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The high resolution spectroscopy of manganese oxide Chandrakumar, Thambirajah 1989

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T H E HIGH RESOLUTION SPECTROSCOPY OF MANGANESE OXIDE by Thambirajah Chandrakumar B. Sc.(Hons), University of Peradeniya, Sri Lanka 1984 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY We accept this as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1989 © T.Chandrakumar, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, t 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 £ H E f/l l S 7 ^-T The University of British Columbia Vancouver, Canada Date K A o w c ^ V ^ §3 DE-6 (2/88) 11 Abstract This thesis reports studies of the electronic spectrum of gaseous MnO. The (0,0) band of the A 6 X + - X 6 I + electronic transition of MnO was recorded by intermodulated laser-induced fluorescence over the range 17770 - 17970 cm - 1 . The hyperfine structure caused by the 5 5 M n nucleus (I = 5/2) is almost completely resolved. Internal hyperfine perturbations between the F3 and F4 electron spin components (where N = J - 1/2 and N = J + 1/2, respectively) occur in the ground state of MnO. These are caused by hyperfine matrix elements of the type AN = AF = 0, AJ = ± 1. Extra lines obeying the selection rules AJ = 0, ± 2 are also induced. Therefore, pCj34, RCj43, p 0 4 3 and R S 3 4 branches appear in the spectrum although they are not allowed in parallel transitions. The reason for the great complexity of the spectra is the occurrence of a large avoided crossing near N = 26 in the A 6 X + v = 0 level by another electronic state, B 6 Z + , with the same multiplicity and symmetry. The perturbation between the A 6 I + and B 6 X + states arises from electrostatic interaction. The selection rules for electrostatic perturbations are AJ = AS = AA = ACl = 0. The perturbing state B 6 Z + state has a considerably longer bond length so that it must come from a "charge transfer transition", possibly by electron transfer either from the 3n to the 4rt orbital or from 80 to 10a. However, the A 6 Z + state has only a small bond length change compared to the ground state so that it comes from a "Valence state transition". The Fermi contact constant b was found to be negative for the A 6 Z + state and this confirms the electronic configuration as being (8a2 3n4 ) 152 4K 2 10a1. The ground state is free of perturbations, except for the internal hyperfine perturbations, and is in nearly pure case (b) coupling. Various satellite branches which iii were observed in the B - X transition confirm the case (a) nature of the B 6 Z + state at low N. The spacing between the main branches and the satellite branches gives values for the spin-spin parameter X and the spin-rotation parameter y of the ground state. iv Table of contents Abstract ii List of Tables vi List of figures vii Acknowledgement ix Theory 1.1 Introduction 1 1.2 Angular momentum 3 1.3. Hund's coupling cases 5 1.3.1 Case (a) 5 1.3.2 Case(b) 7 1.4 Selection rules 9 1.5 Perturbations 10 1.6 Electrostatic Perturbations 13 1.7 Matrix elements 16 1.7.1 Rotational and fine structure Hamiltonian 16 1.7.2 Nuclear Hyperfine structure Hamiltonian 21 1.7.3 Electric Quadrupole Hamiltonian 24 EXPERIMENTAL DETAILS 2.1 Description of the source 27 2.2 Laser-induced Fluorescence 27 V 2.3 Intermodulated Fluorescence 28 2.4 Wavelength Resolved Fluorescence 33 ROTATIONAL AND HYPERFLNE STRUCTURE ANALYSIS OF MnO 3.1 History 38 3.2 Description of the spectra 38 3.3 Ground state energy level pattern 41 3.4 Rotational analysis 43 3.5 Satellite branches 49 3.6 Hyperfine structure analysis 51 3.7 Internal Hyperfine Perturbations 61 DISCUSSION 4.1 Electron configuration 67 4.2 Future work 72 BIBLnOGRAPHY 74 APPENDIX 76 <, vi List of Tables 3.1 F6 (A-X) transitions 55 3.2 F i (A-X) transitions 56 4.1 Molecular constants for the X 6 2 + ground state of MnO 69 4.2 Ground states of the 3d transition metal monoxides 72 vu List of Figures 1.1 Vector diagram of Hund's coupling case ap 6 1.2 Vector diagram of Hund's coupling case bpj 8 1.3 Avoided crossing of two perturbing states 12 1.4 Diabatic potential curves of A 6 L + state and B 6 Z + state cross (solid lines) and cause the adiabatic curves (dotted lines) to avoid crossing by 2H e. 14 2.1 Velocity distribution curves of lower state. Excitation of molecules creates "holes" in the lower state. These holes are known as "Bennet holes". 30 2.2 Two Bennet holes which converge at zero velocity to form a Lamb dip in the profile of fluorescence intensity versus laser tuning frequency. 31 2.3 Schematic diagram of the intermodulated fluorescence experiment used in this laboratory. 32 2.4 Energy level patterns for wavelength resolved fluorescence. 34 2.5 Wavelength-resolved fluorescence spec rum of MnO. 36 3.1 Broad band laser excitation spectrum of the MnO A 6 Z + - X 6 Z + (0,0) band. 40 3.2 Electron spin-spin splitting of a rotational level N of a 6 Z electronic state in case (b) coupling (X is assumed to be positive). 42 3.3 MnO: A 6 E + - X 6 Z + (0,0); the six R(26) lines. 44 3.4 Low J level structure of the MnO A 6 Z + , v=0 and B 6 Z + , v=0 levels. 46 3.5 Figure 3.5 Upper state term values of the Fi and electron spin components of the MnO A 6 Z + , v=0 and B 6 X + , v=0 levels plotted against N(N+1). The term values have been scaled by subtraction of a quantity 0.45N(N+1) cm-1 to magnify the details. 47 3.6 Energy level diagram indicating the satellite branches induced by excitation of the F6 spin component of the upper state. 50 3.7 Hyperfine patterns in the six electron spin components of the (0,0) band of the A6Z+ - X 6 2 + transition of MnO. 53 3.8 Hyperfine patterns in the Ri(24) and R6(24) lines of the A 6 E+-X 6 2; + (0,0) band of MnO. 54 3.9 (a) Upper state term values and (b) effective Fermi contact parameters of the Fi electron spin component of the v=0 level of the A 6 Z + state of MnO. The term values have been scaled by subtraction of 0.47N(N+1) cnr 1 to magnify the details. 58 3.10 (a) Upper state term values and (b) effective Fermi contact parameters of the F6 electron spin component of the v=0 level of the A 6 L + state of MnO. The term values have been scaled by subtraction of 0.46N(N+1) cnr 1 to magnify the details. 59 3.11 A portion of the A6Z+ - X6Z+ (0,0) band of MnO; among the six components of the P(29) line is a line with zero hyperfine splitting, which has been identified as a P5 (19) transition. 60 3.12 Electron spin structure of the v=0 level of MnO, X 6 L + . The quantity plotted is the electron spin contribution to the total energy as a function of the rotational quantum number N. 62 3.13 Mechanism of an internal hyperfine perturbation. 64 3.14 Internal hyperfine perturbation in the R(26) lines of MnO A 6 Z + - X 6 Z + . 65 4.1 The relative energies of the molecular orbitals of MnO, formed from the linear combinations of the atomic orbitals of Mn and O atoms. 68 4.2 Energy Level diagram of MnO [22]. Thick lines indicate the calculated states. Dashed line indicates the experimental A 6 L + state. 