 4 10* 23 08G . 597* 104 . 988* 105 .026 24 086 . 422* 104 . 585 25 081 . , 188* 102 . 78 1 * 26 08 1 489* 102 . 428* 27 075 . 340 ( 100. 319* ) 28 076 . 326* ( 100. 319* ) 29 069 . 145* 097 . 578* 30 070 . 986* (097 . 727* ) 31 062 . 636 094 47 1 32 065 .063* 095 . 123* 33 055 .675* 090 996* 34 059 . 070 092 . 3 16* 35 (048 .520*) 087 162* 36 052 . 556* (089 431*) 37 040 . 745* 083 09 1 ' 38 045 . 794* 086 .14 1* 39 032 .600* (078 653* ) 40 038 . 309* 082 . 652* 41 024 . 180 07 3 646 42 030 . 553* 079 .011* 43 015 . 342* 068 . 396' 44 022 . 553* 075 .005* 0011 0022 OP 11 OP22 40106.600* 40107.513* 106.410* 106.096* 106.600* 40102.159* 40102.781* 105.614* 106.096* 100.992* 101.660* 105.614* 099.745* 100.319* 098.400 098.762 097.083* 097.279* 095.433* 095.915* 093.879 094.173 092.179* 092.454 090.466* 090.743* 088.680 088.881 086.697* 086.884* 084.878 084 878* 082.858* 082.858* , 080.777* 080.777* (078.653*) g 076.326* I (073.646*) 071.620* 068 . 933* 066.709* 063.635* 061.509* 058 . 165 056.084 052.262* 050.478* 046.064* 044 .589* 039 597* 038 627* 032.8 10* 032.496* 025.654 026.080 019 504 010.282 (012.480) 002 060* 005.490* TABLE III (CONTINUED) K = 4 K = 5 OR 1 1 OR22 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 40086. (086 . 085 . 085 . 085 . (085. 084 . 084 . 083 . 082 . 082 . 08 1 . 080. 079 . (078 . 077 . (076 074 073 072 (070 069 067 065 (064 (062 060 14 1 * 14 1*) 9 11* 791 439* 1 10* ) 668* 049* 522 .858* . 103 . 188* . 443* .415 . 352* ) . 372* . 326*) . 848* .646* .096* .868*) .041* . 732 . 675 . 458* ) .013*) .696 058 40086. 086 . 086 . (086 . 085 . 085 . (085 . 084 . 083 . 083 . 082 . 08 1 . 080. 079 . (078 . 077 . 076 . 075 073 072 070 069 067 065 064 062 060 165* 697* 597* 422* 14 1*) 791* 439* 110*) 426 891* 091* . 378 . 489* . 64 1 • . 639 . 651*) .421* . 326* .005* .930* .096* .867* . 145* .989* . 798* .458* . 2 15 . 886 053.908* 052.916* (049.503*) 048 .7 14* (044.611*) (044.404*) 039.310* 001 1 40080.777* 080.64 1* 080.224* 079.639* 0022 40082.103* 08 1 .489* (080 443*) 079.639* 078.998 0P1 1 OP22 0R1 1 QR22 40074. 07 2 . 07 1 . 069 . 068 . 066 . 065 . (063 . 06 1 . 059 . 057 . 055 052 050 048 045 04 3 040 038 035 032 029 026 023 020 017 014 (01 1 007 004 39997 993 2 16 900 645* 984 392 709* 07 1* 236* ) 280* 284 201 1 1 1 912* 62 1 * 239* 785* 34 1 * 760* 090 398 618* 756* . 778* . 816 + .656* .619* . 234* .093*) . 572* . 466 . 485* . 206 (990 985 982 977 975 969 967 40075. 017* 07 3 . 665* 072 . 100* 070. 5 14* (40053. 769* ) 40054. 347* 068 . 933* 053 . 577* (054 . 347* ) 067 . 209 053 . 488* 054 . 105* 065 . 4 1 1 053 . 204* 053 . 879* (063 . 643* ) 052 . 929* 053 . 483* 06 1 . 607* 052 . 572* (053. 100* ) 059 . 626 052 . 044* 052 . 573* 057 . 568 051 . 539 052 . 044* 055 . 427* 050. 910* 051 . 450* 053 . 146* 050. 275* 050. 625* 050. .903 049 . 508* 049. 907 048 545* (048 . 552* ) 049 . 037* 046 087 04 7 . 733* 048 . 169* 04 3 . 556 046 765* 04 7 . 120 040 . 945 045 775* (046 064* ) 038 . 308* 04 4 .600* 044 . 843* 035 . 562 04 3 .357 043 . . 558* 032 .711* 042 .052 042 . 355 029 .913 040 . 767* 040 .947* 026 .964* 039 .277* (039 . 575* ) 023 .994* 037 .74 1 (038 .091 *) 020 . 8 16* 036 . 151 036 . 353 017 . 168 034 .470 034 . 747 (014 .402* ) 032 .718* 032 .923* 01 1 . 288* 030 .947* 037 . 162* 007 .7 16* 029 .004 029 . 304* 004 .588 027 .114* 027 . 278* 39997 . 637* 023 .047* 993 . 360* 020 .827* 020 . 94 1 • 450* ) 018 . 490* 018 . 764* 64 1 016 . 101 016 . 301 .935* 013 .92 1 * . 739 01 1 . 250* I ro oo I 337 587* 676* TABLE III (CONTINUED) K = 5 K = 6 001 1 0022 OP 1 1 QP22 0R1 1 QR22 001 1 0022 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 40047.583* 047.119* 046.589* 046.064* 40049.498* 048 . 927* 048.242* 047.583* 046.760* (046.064* ) 045.134* 40042. 04 1 . 040. 038 . 037 . 035. 034 . 032 . 030. 028 . 026 . 024 . 022 . 020. 018 015 . 013 010 008 005 002 39999 997 994 990 987 984 98 1 977 974 97 1 (967 963 960 (956 952 948 944 936 932 962 86 1 * 469* 993* 54 1 927 256 . 533* .670* .761 . 780* .7 12 .550* . 365 .009 .653 .2 12* . 704* . 143* .475* .664 . 907* .035 .013* .972* .816 . 542* . 330 .963 . 529 .107* . 486* ) . 885* .073* . 280* ) . 389* . 504* .538* . 308* . 294* 40044. 04 2 . 04 1 . 039 . 038 . 036 . 034 033 . 03 1 . 029 . 027 . 025 . 023 . 020. 018 . 016 . 013 . 011. 008 . 005 . 002 . OOO. 39999 (994 99 1 988 984 98 1 978 974 97 1 967 964 394* 962 436 849* 310 625 946 132 242 303 278 200 037 8 18* 479* .089 . 594 * 095 .411* . 79 ) 946* . 148 . 258 . 337* ) . 249* .106* 892 . 573 + . 224 . 782 . 245 + .665* . 024 • 956.585 952.675+ 948 . 798 + (940.763+) 40014. 014 . 014 . 013. 013 . 013. 012 . 012 . 011. 010. 010. 009 . 008 . 007 . 006. 005. 003. 002 . 000. 39999 . 998 996 994 992 990 (989 987 985 982 980 978 975 973 970 968 965 237* 237* 237* 9 10* 594 • 2 14* 709* 188* 492* 84 1 * 062* 252 * 4 1 1 * 300* 227* 049 799 503* . 992* .628* .078*. . 526* .687 . 942 * . 990* .05 2* .083 . 097 . 94 1 • .6 16 .224* . 899 + . 493 887 + 272 + 596* 40015 015 015 014 014 013 (013 012 . 012 . Oil. 010 . 009 . 008 . 007 . 006 . 005 . 004 . 002 . 001 . 39999 998 996 995 993 (99 1 989 987 985 983 980 978 976 973 (97 1 968 (965 344* 344* 04 2 722 370* 91 1 594* ) 850 187* 492* 704 + 806 84 1 7 19* 6 19 .477* . 135 .917* . 465 .908* . 377 . 789 .015 . 208* . 249* ) .4 19* .414* . 339* . 147* 949* 558* 232* 790* 222* ) 6 18* 956* ) 40008.137* 007.802* 007.280* 006.6 14* 005.344* 005.045* 40010.061* 009.252* 008.594* 007.7 18* 006.977* 006.04 1 • 055.045* I oo tn i TABLE III (CONTINUED) K = 6 K = 7 QP1 1 OP22 OR 1 1 OR22 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 (40002. 000. 39999. 998 . 996 . 994 . 993 . 99 1 . 989 . 987 . 985 . 983 . 980 . 978 . 976 . 973 . 97 1 . 968 . 965 . 963 . 960. 957 954 951 948 944 94 1 938 934 931 927 923 920 9 16 9 12 908 904 900 057) 994* 630* 080* 528* 866* 2 10* 251 • 42 1 4 16* 34 1 • 148* 95 1 * 560 237 . 79 1 . 23 1 * .6 18* .953* .122* . 409* . 390 . 375 . 296 . 148* .876* . 507* . 177* . 773* . 153 .606* .912 . 095 . 240* . 355* . 370* . 285* .053* 40004. 002 . 000. 39999. 997 . 995 . 994 . 992 . 990 . 988 . 986 . 983 . 98 1 . 979 976 974 97 1 969 966 (963 960 957 954 95 1 948 945 94 1 938 934 93 1 (927 924 920 916 912 908 055* 508 996* 353 644 915 018* 190 260 1 10 070 863 .577* . 