71 ix Acknowledgement I would like to express my sincere gratitude to my research director, Professor Anthony J.Merer, for his guidance, invaluable advice and constant encouragement throughout my work. I am very much grateful to Dr.Y.Azuma for his assistance and encouragement with the Manganese oxide project. I would like to thank Dr.A.Adam, Dr.V.Srdanov and Mr.C.Chan for their useful and interesting discussions. Thanks also go to my labmates. X To my parents Mr&Mrs.Thambirajah 1 CHAPTER 1 1.1 INTRODUCTION The energy levels of a molecule are given by the eigen values of the time-independent Schrodinger equation. H\|/ = Ey (1.1) where H is the total Hamiltonian operator, \|f is the eigen function associated with a stationary state and E is the energy of this state. The Hamiltonian H can be divided into parts, H = H Q + Hrot + Hsp + Hhfs (1.2) In this equation (1.2), Ho represents the electronic and vibrational Hamiltonian of the non-rotating molecule, H r o t represents the rotational motion of the molecule, and H s p represents the electron spin fine structure of a high multiplet electronic state, which can be accounted for by two interactions, spin-spin (parameter X) and spin-rotation (parameter y); Hhfs contains all the electric and magnetic terms for nuclear spin that cause hyperfine structure. The non-relativistic terms in H, namely Ho+H r o t, can be separated by the Born-Oppenheimer approximation into a nuclear and an electronic part. In this approach, it is assumed that masses of the nuclei and electrons cause their relative velocities to be so different that the electrons adapt instantaneously to the positions of the nuclei. This is equivalent to factorizing the total wave function \|/ into a nuclear part which depends only on the nuclear coordinates, and an electronic part that depends on the coordinates of both electrons and nuclei. The Hamiltonian Ho is then the sum of an electronic part and a nuclear part, Ho = Helec+ Hvibration (1.3) 2 As a result of the Born-Oppenheimer approximation, two types of quantum numbers can be distinguished. (1) electronic state quantum numbers, corresponding to solutions to the electronic part of the Hamiltonian. (2) nuclear motion quantum numbers, which can be separated by purely classical mechanical arguments into those for the vibration and rotation of the molecule in a particular electronic state. It is impossible to solve equation (1.1) analytically. In practice, one chooses a convenient finite basis set <|>j and expands in terms of N V i = X a j < l ) j ( L 4> j=l then reduces the solution of equation (1.1) to the problem of finding the roots of the secular determinant. I H - E5ij I = 0 (1.5) where H is the matrix of the Hamiltonian, whose elements, in the Dirac notation, are given by Hjj = <(|>i I H I <)>j> (1.6) The choice of the complete basis set, <|)j, j=1,2,3, n is such that the diagonal elements of the Hamiltonian matrix most nearly represent the energy levels of the system, in other words correspond most closely to the physical situation. In rotational problems, degenerate perturbation theory [1] is used to take account of matrix elements linking a particular vibronic state with nearby vibrational or electronic states. This leads to an effective Hamiltonian operator [2] which operates only within the rotational subspace of the vibronic state of interest. Then a matrix representation of the effective Hamiltonian can be calculated. The final step is the diagonalisation of the Hamiltonian matrix and the determination of the eigen values and eigen functions. 3 1.2 Angular momentum Angular momentum operators appear in the full mathematical form of the Hamiltonian, and can be divided into two types, internal and overall angular momenta. Internal angular momenta result from the particle motions within the molecular frame, while overall angular momenta correspond to the molecular rotation. In quantum mechanics P is defined as an angular momentum operator if it obeys the commutation rules [3]: A A . A [P x , P y ] = i t P z (1.7) A A A A where P x , P y and P z are the cartesian components of the operator P in a fixed coordinate system. When referred to a molecule fixed coordinate system the internal angular momenta such as I, L and S obey commutation rules analogous to (1.7), but the total angular momentum J, which is seen by photons that are directed towards detectors in space-fixed coordinates, is quantized in both the space- fixed and the molecule-fixed coordinate systems; it obeys the "normal" commutation rules of (1.7) in space-fixed coordinates but has the "anomalous sign of i " [4] in molecule-fixed coordinates. A A i A [ P x , P y ] = - i t P z (1.8) A Angular momenta are important in quantum mechanics because their squares, P 2 and their z-components, P z , commute with the Hamiltonian operator H; therefore they possess simultaneous eigen functions. As a result the matrix elements of the Hamiltonian operator can be calculated using angular momentum basis functions. For this purpose it is convenient to write the Hamiltonian operator in terms of angular momentum operators and their components rather than differential operators. The basis functions are defined in terms of quantum numbers relevant to the individual angular momenta rather than as explicit wavefunctions. In general, for any conserved angular momentum P, there are well known relations [3], P21 P M p > = P(P+1) n 2 IPMp > PzIPMp> =MplilPM p> (1.9) The ladder operators P± for an angular momentum P have the cartesian form A A A P±=P x ±iPy (UO) A A P + has the property of transforrning IPM p > into a multiple of IPMp+i>. Similarly P_ has the property Of transforming IPM p > into a multiple of IPMp.i >. In a molecule-fixed coordinate system the internal angular momenta such as I , L and S obey A 1/2 P + IPM p > = exp(i<j>) [ P(P+l)-Mp(Mp+l) ] IPMp+i > ( U i ) where exp(i<|>) is an arbitrary phase factor, usually chosen as +1. However, in the molecule-fixed axis system, those angular momenta such as J and F, which are also quantized in the space-fixed axis system, have the "anomalous sign" [3] to reverse the sign of the equation. A 1/2 P. I P M p > = exp (i<|)) [ P(P+l)-Mp(Mp-l) ] IPMp.i > ( u 2 ) A A The dot product of two angular momentum operators Pi and P2 is given by [3] A A A A A A A A Pl-P2 = PlZP2z + d/2) (Pl + P 2- + P i - P 2 + ) (1.13) The addition of angular momenta Ji and J2 to form J results in the coupled eigen function [4] IJM > J, -J„+M |JM> = 2, (-D 72J+T Jl J 2 J Mj M 2 - M IJ 1M 1>U 2M 2> (1.14) where IJ1M1 > and IJ2M2 > are the uncoupled eigen functions. The term in brackets is a coefficient called a Wigner 3-j symbol, and must satisfy the requirement that Mi+M 2 =M and IJ1-J21 ^  J ^ (J1+J2) The derivation of expressions describing the coupling of angular momenta, particularly those for the magnetic hyperfine and quadrupole hyperfine interaction, is often best approached using irreducible spherical tensors, since first rank tensor operators transform like angular momentum vector operators. 5 best approached using irreducible spherical tensors, since first rank tensor operators transform like angular momentum vector operators. 1 i A A 1 A T ±j (J) = T ^ | ( J X ± LJy) , T Q(J) = J Z (1.15) 1.3 Hund's coupling cases The Hamiltonian for a molecule can be divided up into two terms H=Ho +Hi (1.16) where Ho contains all the electronic and vibrational terms, apart from the spin-dependent interactions, and Hi is the rotational and spin Hamiltonian, given by Hi = Hrot + Hso + Hsr + Hss +Hhfs (117) In working out rotational problems we are not concerned with the form of Ho or its eigen functions. For the rotational and spin fine structure resulting from Hi we must choose suitable basis functions, which depend on the influence of the rotational and electronic motions on each other. The choice is usually that which gives the most nearly diagonal matrix representation for H, corresponding most closely to the physical situation. The diagonal matrix elements can be used as a working approximation to the real energy levels. But sometimes it may be an advantage to use a basis where the calculation of the matrix elements is easy but the matrix is far from diagonal . The choices of angular momentum basis functions for molecular problems in field-free space are called the Hund's coupling cases. Hund distinguished five types of coupling, labelled case(a) to case(e) [6,7]. We shall only need to describe the first two in this thesis. 