257 . 836 . 36 1 . 789 .117* .411* .646*) . 729 .818 .810* . 703 .500* . 266* .913 . 497* .942* . 487 . 822* ) .18 1 . 440* .519* .525* . 640* 39968. (968. 968 . 968 . 967 . 967 . 967 . 966 . 966 . 965 . 964 . 964 . 963 . 962 . 96 1 . 960. 958 . 957 . 956 . 954 . 953 . 951 950 ( 948 946 944 942 940 938 936 933 93 1 928 926 923 92 1 9 18 9 15 287* 176* ) 176* 008* 860* 630* 137* 668 153 592* 804* 023* 123* 226 147 079 903* 567 . 279 . 809* . 307 . 702* . 044 . 352* ) . 485* . 532* .581 . 554 358* .230* . 8 10 . 49 1 • .996 .540 .917* . 199 . 46 1 * . 587* 39969. 969 . 969 . 969 969 . 968 968 967 967 966 965 964 963 926 96 1 960 959 958 956 955 953 952 950 948 946 944 94 3 940 938 936 934 93 1 929 926 924 92 1 918 916 9 1 1 * 773* 590* 306 ' 116* 619 + 176* 669* 137 * 4 IO ' .592* . 804 + 886 + . 948 • . 844 . 728* . 467 • .14 1 . 776 . 368 . 769 . 207 .527* .8 17 + . 982 + . 974 * .012 . 938 .807 . 534 . 250 . 86 1 . 450 .910 . 191 + . 634 .847* . 007 • 0Q1 1 39961.699 961.152< 959.831 0022 0P1 1 0P22 958 957 956 954 953 952 951 949 94 7 .105 + . 152 + .009 + . 808 + .677* . 400* .042 + . 503 . 950 + 39962. 228* (39962. 665* ) 39955. 374 « 96 1 . 354 952 . 13 1 1 * 953 . 764 * 960. 420* 950. 523* 951 . 987* 959 . 473* 948 . 813* 950. 253 947 . 1 10 948 . 49 1 * 957 158 945 . 26 1 * 946 . 48 1 * 956 009* 943 . 398 944 . 528* 94 1 . 407 942 . 442 953 .321* 939 . 3 12 940. 285 ( 95 1 .706*) 937 . 146 938 . 129* 950 . 268* 934 . 936* 935 . 782 948 .809* 932 . 570 933 . 4 14 930. 209 931 . 026 927 . 75 1 * 928 . 478 925 . 180 925 . 862 922 . 523 923. 2 1 1 919 .814 920 434 9 16 . 980* 917 . .606* 914 . 164* 9 14 . 743* 91 1 . 174* 911 .671* 908 . 102 • 908 . 643* 904 . 998* 905 .635* 901 . 799 (902 . 386) 898 . 509 898 . 990* 895 . 190 895 .6 14* 89 1 . 724* 892 . 235* 888 .17 1* 888 . 780 + 884 . 580 885 . 150* 881 .02 1 * 88 1 . 392 877 .12 1 877 .690* (873 . 340*) 873 . 763 869 . 536* 869 .950* ( 865 . 460) 865 .918 86 1 . 48 1 * 86 1 .888* (857 . 293*) 857 . 752* I ro oo CTi i 887.2 13 910.029 TABLE III (CONTINUED) K = 8 K = 9 N 0R1 1 QR22 8 9 10 (39915. 775* ) 39917 . 809* 1 1 915 . 777* 917 . 613* 12 (915. 680* ) 917 . 285 13 (915. 530* ) 916 . 991 * 14 9 15. 1 10 916. 520* 15 914. 746* 916. 012 16 914. 166* 915 . 480* 17 913 . 7 10 (914. 750* ) 18 913. 07 3 9 14. 170* 19 912 . 347 913 . 391 20 911. 564* 912 . 527 21 910. 583* (911. 677* ) 22 909 . 656 910. 587 23 908 . 645* 909 . 428* ^24 907 . 376* 908 . 369* 25 906 241* 907 . 042 26 904 .980* 905 . 64 1 * 27 903 .586* 904 . 283 28 902 . 100* 902 .815 29 900 . 535* 901 .2 12* 30 898 .991* 899 .650* 31 897 . 240 897 .898 32 895 .433* 896 .074 33 893 .576 894 . 185* 34 89 1 . 748* 892 . 243* 35 889 . 597 890 . 238* 36 887 .451 888 . 185* 37 885 . 294 885 . 884* 38 882 .870* 883 . 573* 39 880 .656* 88 1 . 166* 40 878 . 279 878 .728* 4 1 875 . 749 876 .18 1* 42 (873 .081 ) (873 .549*) 43 001 1 39908.637* 908.116* 907.382* 906.574 905.638* 904.703* 903.581 * 902.394* 901.203* 0022 OP 1 1 QP22 QR1 1 QR22 399 1 1 910 909 908 907 906 (904 903 902 177* 327* 426 37 1 * 382* 238* 973* ) 783 . 394* 39899 (898 . 896 . 894 . 892 . 890. 888 . 886 . (884 . 882 . 880. 877 . 875 . 872 . 869 . (867 . 864 . 86 1 . 858 855 852 849 (845 (824 838 825 831 827 824 820 8 16 812 633* 14 3*) 395 7 20 907 979 945 872 7 14) 472 120 690 208 626 . 948 .2 18) . 370 .473 . 502* .516* .231* .156* .626* ) .431*) .929* . 298* .692 . 98 1 . 226* . 274 * . 338* . 287* 39901 900 898 . 896 . 894 . 892 . 890. 888 . 885 . 883 . 88 1 . 878 . 876 . 873 . 870. 868 . 865 . 862 . 859 . 856 . 852 . 849 . 846 . (843 839 835 832 828 824 820 8 16 8 1 2 792* 051* 152 402* 380 239* 234 18 1 * 88 1 570 162* 725 . 170 .540 . 84 1 .052 .2 14 . 236 . 246 . 133 .968 .620* . 394 . 899* ) . 499* .931* . 346* . 563 . 785 782 838* 916* 39856. 856 . 856 . 855 . 854 . 854 . 853 . 852 . 851 . 850. 849 . 848 . 847 . 846 . 844 . 843 . 84 1 . 839 . 838 . (836. 834 . (832 . 830. 828 826 823 (821 8 18 816 (8 13 807 506* 130* 524* 999* 39 1 607 927* 936* 898* 730* . 727* . 360* .086* . 596 . 126* .601 .897* . 179* . 216* ) . 425* . 573*) . 372* . 263* .058* . 797* . 207* ) . 759* . 347* .624* ) (39858. 858 . 857 . 857 . 856 . (855 . 854 . 854 . 853. 852 . 850. 849 . 848 . 847 . 845 . 844 . 842 840 838 837 835 833 831 828 826 824 82 1 8 19 816 468* ) 468* 755* 165* 506* 699* ) 999* 075 036* .014* .898* . 730* . 437* .050* .625* .155*. . 446* . 768* .938* . 153 . 297* . 203 . 129 .968 . 786 . 339* .907 . 503* . 836* ro 00 I 804.034 804.524 TABLE III (CONTINUED) K = 9 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 001 1 0022 " 0P1 1 QP22 39849. 434 39852.236* 7 1 1 * 848 . 727 851.251 3984 1 . 011* 39843. (848 049* ) 850.256 839 . 487* 84 1 . 84 1 847 .050* 849. 162* 837 . 908* 940. 030* 846 086 848.049* 836 . 07 1 • 838 . 179* 844 . 956* 846 . 863 834 . 335* 836 . 2 16* 843 .828* 845.625* 832 . 330* 834 . 122* 844.155 830 . 358* 832 . 017 84 1 .281* 827 . 263 829. 849 839 .897* 841 .281 * 826 . 039 827 . 549 823 . 794 825 . 184 836 . 745 . 82 1 450 822 . 772 835 .098* 8 18. 999 820. 284 (816 . 357* ) 817 685 813 .834 " 815 .027 811 . 166 812 . 274 808 .4 17 809 . 470 805 . 573 806 . 605 802 .628 803 .6 15 799 .674 800 . 585 796 .518 797 . 428 793 . 366 794 . 188 ( 789 . 925) 790 .988 786 .722 787 . 526 783 . 367* 784 . 139 779 .807* 780 . 503 776 .2 19 776 . 966 SRO F 1 + F2 SR 1 F 1+F2 ASYMMETRY INDUCED 40158.837* 158.357 157.768 157.045 156.279* 155 . 2 1 1 154 . 156 152 . 727 151.229 149.580 147.685* 145.550 40167.552* 166. 174 164.740 162.971* 161.020 158.837* 156.279* 153.706 150.834 147.680* co co TABLE IV ASSIGNED RO I ATI ONA L LINES OF 1 HE A *H - X^'fO.OI BAND OF VO PI 01 R1 P2 02 R2 P3 03 8 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 9445.053 9443.808 9442.432* 944 1 . 9439 . 9438 . 9436 . 9434 . 9432 . 9430. 9428 . 9426 . 9424 . 9422 . 9420. 94 18 . 94 15 . 9413. 94 10. 9408 . 9405 . 9402 . 9399 . 9396 . 9393 . 9390 9387 9384 938 1 9378 9374 9371 9367 9364 9360 9356 9352 07 7 591 053 339* 673 846* 936* . 994* .963 . 883* .725* . 476* . 133 . 7 14 . 245 .705* .076 . 384 .600* . 787 .867 . 899 . 879* . 752 .577 . 