1.3.1 Case (a) In Hund's case(a) coupling it is assumed that the interaction of nuclear rotation with electronic motion (spin as well as orbital) is very weak, whereas the electronic motion itself is coupled very strongly to the internuclear axis. Figure (1.1) is a vector diagram Figure 1.1. Vector diagram of Hund's coupling case ap 7 representing Hund's coupling case(a). Both the total electronic orbital angular momentum L and the total spin angular momentum S are strongly coupled to, and precess rapidly about the internuclear axis (the z axis) with constant projections L z and Sz. L and S are coupled to the nuclear rotational angular momentum, R, to form the total angular momentum J excluding the nuclear spin . J = S + L + R (1.18) The vectors S, L , R, J and their z components L z and S z are represented by respective quantum numbers S, L, R, J , A and Z in the figure (1.1). The projection of the angular momentum J along the molecular axis is represented by the quantum number Q, where Q = A + Z (1.19) The quantum number S and the quantum number J are integral or half-integral depending on whether there are even or odd numbers of electrons present in the molecule. Also the quantum number J cannot be smaller than the corresponding component quantum number Q , i.e. J > Q. The quantum number Z can take the 2S+1 values ranging from S.S-l, -S. In the usual spectroscopic nomenclature, the spin multiplicities (2S+1) are written as left superscripts to the electronic orbital angular momentum symbols, which are labelled as Z, n , A, O,... [7] corresponding to the A values of 0,1, 2, 3 The basis functions in Hund's case (a) are defined in terms of the good quantum numbers described above, and are written as IA; SZ; JQ> where the symbol A refers to the orbital angular momentum of the electronic state. 1.3.2 case(b) In Hund's coupling case(b) the spin angular momentum S is only weakly coupled to the internuclear axis. Case(b) coupling applies in Z states where the spin angular momentum is weakly coupled to the internuclear axis by second order spin-orbit 8 Figure 1.2. Vector diagram of Hund's coupling case bpj 9 interaction. Even if the orbital angular momentum is non-zero, the electron tends to uncouple from the intemuclear axis with increasing rotation. Spin uncoupling [8] takes place when BJ >A for A ^ O cases BJ > A«ff (yet to be defined in section 1 . 7 . 1 ) for A = 0 , Z states. ( 1 2 0 ) Therefore, given large enough J values any state in case(a) coupling will uncouple towards case(b) because of spin-uncoupling effects. A basis function for case(b) is designated as INASJ> where the total angular momentum J is given as [ 7 ] R + L = N ; N + S = J ( 1 . 2 1 ) Figure ( 1 . 2 ) represents the case(b) coupling vector diagram. Because X is not a good quantum number in Hund's case(b), the quantum number Q. is also not defined. If the nuclear spin is included, it is very unlikely that the nuclear spin will be strongly coupled to the intemuclear axis because of the small size of the nuclear magnetic moment. Then the coupling schemes ap or bpj are expected, where PJ indicates that the nuclear spin is not coupled to the intemuclear axis but to the angular momentum J. The total angular momentum F is given by F = J + I ( 1 . 2 2 ) The basis functions for case ap and bpj are designated by IA; SE; jnLF> and INASJIF>, respectively [ 8 , 9 ] , where the quantum number F takes the values IJ-II, IJ-I+ll , (J+I). 1.4 Selection rules An electronic transition occurs between states only if there are non-zero matrix elements of the electric dipole moment operator ji., which allows the interaction with electromagnetic radiation. The electric dipole moment for a total of N atoms is given by [ 7 ] 10 N ^ = Xeiri (1-23) i=l where ej is the effective electric charge on the i * particle, which has the coordinate ri. An electronic transition can occur only if the electronic transition moment Rnm is non-zero. If the wavefunction can be factorized into a product of electronic, vibrational and rotational wavefunctions, the transition moment becomes Rnm = <en I ri° I em > <vn lvm> < r n I <t>zg lrm> (1.24) where <t>zg is the direction cosine between the space-fixed axis system of the photons and the molecule-fixed axis system. The matrix elements of (|>zg give the rotational selection rules: AJ =0, ± 1 , except AJ = ±1 only, when AA=0 and A'=A" =0 . Since the hyperfine interactions split a rotational level with quantum number N into several hyperfine energy levels with different F quantum numbers, each rotational transition in fact consists of a number of hyperfine transitions whose selection rules generally follow those for the rotational transitions: AF = AN = AJ = 0, ± 1 1.5 Perturbations An irregularity in the vibrational and rotational structure of an electronic state, where the standard formulae are not obeyed, is called a perturbation. In diatomic molecules perturbations arise from terms neglected in the Born-Oppenheimer approximation. If two levels belonging to two different electronic states have their energies close together, then these two states perturb each other. The energy matrix for the interaction of the two states A and B can be written as: 11 1 A> IB> ' E A , J H 1 2 H' = H1 2 =41 (1.25) where E A and E B are the unperturbed energy levels of states A and B and H12 is the interaction matrix element. The energy values of the perturbed states Ei and E2 are obtained by diagonalising the matrix H'. 12 (1.26) E, =• &4 ( E A - E B ) (1-27) l12 The result is always in the sense of "repulsion"; the higher level is displaced upward and the lower downward by an equal amount. The combination difference relations (yet to be defined in section 3.4) must hold exactly even if strong perturbations are present, and therefore, analysis is always possible in principle. The eigen functions belonging to the two perturbed states can be written as = c IB > + s IA > (1.28) Y2 = s l B > - c l A > (1.29) where c and s are the cosine and sine of the rotation angle. As a result of mixing of the eigen functions, the perturbed level assumes some of the properties of the perturbing level, and vice versa. One of the most important consequences of the mixing, besides the shifting of the level positions, is the appearance of extra lines with intensity irregularities in the vicinity of the crossing region. Extra lines are induced, involving states which otherwise carry zero or very small oscillator strength of their own for emission to the ground state. There are two types of perturbations, which can be classified by the J-dependence of H12. If H12 is independent of J, the perturbation is called "homogeneous1, and the selection rule Figure 1.3. Avoided crossing of two perturbing states 13 on Q is AQ = 0. The selection rule AD = ± 1 implies a "heterogeneous" perturbation. The important feature of such perturbations is that the interaction matrix element depends on J (usually approximately proportional to J). If the unperturbed states actually cross each other, say at at J = Jc, the energy levels are E i = E A + H42 and E2 = E ^ - H12. As shown in Figure (1.3), the separation of the two perturbed levels is 2H12 at the avoided crossing point. Dotted and thick lines in Figure 1.3 represent the unperturbed and perturbed states respectively. At the crossing point, the intensities of the main line and the extra line become equal and the lines have minimum energy separation. (i.e) I Emain -E e x t ra ' =21 Hi 21 (1.30) Continuing to higher J values the regular transition fades away entirely and the other transition becomes the main one and carries all the intensity. By means of the perturbed eigenfunctions, we can calculate the intensities of perturbed lines. It can be shown that the intensity of the unperturbed line is reduced compared to what it would be in the absence of any perturbation[ll]. The sum of the intensities of each pair of perturbed lines is always equal to the unperturbed line intensity. In this thesis, electrostatic and internal hyperfine perturbations [12] (see section 3.7) will be discussed. 