333 .035* .641 . 193 .675 . 1 10 455* 743 965 9463.781* 9464.329 9464.857 9465.522* 9465.573* 9465 . 9464 . 9464 . 9464 . 9463 . 9462 . 9461 . 9460. 9460. 9459 . 9457 . 9456 . 9455. 9454 . 9452 . 945 1 . 9449 . 9447 . 9446 . 9444 9442 9440 9438 9436 9434 9431 9429 9427 9424 9422 94 19 94 16 94 14 94 1 1 484* 989* 642 051 383* 702 * 873* 968* 014* .029* .860* . 680* . 394* .010 . 588 .04 1 . 435 . 786 .052* .2 14 .319 . 36 1 . 324 .2 18 .042 . 796 . 482 .096 .644 . 120 . 535 . 869* . 154 . 366 9475 .676* 9476 .479* 9477 . 192* 9477.857* 9478 . 9478 . 9479 . 9479 . 9480. 9480. 9480. 9480. 9480. 9480. 9480. 9480. 9479. 9479 . 9478 . 9478 . 9477 . 9477 . 9476 . 9475 . 9474 . 9473 . 9472 . 9471 . 9470. 9469 . 9468 . 9466 . 9465 . 9463 . 9462 . 9460. 9458 . 9457 474* 989* 453* 817* 1 16* 343* 501 626* 626* 572* 343* 1 16* 817* 453* 989* .474* . 857* . 192* . 479* .676* . 779 . 838 .819 . 748 . 590 . 360* .058 . 708 . 275 . 78 1 . 220 . 587* .895 . 132 9459.359 9457.662 9455. 9454 . 9452 . 9450. 9448 . 9446 . 9444 . 944 1 . 9439 . 9437 . 9434 . 9432 . 9430. 9427 . 9424 . 9422 . 94 19 . 94 16 . 94 13 . 94 10. 9407 . 9404 9401 9398 9395 9391 9388 9385 938 1 9377 9374 9370 9366 9363 897 010 210* 217* 2 10* 150* 080 903 .537* . 343 . 997* . 560 .044 .502 . 883* . 199 . 535* . 649 . 773 . 834 . 808 . 777 .649 . 462 . 189 . 88 1 . 488 . 048 . 535* .964' . 325 . 603 . 856 .009 9479 .218 9478 . 9477 . 9477 . 9476 . 9475 . 9474 . 9473 947 1 . 9470 9469 9467 9466 9464 9463 9461 9460 9458 9456 9454 9452 9450 9448 9446 9443 944 1 9439 9436 9434 9431 9428 9426 9423 9420 94 17 572* 857* 04 2 164 195 189 095 922 674 360 977 521 .989 . 383* . 723 .014* .218 . 360 .426 . 434* . 382 . 237 .052* . 808 .491 .113 .667 . 157 .588 . 948 . 253 . 490* . 660 . 755 9493 . 9492 . 9492 . 9492 . 9491 . 9491 . 9490. 9490. 9489 . 9488 . 9488 . 9487 . 9486 . 9485 . 9484 . 9483 . 9482 . 948 1 9479 9478 9476 9475 9473 9472 9470 9468 9467 023 829 6 16 253 868* 4 14 920* 334* 693 .950* . 220 . 380* . 472* . 578 .483* .4 15* . 232 .006 . 723 . 358 . 932* . 503 .898* . 292* . 590* .870* .046 9462 . 9459. 9457 . 9454 . 9451 . 9448 . 9446 . 9443 . 9440. 9437 . 9434 . 9431 . 9427 . 9424 . 942 1 . 94 17 . 94 14 . 94 1 1 . 9407 . 9403 . 9400. 9396 . 9392 . 9389 9385 9381 9377 440 84 1 208 513 765 978* 052* 213 . 251 . 235 . 157 .023 . 851 . 644* .311 .932 .516* .067* . 538 .951* . 301 . 600 .844 .018 . 143 .213 . 203 9499 . 9498 . 98 . 9497 . 97 . 9496 . 9494 . 9493 . 9492 . 9490. 9489. 9487 . 9486 . 9484 . 9482 . 948 1 . 9479 . 9477. 9475. 9473 . 947 1 . 9469 . 9467 . 9465. 9463. 9460. 9458 9456 9453 945 1 9448 9446 9443 9440 9437 9435 9432 9429 9426 634* 535) 348) 349) 147) 124 847 540 169 751 272* 782* 195* .581 915 . 180 .411 .573 .676 . 750 .748* . 707 .606 .448* . 238* .968* .634 . 247 .8 12 .317* . 764 . 150* .492 . 773 .992 . 145 . 255 , 303 . 295 I ro 00 i TABLE IV (CONTINUED) P 1 01 R1 P2 02 R2 P3 03 5 1 9349 . 124 9408 . 512 9455 . 3 13 9359 . 144 52 9345 . 2 18 9405 . 596 9453 . 403 9355 . 179 53 934 1 . 248 9402 . 600* 945 1 . 458 9351 . 164 54 9337 . 209 9399 . 548 9449 . 435* 9347 . 089 55 9333 . 103 9396 . 427 9447 . 330* 9342 949 56 9328 . 940 9393 . 239 9445 . 170 9338 .733* 57 9324 . 708 93B9 . 992 9442 . 940 9334 . 482 58 9320 4 12 9386 . 677 9440 648 9330 .137* 59 93 16 06 1 • 9383 . 295 9438 32 4* 9325 . 763* 60 93 11 . 634 9379 . 847* 9435 887* 932 1 . 306 61 9307 . 134 9376 . 340 9433 386 93 16 . 793 62 9302 . 600 9372 . 77 1 9430 . 837 63 9297 . 979* 9369 . 130 9428 .2 12 64 9293 . 298 9365 . 424 9425 . 497 65 9288 .569 936 1 658 9422 . 785 66 9283 . 764 9357 8 1 1 * 94 19 . 979 67 9278 .906 9353 937 68 9273 . 976 9349 976 69 9268 . 980 9345 925 70 9263 . 925 934 1 . 849 7 1 9258 . 8 10 9337 .7 15 72 9333 . 503 73 9329 . 220* 74 9324 . 885 75 9320 . 490 76 93 16 . 024 77 931 1 . 484 78 9306 . 899 79 9302 . 240 80 9297 .529 81 9292 . 724 82 9287 .905 83 9282 . 998 84 9278 .036 85 9273 .010 86 9267 .920 87 9262 . 779 88 9257 .507 94 14 . 8 18 9465 . 175 9373 . 172 9423 . 236 94 1 1 . 8 12 9463 . 238* 9369 . 059 9420. 106 9408 . 7 19 946 1 . 240 9364 . 889 94 16 . 869* 9405 . 596* 9459 . 180 9360. 665 94 13 . 659 9402 . 396 9457 . 04 7 9356 . 385 94 10. 379 9399 . 145 9454 . 856 9352 . 04 3 9407 . 024 9395 . 8 12 9 152 . 588* 9347 . 637 9403 . 604 9392 . 429 9450 . 273* 9343 180 9400 . 126 9388 . 982 9447 . 863* 9338 663 9396 . 600* 9385 . 467 • 9445 437 9334 .087* 9393 001 938 1 . 901 9442 .940* 9329 . 456 9389 348 9378 . 277 9440 . 36 1 • 9324 . 756 9385 643 9374 573 9437 .7 12 9320 .012* 938 1 .901 * 9370 .820 9434 .997* 93 15 . 194* 9378 .035 9366 . 984 9432 . 233* 9310 . 325* 9374 . 158 9363 . 124 9429 . 386 9305 .433* 9370 . 209 9359 . 184 9426 . 454 * 9300 .4 15* 9366 . 201 9355 . 179 9423 . 490* 9295 . 368 9362 . 139 935 1 . 123 9290 . 247 . 9358 .013 9346 . 994 9353 .827 9342 .808 9349 . 565 9338 . 563 9345 . 277 9334 .248 9340 .922 9329 . 879 9336 .500 9325 . 454 9332 .017 9320 .963 93 16 . 4 10 93 1 1 . 793 TABLE IV (CONTINUED) R3 P4 04 R4 PQ13 SR32 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 7 1 72 73 74 75 76 77 78 79 80 8 1 82 83 84 85 86 87 88 ' 89 90 91 9478 . 9476. 9474 . 9472 . 9470. 9467 . 9465 . 9463 . 9460 9457 9455 9452 9449 9447 9444 944 1 859 779 642 422 . 167 .836 . 448* .006 .511 .860* .9 13* . 588* .857 .071* . 2 14* .311* 9386 . 9382 . 9377 . 9373 . 9368 . 9364 . 9359 . 9354 . 9350. 9345. 9340. 9335 9330 9325 9320 9315 394 035 ,667* . 172 . 694 . 1 10* . 534 .888 . 189 . 365* .672* .757 .827 .819 . 823 .761 9438 9435 9431 9428 . 9424 . 9421 . 94 17 . 9413. 94 10. 9406 . 9402 . 9398 . 9394 . 9390. 9386 . 9382 . 9378 . 9374 . 9370. 9365 . 936 1 . 9357 9352 9347 9343 9338 9334 9329 9324 9319 9314 9309 9304 9299 9294 9288 9283 9278 9272 9267 9261 425 104 705 260 770 2 13 620 962 254 4B6 664 79 1 869 879* 844 762 609 407 . 143 .828 . 447 .012 .527 . 977 . 379 . 733* .000 . 228* . 393 . 494 . 542 .530 .461 . 336 . 143 .901 . 597 . 224 . 797 .312 . 786 9491 . 9489. 9486 . 9484 . 9482 . 9479. 9476 . 9474 . 947 1 . 9468 . 9466 . 9463 9460 9457 9454 945 1 9448 9444 944 1 9438 9435 9431 9428 584 272* 908 483* 031 453* 932* . 295* .619 . 870* .090 . 238* . 345* .4 17* .361* .291* . 164* .980* . 