1.6 Electrostatic Perturbations The principal term left out in the Born-Oppenheimer approximation results from the nuclear kinetic energy operator, T N , acting on the electronic part of the wavefunction. T N acts as a one electron operator. The potential energy curves for the interacting electronic states then have to be drawn as if they avoided each other at the energy where the interaction occurs. This representation is known as the adiabatic or non-crossing Figure 1.4. Diabatic potential curves of A 6 I + state and B state cross (sol id lines) and cause the adiabatic curves(dotted lines) to avoid crossing by 2H e . 15 representation [5]. It is very difficult to calculate the vibrational and rotational energy levels in such a basis. Another representation of the same effect, called the diabatic representation, allows two Bom-Oppenheimer potential energy curves to cross (Figure 1.4), and the interaction between the states takes the form of an electrostatic interaction. The operator responsible for interaction between the two states, which causes the vibrational and rotational level structures of the two states to be disrupted, is a two electron operator, H e l = Ze2/rjj. The selection rules for electrostatic perturbations are AJ = AS = AA = AQ =0 [5]. For diabatic functions <f>i, < § i \ TNI <{)2 > = 0 (1-31) «l>ilHdl<l)2> = H 12 (1-32) The single configuration approximation (where an electronic state is assumed to arise from a single electron configuration) is an adequate zero-order representation for diabatic electronic functions. Electrostatic perturbations occur frequently between states whose configurations differ by up to two spin orbitals. The interaction between a level vi ,J of the diabatic potential curve V i and the level V2 ,J of another diabatic curve V2 is given as: Hi,Vl,J;2,v2,J = Hi2<vi,J lv2,J > = H!2<vi lv2 > 8jT (1.33) The magnitude of the perturbation interaction is determined by the product of the electronic and vibrational factors. The electronic factor is nearly independent of vi and v 2. This factor can only be computed by ab initio methods or determined as a result of a full "deperturbation"[5] analysis. The magnitude of the vibrational overlap factor requires a detailed knowledge of the potential energy curves of the two electronic states. Since we do not have this information available in the MnO spectrum, this will not be considered further. 1 6 1.7 M A T R I X E L E M E N T S Internal hyperfine perturbations occur when the energies of two of the electron spin components happen to be accidentally degenerate, and interactions between their hyperfine sub-levels become important. The matrix elements responsible for the internal hyperfine perturbations are off-diagonal in both the case ap and bpj basis sets, but in case ap the spin-uncoupling is also off-diagonal. The spin-uncoupling matrix elements, which arise A A from the x and y components of the operator -2BJ *S , are very much larger than the internal hyperfine perturbation matrix elements. This gives trouble with the energy ordering of the eigen values in computer calculations, near the regions of the internal hyperfine perturbations. The calculation is best done in a case bpj basis, even though the matrix elements in case bpj coupling are algebraically much more complicated than those in case ap coupling. The 6L+states of MnO are close to case (b) coupling, and it turns out that there is an advantage to the case(b) formalism because the hyperfine structure can then be included simply. 1.7.1 Rotational and fine structure The rotational Hamiltonian is given by H r o t = BR 2 - DR 4 (1.34) For £ states.where A=0, R is equal to N , so that H r o t = B N 2 - D N 4 (1.35) The matrix elements of the rotational Hamiltonian are given by [13] < NSJ I B N 2 - D N 4 INSJ > = BN(N+1) - DN2(N+1)2 (1.36) When a molecule has one or more unpaired electrons (or in other words is a free radical) the rotational structure of its spectrum shows further splittings, referred to as the fine structure. These arise from couplings of the total spin angular momentum S and the electron orbital angular momentum L with each other and with the molecular rotation. 1 7 Hfs = H s o + H s r + H s s + Heic.d (1-37) where H s 0 , H s r , H s s and Heic.d are the energy operators for the spin-orbit, spin-rotation, and spin-spin interactions and their respective centrifugal distortion corrections. The spin-rotation Hamiltonian is: H ^ N - S (1.38) where y is the spin-rotation interaction parameter. The spin-rotation Hamiltonian is diagonal in a case(b) basis. Its matrix elements are given by[13]: < NSJI yN-S INSJ > = -i-y[N(N+l) +S(S+1)- J(J+1)] = jyW(JSN) (1.39) where W(abc) = a(a+l)-b(b+l)-c(c+l). The Hamiltonian for the spin-spin interaction is given as[5]: HSS = | M 3 S J - S 2 ) (1.40) The spin-spin interaction originates from two mechanisms [5], a contribution to X from the dipolar-dipolar interaction of two unpaired spins, and a contribution from second order spin-orbit coupling(which is usually the larger in heavier molecules). Xfr = X + X& (1.41) eff ss so v The diagonal matrix elements of the spin-spin interaction are given by[13]: <NSJlfX(3S>-&) I NSJ > = - ^ ^ ^ ^ " ^ ^ l ^ l f '0^1^ 1 8 Its off-diagonal matrix elements are given by[13]: <N-2 , S J I il (3S?-S2) I N S J > = , 1 x 3 2 2 ( 2 N - l ) V ( 2 N + l ) ( 2 N - 3 ) V ( 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 ) a^ (1 .43) The effective Hamiltonian for the spin-orbit interaction, including the high-order parameters is given by [14] Hso = A L Z S 2 + (2/3) X® ( 3 S 2 - S2)+ T ^ S Z [ S 2 - ( 3S 2 + 3 S 2-l)/5) ] J _ 12 + T T 6 [ 3 5 S * - 3 0 S 2 S 2 + 2 5 S , - 6 S 2 + 3 S 4 ] (1.44) For a Z state, the first and the third terms vanish (A=0), and the dominant contribution to the X term is the second order contribution of the spin-orbit interaction (i.e. second order perturbation theory applied to the matrix elements of the spin-orbit interaction). The spin orbit operator is written as a single particle operator [5]: H = 1 a i l r s i (1.45) so < Af f IH ( 2 ) IAQ > = I <AQ1 H s o lA'Q > <A'OJ H s o IAfl> 8 0 A' E(A)-E(A') (1.46) The main second order spin-orbit interactions will be due to the nearest states. Often these are the states that belong to the same electron configuration as the state under consideration. The fourth term in Hso is the fourth order contribution of the spin-orbit operator. There is no simple algebraic form for its matrix elements in case (b) coupling, though in tensor form they can be written for £ states as[13] 1 9 / N' 4 N < N'SJ I H ( 4 ) I NSJ > = — (-1 f J(2N+1) (2N"+1) J 24 V \J) 0 0 fj s (_ 1 ) N + S + V f -/(2S-3)(2S-2)(2S-l)2S(2S+l)(2S+2)(2S+3)(2S+4)(2S+5) 14 N' S (1.47) Since the third term in H s o Hamiltonian vanishes, the third order contribution to the energy given by the spin-orbit operator arises when the spin uncoupling operator A A A A A A -2B (JXS x+JySy) is taken with the spin -orbit operator X a j l j - S j twice. This term is i called the spin-orbit distortion of the spin-rotation interaction! 