740* .442* 088* 676* 207* 9421.117* 94 17.452 9413.773* 9406.190 9402.337* 9398.380 9394 . 382 9390.303 9386.202 9382.035* 9377.794 4 9448 . 699 5 9448 . 303 6 9447 , 863* 7 9447 . 330* 8 9446 . 73 1 9 10 9445 . 276 1 1 9444 .427 12 9443 . 497* 13 14 15 16 17 18 19 20 2 1 22 23 24 9531 953 1 9532 9533 9534 9535 9535 9536 9537 9537 9538 .01 1 .956 .856 .688 .481 .213 .880* .525 . 105 .649* . 104 TABLE IV (CONTINUED) R3 P4 04 R4 QP43 RQ43 SR43 4 5 6 7 8 9 10 1 1 12 13 14 15 16 9515 . 097 17 9514 . 934 18 9514. 842 19 9514 . 596 20 9514 . 365 2 1 9514. 067 22 9513. 7 16 23 9513 . 335 24 9512 . 830 25 9512. 313 26 951 1 . 738 27 95 1 1 . 148* 28 9510. 446 29 9509. 703 30 9508 . 911 9467 . 217* 31 9508 . 062 9463 . 781* 32 9507 . 159 9460. 348* 33 9506 . 199 9456 . 809* 34 9505 . . 178 9453 . 282* 35 9504 . . 100 9449. 685 9521 . 901 • 36 9502 .964 9446 . 052 9520. 250* 37 9501 .772 9442 . 436* 9480. 1 16* 9518. 7 18* 38 9500 .521 9438 . 691 9477 . 400* 9517 . 073* 39 9499 . 205 9434 . .928* 9474 . 642 9515. . 406* 40 9497 .835 9431 146 9471 . 922 9513. 716* 4 1 9496 .408 9427 349 9469 . 112 951 1 . 923* 42 9494 .918 9423 .490* 9466. 306 9510 . 126* 43 9493 . 376 9419 .535* 9463 . . 383* 9508 . 293* 44 9491 . 777 94 15 . 536 9460 .445* 9506 . 33 1 45 9490 . 103 94 1 1 . 499 9457 . 464* 9504 . 387* 46 9488 . 378 9407 . 446 9454 . 426* 9502 . 375* 47 9486 .591 9403 . 330 9451 .317* 9500 . 323 48 9484 . 752 9399 . 199 9448 . 172 9498 .2 13 49 9482 .838 9395 .007 9444 .985 9496 .028* 50 9480 .877 9390 . 728 944 1 . 746 9493 . 844* 9552 . 88 1 9553 . 306 9546 . 923* 9553 . 904 9546. 320* 9554 . 335 9545 . 672* 9554 . 829* 9545 . 003* 9555 . 197* 9544 . 291* 9555 . 563* 9532 . 315 9543 . 567 9555. 897 9530. 485* 9542 . 805 9556 . 205 . 9528 . 632 9542 . 020 9556 . 476 9526 . 736 9541 . 188* 9556. 712* 9525 . 020) 9540. 533) 9556 . 930) 24 . 806 ) * 40. 311)* 57 . 183) 9539 . 649 9557 . 314 952 1 . 083 9538 . 740 9557 . ,465 9519. .094 9537 . 793 9557 . .583* 95 17 .073* 9536 . . 84 1 9557 . 705* 9535 . ,880 9557 . 795* 9512 . 963* 9534 . .857 9557 .823* 9510 .867 9533 .818 9557 .823* 9508 .741 9532 .750 9557 .823* 9506 .581 9531 .644 9557 .795* 9504 . 387 9530 .485* 9557 . 705* 9502 . 169* 9529 .314 9557 .583* 9499 . 901 9528 .12 1 9557 . 428 9497 .616* 9526 .880 9557 .231 9525 . 597 9556 .997 9524 . 274 9556 .712* 9522 .908 9556 .404 9521 .514 9556 .057 9520 .077 9518 .593 9517 .073* 95 15 .501* 9513 .899* 9512 . 223* 9510 .534* I ro ro i TABLE IV (CONTINUED) (0.1) BAND OF VO A-X SYSTEM p1 Q1 R1 p2 02 03 R3 R043 SR43 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 8434 8432 8431 8429 8427 8425 8423 842 1 84 19 8417 84 14 84 12 .525* .863* .115 .321* .481* . 52 1 .567 .507 . 354* . 136 .913* . 544 8407 . 8405 . 8402 . 8399. 8397 . 8394 . 839 1 . 8388 . 8385 . 8382 . 8379 . 8376. 8373 8370 8366 8363 787 228* .670* . 992 .276* .513 . 7 10 .844* .855 .857 . 777 .656 . 467 . 208* .915 .533 846 1 8462 8462 8463 8463 8464 8464 8464 8464 . 506 . 320 .959 . 595 .987 . 292* .613* .803* .919* 8464 . 8464 . 8464 . 8464 . 8463 . 8463 . 8462 . 8462 . 846 1 . 8460. 8459. 8458. 