15]. It arises when other nearby electronic states interact strongly by second order spin-orbit coupling. A A A A A A ^ 0 ) V <X I J -S I a x a I L-S I b x b I L-S I X> ( ] 4 g ) a b ^ a X ^ b X where a and b are the interacting electronic states. The diagonal matrix elements for the spin orbit distortion of the spin-rotation are given in terms of a parameter Ys[l 3]: <NSJ I C T3(L2 , j )• T3(s\sAi) I NSJ> = - ^ ^ 5 ^ (2W(NSJ) W(JNS) x [5N(N+l)-2] +2W(SNJ) S(S+1) [2W(JNS) +1] + 5W(SNJ) W(JNS) W(NSJ) + 4W(JNS)J(J+1) +4W(NSJ)N(N+1)S(S+1)} (1.49) where C < A I f 2 (L 2 ) I A > = - ^ r s 0V y[6 20 Its off-diagonal matrix elements are[13]: <N-2,SJ I C T 3 ( L 2 , j )-T3(S ,S , S ) I NSJ> = Y s 8(2N-l)V(2N+l)(2N-3) [5J(J+1)-5N(N+1)+S(S+1)+2(5N-1)] Y(JSN) Y(JS,N-1)] (1.50) where Y(abc) = V(a+b+c+1 )(b+c-a)(a+c-b)(a+b-c+1) (1.51) The centrifugal distortion corrections to the spin-rotation and spin-spin interactions, YD and XD, must also be considered. The Hamiltonian is Heic.d = Y D ( N - S ) N 2+(l/3) Xd[ ( 3 S 2 - S * 2 ) N 2 + N 2 (3S 2 - S 2 ) ] (1.52) Its matrix elements are given as[13]: <NSJ I Hek.d I NSJ> = - j Y D N(N+1) [ N(N+1) + S(S+1) - J(J+1) ] K (3W(JSN) [W(JSN)+1] - 4S(S+1)N(N-H)} - 3 AD (2N-l)(2N+3 ) x I S ( I S + 1 ) (1.53) <N-2, SJI H e k d I NSJ> = JXD ^ ^ ^ " ^ " ^ L r x Y C J S N ) Y(JS , N-l) 2 u(2N-lW(2N+l)(2N-3) (1.54) In the formalism of Hund's case (b) the fine structure, which splits each rotational energy level into six electron spin components Fi(J=N+5/2), F2(J=N+3/2), F3(J=N+l/2), F4(J=N-l/2), F5(J=N-5/2), F6(J=N-5/2), arises mainly from spin-orbit interactions of the states of interest with neighboring states according to the selection rules A A =0, ±1, AS =0, ±1. These interactions have the same form as the direct internal spin-spin (X,) and spin -rotation interaction (Y). The relative effects of X and y are very easy to distinguish when the states are in pure case (b), which always occurs for the high values of N. 21 1.7.2 Nuclear Hyperfine Structure Hamiltonian If the molecule contains a nucleus with non-zero nuclear spin, the nuclear spin angular momentum I couples with S and L to produce additional structure in the spectrum, called magnetic hyperfine structure. The nuclear spin angular momentum I can generate hyperfine structure also in molecules without unpaired electrons, either through nuclear electric quadrupole interaction (if I £ 1) or nuclear spin-overall rotation interaction. For our purposes the hyperfine Hamiltonian is written Hhfs = Hmag.hfs + H Q (1-55) where Hmag.hfs is m e magnetic hyperfine Hamiltonian, which arises from the interaction of the nuclear spin magnetic moment with the other magnetic moments in the molecule, and H Q is the electric quadrupole Hamiltonian, which arises when nuclei with spin I > 1 (which possess electric quadrupole moments) interact with the non-spherical electron charge distribution in the finite volume of each nucleus. A A A A A 1 A A A A A i A, Hhfs =acI'S + c ( I £ * - y I - S ) + Q W +e 2 Qo < ) (3£P) /41(21-1) + C h T l (I) -T 1 [T*2(L2) (S ,S ,S)] (1.56) where ^ j ? b s = C h <A lf^ (L2) I A> The first term is called the Fermi-contact interaction [16,17], 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.The Fermi contact interaction is isotropic, and has the form 22 HFenni-cont= X \,i 1 S l ~ % 1 S d-57) where a c j = 8 l C f l l | W w ^ ( 0 ) (1.58) The diagonal matrix elements are given by [13] <NSJIFI ac I • S I NSJJF> = - 4 J ( ^ + 1 ) W(FIJ)W(NS J) (1.59) Its off diagonal matrix elements are given by [13] <NS,J-1,IF I ^ I-S I NSJIF > = — , a ? Y(NSJ) Y(FU) (1.60) 4JV(2J-1)(2J+1) where, as before, Y(abc) = V(a+b+c+l) (b+c-a) (a+c-b) (a+b-c+1) (1.61) The probability y 2(0) of the electron being found at the nucleus is usually negligibly small for an electron in a p or d atomic state. Whenever there is an appreciable amount of s character in the wavefunction of an unpaired electron, the Fermi contact interaction, which is proportional to Y 2(0), may be expected to dominate. Sometimes the observed sign of ac is negative; this is explained by the mechanism of spin polarization [18]. The second term is the dipolar (I,S) interaction between the nuclear magnetic moment and the magnetic field produced at the nucleus by the unpaired valence electrons. Its diagonal matrix elements are given by [13]: 23 <NSJTF I ^(31,5 j - I -S ) INSJD?> - 1 » " T - T i t [3W(JSN)W(SNJ)-2N(N+1)\V(NJS)] - j c w ( n j ) 4JU + D(2N-l)(2N+3) (1.62) The off diagonal matrix elements are given by [13]: <NS,J-1,IF I 5C(3IZSZ - I-S) INSJIF> - I c IN(N+1)+3S(S+1)-3J2] Y ( N S J ) Y ( F I J ) ( L 6 3 ) ~4J(2N-1)(2N+3)V(2J+1)(2J-1) <N-2,S,J+1,IF I ^ c ( 3 I z S j - l - S )INSJIF> 8(2N- 3 )(J+1 )\'(2N+l)(2N-3)(2J+l)(2J+3) •Y(F,1,J+1)Y(JSN)X(NSJ). where X(NSJ)= V(N+S-J-2)(N+S-J-l)(J+S-N+2)(J+S-N+3) (1.64) v o c n r i 1 c i c \ i v c n r s . W(FIJ) Y(JSN) Y(JS.N-l ) <N-2,S,JIF I TC (3 I Z S z - I S ) INSJIF> = c— — ^—— , 3 8J (J+1 )(2N-1) V (2N+1 )(2N-3) (1.65) v i n i n : i l ft* Y(FIJ) Y(JSN)Z(NSJ) <N-2,SJ- l , IFI^c(3I z S z - I-S )INSJIF> . c W ( 2 J _ ] ) ( 2 J , ] ) ( 2 N > ] j ( 2 N _ 3 ) (1.66) where Z(NSJ) *W(N+S+J)(N+J-S-l)(N+S+J-l)(N+J-S-2) (1.67) A A Since the I .S terms from the Fermi contact and dipolar-dipolar interaction are indistinguishable, the hamiltonian can be rewritten as: A A A A H sb I*S + c I z S x (1.68) 24 The third term is the nuclear spin-rotation interaction. This operator is diagonal in a case(b) basis, and has matrix elements : <NSJIFICi I-J INSHF> = jCi\V(FIJ) (1.69) 1.7.3 Electric Quadrupole Hamiltonian The electric quadrupole Hamiltonian comes from the interaction between the nuclear electric quadrupole moment and the electric field gradient produced by the surrounding electrons. The electrostatic interaction between the particles making up a nucleus, at positions T}, and an electron at the point r, is given by Coulomb's law as: Te2 Ze2 H QS- I f r " - I . - -.. (1-70) J I»J J 1 £ I - £ J 1 Electrons are point charges but nuclei have a finite size for their electric charge distribution. Expressing equation (1.70) as a series of Legendre polynomials we get: k HQ = -^ 2 X^ P k(C 0 SV <L71> j - k lij where Pk (cos 8ij) is the Legendre polynomial of order k and 6y is the angle between rj and rj. The first term in equation (1.71), k = 0, represents a monopole-monopole interaction which is already included in the electronic Hamiltonian Ho- The second term (k = 1) represents a nuclear electric dipole interaction which can be shown to be zero, as are the higher electric multipole moments of odd order. The term with k=2 corresponds to the electric quadrupole interaction. HQ= ef 2(Q ) - f 2(VE) (1.72) 25 The diagonal matrix elements of the electric quadrupole interaction are given by [13]: <NSJIF I ef2(Q)-T2(VE)INSJIF> [3W(FIJ) [W(FIJ)+1] - 4I(I+1)J(J+1)} --e^q0 41(21-1) x {3W(SNJ)[W(SNJ) + 1] - 4N(N+1)J(J+1)} , 4J (J+l) (2J-1) (2J+3) (2N-1) (2N+3) { L , j > ) For a linear molecule in a £ electronic state the tensor T^VE) has only one independent component which is called the "quadrupole coupling constant ", q0) and defined as q = 2<A I T 2 (VE) IA>. The off-diagonal matrix elements of the electric quadrupole interaction are given by [13]: <NS,J-1JF I ef 2(Q)-f 2(VE) INSJIF> = - L^Pfox Y(FIJ) 41(21-1) 4 J(J+1) (J-1)V(2J+1)(2J-1) [W(FU) + (J+1)][W(SNJ) + (J+1)] (2NIix2N+3) <L74) <N-2, S, J+l, IF I eT2(Q).f2(VE) I NSJIF> = - M r j ¥ x SJaTlKJ+LaN-41(21-1) A8J(J+1)(J+2)(2N-1) Y(F,I,J+1)Y(JSN)X(NSJ) (1.75) V (2N+l)(2N-3)(2J+l)(2J+3) <N-2, S, J, IF I ef2(Q)-T2(VE) I NSJIF> = l^i) Y(JSN) Y(J, S, N-l) [3W(FIJ){W(FIJ) + 1}-41(1+1 )J(J+1)] ? 6 8J(J +l)(2N-l)V(2N+l)(2N-3) X (2J-l)(2J+3) 2 6 <N-2, S, J-l, IF I ef 2(Q)-f 2(VE) I NSJIF> = li^-i) [W(FIJ)+(J+1)] Y(FIJ)Y(JSN)X(NSJ) 8J (2N-1)V(2N+1)(2N-3)(2J-1)(2J+1) (J-1)(J+1) 27 CHAPTER 2 Experimental details 2.1 Description of the source Gaseous MnO was prepared by the action of a microwave discharge on a flowing mixture of argon, N2O and the vapour from a volatile manganese compound. The chosen compound was manganese cyclopentadienyl tricarbonyl (C5Hs)Mn(CO)3, which is a commercially available yellow solid; this was placed in a U shaped tube. One end of the U tube was connected to the gas handling system and the other end to the microwave discharge tube. Argon gas mixed with nitrous oxide was used to entrain the (CsH.5)Mn(CO)3, and this gas mixture was pumped rapidly through the microwave discharge region and into a cube-shaped fluorescence cell The total pressure of the system was around 1 Torr. A 2.45 GHz microwave discharge was used to break up the molecules into their constituent atoms. Manganese and oxygen combined in the low pressure region, and a purple-colored flame of MnO chemiluminescence could be seen a few centimetres downstream from the discharge region. 2,2 Laser-induced Fluorescence A laser beam was directed into the fluorescence 'cube' such that it intersected the discharge flame perpendicularly. The resulting molecular fluorescence was observed at right angles to both the laser beam and the stream of molecules. The laser used for the experiments was a tunable dye laser (Coherent model CR 599-21) pumped with 3W of 514.5 nm radiation from an argon ion laser (Coherent model I 90-6). A narrow band width laser beam with an output power of 50-100 mW was obtained from the dye laser. The power obtained depended on the lifetime of the dye solution and the position in the dye gain curve that was being selected. The dye Rhodamine 110 was used to take MnO spectra over the 17770 cnr 1 - 17970 cnr 1 region. 28 A small portion of the cw laser beam was directed to an iodine absorption or fluorescence cell for absolute frequency calibration. Another small portion was sent to a Tropel fixed-length semiconfocal Fabry-Perot interferometer with a 299 MHz free spectral range to provide interpolation markers for the spectra. The beam containing the majority of the laser output power was sent to the fluorescence cell. A photomultiplier tube, equipped with a sharp cut-off filter to reduce scattered light, and powered by 300-500 V from a high voltage power supply, was used to monitor the fluorescence. Phase sensitive detection was used to increase the signal-to-noise ratio. The signal was modulated by mechanically chopping the laser beam, and a PAR model 128 Lock-in amplifier and a suitable reference signal was used to detect the modulated sample signal. The MnO signal, the I 2 reference spectrum and the interpolation markers were recorded simultaneously by a computer as the laser frequency was scanned. The resolution of the spectra was limited by Doppler broadening so that most of the rotational structure in the spectra was resolved but the hyperfine splittings remained unresolved. For the study of the hyperfine structure of MnO, sub-Doppler spectra were recorded using intermodulated fluorescence (IMF) [18], a high resolution technique capable of eliminating the Doppler broadening. In the next section the theory and experimental arrangement of this technique will be considered. 2.3 Intermodulated Fluorescence Suppose a laser beam is passed through an absorption cell. The molecules which have a component of velocity in the direction of the laser beam will have a Doppler shift where the shift is given by 29 f i = f 0 ( l + v / c ) (2.1) where v = velocity of the molecule c = velocity of light. fo = resonance frequency, fl = absorption frequency. A Doppler broadened spectral line is the sum of all the individual Doppler shifted absorptions. This effect consequently produces a Gaussian line shape corresponding to the Gaussian velocity distribution of the molecules in the cell. The center of the line corresponds to the transition frequency of the molecules which have zero velocity in the laser direction. This is therefore the transition frequency fo- Because of the high power used in the experiment, the velocity distribution of the molecules in the lower state can actually be distorted. A depletion of population of the molecules at one particular velocity takes place, and the effect is known as 'hole-burning' [19]. When two laser beams of the same frequency are passed through the sample cell in opposite directions, as shown in Figure 2.1, two symmetric holes are burned, one on either side of the centre of the Doppler profile. This is the result of the opposite Doppler shifts that are excited by the laser beams. As the laser frequency is tuned across the transition, the two holes move towards each other, cross and separate towards the opposite side of the velocity distribution curve. If the intensity of the total fluorescence is monitored as the laser is scanned through the transition frequency, the intensity will increase to a maximum and then decrease, because of the increasing and decreasing numbers of molecules excited in accordance with the Gaussian velocity distribution. However, a small dip will be seen at the actual transition frequency. This is called a Lamp Dip [20]. It arises because both laser beams are now interacting with only one group of molecules, those with zero velocity in the laser direction, and therefore the total fluorescence decreases. This is 30 GAS SAMPLE C E L L LASER LIGHT FREQUENCY Figure 2.1. Velocity distribution curves of lower state. Excitation of molecules creates "boles" in the lower state. These boles are known as "Bennet holes". 3 1 GAS SAMPLE C E L L LASER LIGHT F R E Q U E N C Y Figure 2.2. T»o Bennet boles which converge at zero velocity to form a Lamb dip in t i u profile of fluorescence intensity versus laser tuning frequency. 3 2 Discharge in flow system PMT Refe Lock-in spectrum Tunable dye laser ence signal PDP-ll/23 Mic ro -computer 3-pen chart recorder *5 PMT calibration Interpolation markers Figure 2.3. Schematic diagram of the intermodulated fluorescence experiment used in this laboratory. 33 illustrated in Figure 2.2. Since the Lamb dip occurs at the transition frequency, it is possible to perform sub-Doppler spectroscopy by monitoring the Lamb dip. In practice, Lamb dip signals are not detected as shown in Fig 2.2. Instead they can be observed using a more sensitive modulation method, intermodulated fluorescence. In an IMF experiment the laser is split by a beam splitter into two nearly equally intense beams. The two beams are chopped at frequencies fi and f2 and are passed into the sample cell collinearly but in opposite directions. The beams produce laser-induced fluorescence signals which are modulated at fi and f2 respectively. When the laser frequency is not at the transition frequency, two different velocity groups of molecules are excited, one modulated at fi and the other at f2. However, at the center of the Doppler profile the two laser beams excite the same group of molecules. Sorem and Schawlow [18] have shown that the resulting fluorescence at this frequency is not only modulated at fi and f2 but also at (fi+f2) and (fi - f 2 ) . If a lock-in amplifier tuned to either the sum or difference frequency is used to monitor the fluorescence, then only the signal corresponding to the Lamb dip will be observed and the rest of the Doppler profile will be suppressed. The signal obtained has a line width much smaller than the Doppler width, and hyperfine components within a rotational line can be resolved. Fig 2.3 shows the experimental intermodulated fluorescence set up used to record the hyperfine splittings of MnO. 2.4 Wavelength Resolved Fluorescence Instead of using just a photomultiplier tube to monitor the total fluorescence signal resulting from excitation of a single rotational transition, the technique of wavelength resolved fluorescence is often used. In this technique a monochromator or spectrograph is used to disperse the total fluorescence. In a monochromator a narrow band of diffracted light is projected onto an exit slit where a photomultiplier tube is situated. A spectrum is obtained by rotating the grating so that the dispersed spectrum 34 N=7 Pumped transition i P(8) R(6) Figure 2.4. energy level patterns for wavelength resolved fluorescence 35 is swept across the exit slit. In a spectrograph, the exit slit and photomultiplier tube are replaced by a photographic plate and the entire dispersed spectrum is recorded at one time. The spectra obtained by either of the above methods give information about the excited energy level since the excited molecules can only emit light to different rotational levels in the ground state according to the selection rules for electronic rotational-vibrational transitions (Figure 3.4). The quantum numbers