8457 . 8456 . 8455 . 8454 . 8452 . 8451 . 8450. 8448 . 8447 . 8445 . 8443 . 844 1 . 8440. 8438 . 8436 . 8434 . 8432 . 8430. 8427 . 8425. 8423 . 8420 . 84 18 919* 855 6 13 292* 840 421* 815* 084* 374* 532* 660* 726* 703* 629 449 250 95 1* 563 131 638 049 450 744 988 . 163 . 273 . 320 .312 . 24 1 . 107 .908 .658 . 336 . 96 1 . 5 16 8477 8477 8478 8479 8479 8479 8480 8480 8480 8481 848 1 8481 8481 8481 8480 8480 8480 8479 8479 8479 8478 8477 8477 .084 .814* . 389* .046* .541* .963* .331* .658* . 859 + .059* . 167* . 167* . 167* .059* .859* .658* .331* .936* .541* .046* . 540 .896 . 229 8475 .677 8474.812 8473.879 8544 . 072* 8553 . 232 8543. 503* 8553 . 720* 8513. 337* 8542 . 869 8554 . 122 8513. 525 8542 . 216 8554 . 542 8513 . 756 854 1 . 537 8554 . 940* 8513. 980* 8540. 840 8555 . 340 8514 . 053* 8540. 8539 . 39 . 8538 . 1 12 560)* 365) 798 8555 . 8555 . 56 8556 639 959) 162 ) .449* 8514 . 250* 8537 . 8537 .999 192 8556 . 8556 . 724 993 8514 . 053* 8536 . . 370 8557 . 238 8475 . 833* 8513. 980* 8535 .543 8557 .439* 8475 . 026 8490. 470* 8513. 865 8534 .672 8557 .64 1 8474 . 157 8489 . 268* 8513 . 712 8533 . 786 8557 .804* 8473. 223 + 8487 . 849* 8513 . 413 8532 .876 8557 .954* 8472 . 2 14 8486 . 513* 8513 122 • 8531 .937 8558 .079* 847 1 . 153 8485 . 040 8512 . 779 8530 .988 8558 .191* 8470. 004 8483 . 568 8512 . 406 8529 .946 8558 . 245* 8468 . 819 8482 . 041 851 1 .969 8528 .969 8558 . 245 + 8433 . 667* 8467 . 559 8480 . 446 851 1 . .447* 8527 .933 8558 . 245* 843 1 . 333* 8466 . 220 8478 . 824 8510 .936* 8526 . 799 8558 . 245* 8429. 051 8464 . 855 8477 . 152 8510 . 3B4 8525 . 732 8558 .191* 8426 . 56 1 * 8463 . 421 + 8475. 437 8524 .611* 8558 .079* 8424 . 102 8461 . .918* 8473 . .668 8509 .075 8523 .4 19 8557 .945* 8421 . 635 8460. . 353 847 1 . 847* 8508 . 329 8522 .211* 8557 . 724 84 19 . 000 8458 726* 8470. .004* 8507 . 557 8520 .976* 84 16 433 8457 .053 8468 . 075 8506 . 725 84 13. .759* 8455 .312 8466 .112 8505 .850 8410 . 979 8453 .513 8464 .098 8504 . 909 8408 . 184 8451 .675 8462 .034 8503 .915* 8405 .313 8449 .673 8459 .929 8502 .894 8402 . 448* 8447 . 776 8457 . 779 8501 .811 8399 .438 8445 . 734* 8455 . 565 8500 .665 8396 .420 8443 .656 8453 . 326 8499 . 478 8393 . 343 8441 .510 8451 .014 8498 . 230 8390 . 246* 8439 .315 8448 .638* 8496 .933 8387 .006 8437 .062 8446 . 261 8495 .604 8383 . 782 8434 .750 8443 .806* 8494 .17 1 8380 .458 8432 . 390 844 1 . 306* 8492 . 729 8377 .098 8429 . 956 8438 . 748 8491 .237 8373 .67 1 8427 8424 . 48 1 . 948 8436 8433 . 145 . 486 8489 8488 .672 .069 I ro CO CO TABLE IV (CONTINUED) 01 02 03 51 52 53 54 55 56 57 58 59 60 6 1 62 63 64 65 66 84 16 . 033 8422 . 356 8430 . 78 1 84 13 . 479 84 19 . 694 8428 . .033* 84 10. 854 84 17 . 016 8425 2 15 8408 184 84 14 253 8422 . 356 8405 . 466 84 1 1 454 84 19 . 444 8402 670* 8408 579 8399 8 19 8405 . 669 8396 . 900 8402 . 670* 8393 .945 8399 .654 8390 . 943 8396 . 560 8387 .855 8393 . 442 8384 .722* 8390 . 246* 8381 . 622 8387 .006 8378 . 287 8383 . 782 8374 . 797 8380 . 458 8371 . 622 8377 . 098* I ro to 4^ 

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