involved in the individual lines or their relationship to other lines as determined by this technique can be invaluable in completing a rotational analysis, especially when the system being studied is massively perturbed. A diode array detector or optical multichannel array detector is now quite often used instead of a photomultiplier tube or photographic plate in the above methods. In fact the diode array detector (DAD) is the electronic analogue of the photographic plate used in spectrographs. The main advantage of the DAD over a photographic plate is that the spectral information is ready to analyze more rapidly and intensity information is available in digital form. In the MnO experiments an EG&G model 1421-1024-G DAD was used to replace the exit slit and detector on a SPEX model 1702 spectrometer. The DAD was interfaced to a computer with an EG&G model 1461 detector interface so that the information could be stored. The DAD has a spectral window of about 600 cm-1 at 600 nm when the spectrograph is used in first order. Resolved fluorescence line measurements are only accurate to about 0.2 cm-1 in this case. Two important types of information can be obtained from wavelength resolved fluorescence: 1) AJ selection rules. If X\ is the exciting wavelength, one of the fluorescence lines in the spectrum will also appear at X\, though with enhanced intensity as a result of scattered laser light. If only two lines appear in the spectrum and the enhanced line is at a shorter wavelength than X\t then the excited line belongs to a P branch (AJ—1). 36 R,(14) (^35) ( b l e n d e d l i n e ) R,(12) RA(15) M5) Po(15) D R 6 07) (17) Pc09) 14 16 F2 F, 18 Figure 2.S. Wavelength-resolved fluorescence spectra of MnO. 37 However, if the second line is at a longer wavelength then the excited line belongs to to an R branch(AJ=+l). 2) Combination differences. The A 2 F " combination difference (described in detail in section 3.4) can be approximated from equation (3.3) as A2F"(N) = 4B"(N+l/2) (2.2) where B" is the ground state rotational constant. Equation (2.2) can be used to give a rough estimation of the B" value for the ground state if several lines with different N values are excited. The DAD spectra are calibrated very simply. After a picture of the MnO spectrum is taken, an Fe-Ne hollow cathode lamp is imaged onto the entrance slit of the spectrograph. Using the same grating and slit settings a DAD spectrum is taken of the hollow cathode discharge. The Fe-Ne atomic lines in the spectrum are used to establish the wavelength scale for that setting. However, determining the frequency for every line is very tedious and time consuming work. Fortunately the quantum numbers of the sample peaks can be determined more simply. The frequency spacing between a P and an R branch line is related to the A 2 F " combination difference through the constant B. This can be related to a measured distance on a calibrated DAD spectrum. For DAD spectra taken with the spectrograph operating in second order the relationship is A2F"(N) = 4B(N+l/2) = dxl.66 (2.3) where d is the distance measured between R and P lines on the plotted spectra, and the proportionality constant is 1.66 mm/cm-1 in this case. For each value of N a corresponding d value was calculated and given this information the quantum number N could essentially be "read off" the DAD spectrum. Two sample DAD spectra showing the wavelength resolved fluorescence of MnO are shown in Figure 2.5. 38 CHAPTER 3 3.1 History The electronic spectrum of the MnO molecule consists of strong structured transitions in the 4500 - 6500 A region [21] and a diffuse absorption near 2600 A [21]. Further bands of MnO were reported in the 7100-7900 A region by Pinchemel and Schamps [22]. The (1,0) band of the strong visible system, A 6 E + - X 6 Z + , which lies at 5360 A, was first studied at high resolution by Pinchemel and Schamps [22]. They used an arc between manganese electrodes in an atmosphere of oxygen. An emission spectrum with wide Doppler widths corresponding to a high rotational temperature was obtained. Later, Gordon and Merer [21] recorded the (1,0), (0,0) and (0,1) bands at slightly higher resolution, using a microwave discharge in Mn03F as the emission source, which gives a much lower rotational temperature. They reported a partial rotational analysis, explaining some aspects of the high electron multiplicity and the manganese hyperfine structure. In the present work, the A 6 £ + - X 6 £ + transition has been recorded with the aid of a Coherent Inc. model CR 599-21 dye laser. The (0,0) band of the transition has been recorded over the range 17770-17970 cm-1. Since the spectra between 17940 and 17970 cm-1 are very weak, it was necessary to obtain spectra with the higher power available from a ring laser (Coherent Inc. model CR 699-21). 3.2 Description of the spectra The (0,0) band of the A 6 £ + - X 6 E + system of MnO molecule is unusually complicated because of the high electron spin multiplicity, the manganese nuclear hyperfine structure ( 5 5 Mn has I = 5/2), and the fact that the spectrum consists of two electronic transitions. Besides the strong A 6 E + -X 6 Z + system, there is a second weaker 39 hyperfine structure ( 5 5 Mn has I = 5/2), and the fact that the spectrum consists of two electronic transitions. Besides the strong A 6Z +-X 6Z +system, there is a second weaker transition between another 6X+state, which perturbs A 6 Z + , and the ground state X 6 Z + . The second transition will be called B - X. The analysis is not complete yet, but it seems that the B - X transition is obtaining most of its intensity from mixing between the upper states A and B, although certain intensity anomalies indicate that the B - X transition may have some intensity of its own. The general appearance of the (0,0) band of the A 6 Z + - X 6 Z + s y s t e m is shown in Figure 3.1. Instead of the "usual" appearance of a red-degraded Z-Z transition, where there is a pronounced band-head formed by the R branches, the structure is very confused. At higher dispersion it can be seen that the strong R branch lines begin to form a head near 5585 A, but their spacing suddenly opens out, and their intensity decreases. The R branches can be followed to about 5580 A, but at this point they die out completely. The intense region round 5585 A (17900 cm - 1 ) is extremely crowded because a second series of R branches forms weak band heads underlying the intense R branches just described. In addition to the above transitions, the spectra has a long series of internal hyperfine perturbations between the F3 (N=J-l/2) and F4 (N=J+l/2) electron spin components of the ground state; these produce characteristic double line patterns where the two groups of lines are separated by about 0.13 cm - 1 . The F3 and F4 branch lines are easily identified because they form a kind of band head within the hyperfine structure as a result of the internal hyperfine perturbations. For a given N value we observe 12 main branch lines ( R 1 - R 6 , P 1 - P 6 )• Other branches appear, which are not expected for a X-Z transition, and detailed analysis shows that these are spin satellite branches obeying the selection rule AJ = 0, ± 1, AN = ± 3, ± 5 instead of the expected main branch selection rule AJ = AN = ± 1. These satellite branches form a very weak head at 5568 A (17970 cirr 1). — 1 ' ' ' : 5575 5600 5625 5650 a Figure 3.1. Broad band laser excitation spectrum of the MnO A'5> - X«2> (0,0) band O 4 1 3.3 Ground state energy level pattern From the argon matrix electron spin resonance spectra of Baumann et al [23], the ground state of MnO is known experimentally to be a 6 Z + state. The rotational and fine structure Hamiltonian for a 6 Z + state is given by equations (1.35), (1.38) and (1.40): H = B N 2 - D N 4 + | X ( 3 S 2 - S 2 ) + y N'S (3.1) The rotational structure of a 6 L + state in the absence of electron spin effects, that is for very small values of the second-order spin-orbit splitting parameter X, and the spin-rotation parameter Y, is exactly like that of a *X state. There is a series of rotational energy levels with increasing spacings, following the energy expression E(N) = BN(N+1) - D N2(N+1)2 (3.2) The effect of the second-order spin-orbit splitting, which depends on the eigen value , 2X.Z2, of the operator 2A,SZ, is to split each rotational level into three pairs corresponding to the possible projection values of the quantum number Z, which are ±1/2, ±3/2 and ±5/2 for S = 5/2. In case (b) coupling (see eq.1.42) the separations of these three pairs are 2X. and 4X (see Figure 3.2). The spin-rotation interaction splits each pair: the splittings are in the ratio 1: 3: 5 for the pairs F3, F4: F2, F5-.F1, F&, as given by eq (1.39). The spin satellite branches described in section (3.5) have been invaluable in determining the ground state spin patterns. After measurements had been made of several sets of main branches and satellite branches, it was found that the spin parameters of the ground state were closely approximated by the values X, = 0.57 cm - 1 , Y = -0.003 cm-1. To establish this, we set up the matrix of the rotational and fine Energy 8/3 X-2/3 X -10/3 X 2X F3 F4 F2 F5 AX FI F6 Figure 3.2. Electron spin-spin splitting of a rotational level N of a 6 Z electonic state in case (b) coupling (X is assumed to be positive) structure Hamiltonian in a case (b) basis I NSJ >. Conservation of parity implies factorization of the matrix for each N into two sub-matrices of order 3, where the basis functions are labelled by even or odd values of N, respectively. Using this Hamiltonian and the parameters above we calculated the approximate ground state energy levels. 3.4 Rotational analysis As described in section (3.6) a rotational line can usually be assigned immediately to one of the three IN—Jl pairs from its hyperfine width, as shown in Figure 3.3. In the absence of hyperfine effects all rotational lines would have the same width, so the hyperfine structure has the advantage of simplifying the assignments of the branches though the overall complexity of the spectrum is naturally greatly increased. The hyperfine patterns can usually be used to identify the electron spin component. The rotational assignments were made by the method of ground state combination differences. Two rotational lines R(N-l) and P(N+1) with a common upper level N' will be separated by the energy difference AjF", defined as AjF'XN) = 4B"(N+l/2) - 8D"(N+l/2)3 (3.3) which is known as a combination difference. Since the ground state is well-behaved, other than at the internal hyperfine perturbations, it is possible to assign the N-values when P and R branch lines that follow eq.(3.3 )can be found. In much of the spectrum the very great line density causes even regular branch structure to be completely unrecognizable. It was then necessary to use wavelength-resolved fluorescence experiments to identify the combination differences and therefore the N values. Fortunately the ground state has quite small spin-rotation coupling, so that the F"= ft(12) F,(12) Pi(U) Figure 3.3. MnO: A 62> - X«2> (0,0); the six R(26) lines 45 combination differences for the six electron spin components are identical at the resolution of the spectrometer used for these measurements. Using the ground state energy levels, the upper state term values were calculated by adding the corresponding transition frequencies. A plot of the upper state term values as a function of J(J+1) is given in Figure 3.4. This figure shows that there are two electronic upper states, whose rotationless energies are about 17894 cm- 1 and 17947 cm-1. Since both the upper states have a number of local rotational perturbations by other electronic states, we have difficulties in obtaining the upper state parameters. The difficulties arise from the fact we are unable to identify the nature of the perturbing states. However, approximate values of the rotational constant B and X values of the upper states A 6 Z + and B 6 L + can be determined graphically. Figure 3.5 is a plot of the upper state energy levels, scaled by subtracting the quantity 0.45N(N+1), of the upper set of F\ and F6 electron spin components against N(N+1). The scaling is necessary to subtract most of the rotational energy in order to magnify the details. In the absence of any kind of perturbation, the graph should have two parallel straight lines. However, what is seen is characteristic of an avoided crossing of two electronic states. The low N structure of Figure 3.5 belongs to one state and the high N structure belongs to the other state. Since the transitions involved in the high N region are very intense, these presumably belong to the expected A - X transition and the weaker transitions in the low N region belong to the B - X transition. The dotted lines in Figure 3.5 indicate the average of the upper term values of the F! and the F6 components. The slopes of the dotted lines in Figure 3.5 give the rotational constants of the two electronic states. The rotational constants of the B 6 Z + and the A 6 L + state were found to be 0.39 cm - 1 and 0.45 cm - 1 respectively. Since the lower of the two states, A 6 L + , has a much larger rotational constant B than the upper state, B 6 Z + , its level structure catches up to it with increasing rotation. Figure 3.4. Low J level structure of the MnO A«L+, v=0 and B«2> states. 17980 17970 -17960 -17950 -17940 -17930 800 N(N+1) Figure 3.5 Upper state term values of the Fx and F« electron spin components of the M n O A 6 X+, v=0 and B 62> states plotted against N(N+1). The term values have been scaled by subtraction of a quantity 0.4N(N+1) c m - 1 to magnify the details. 48 From the intersection of the dotted lines, we can find the corresponding upper state energy at which the states A and B cross each other in zero order. As explained in section (1.5) and (1.6) there is an avoided crossing between the two electronic states, and therefore the term values are pushed up from the zero order crossing point. The shift gives the experimental value of the interaction matrix element (see eq.(1.30)) as approximately 10 cm - 1 . This interaction matrix element value is useful in determining the deperturbed electronic energies of the A 6 1 + and the B 6 Z + state. We can use equations (1.26) and (1.27) to calculate the deperturbed electronic energies, since the E i and E 2 values are found to be 17947 cm - 1 and 17894 c m - 1 respectively. The deperturbed electronic energies of the A 62> and B 6 £ + states are therefore 17945.0 cm-' and 17896.0 cm - 1 . The rotational quantum number N at the maximum of the avoided crossing is approximately that at which the A 6 I + - X 6 Z + (0,0) band should form its R heads, and the result is that the normal patterns around 17900 cm - 1 are considerably disrupted and crowded. To determine the X value of the upper state experimentally, we consider the splittings of the fine structure at J=0 caused by the spin-spin interaction. The splitting caused by the spin-spin interaction follows the case (a) diagonal matrix elements [21] 2 A A of the Hamiltonian jA, (3S z-S 2 ) . Omitting the term in S(S+1), the eigen values are 2M 2(see section 3.3), where I takes the values ± 1/2, ±3/2 and ±5/2 for S = 5/2. The separation of the substates with 111 = 5/2 and 1/2 should be 12A,. Experimentally, the X, value for the B 6 X + state was found to be = 3 c m - 1 . Since all six electron spin components of the A 6 Z + state converge to 17894 cm - 1 , the X value for the A 6 I + state must be = 0 cm - 1 . 3.5 Satellite branches The identification of some of the extra lines as satellite branches follows directly from the wavelength-resolved fluorescence experiments, where laser excitation of some particular line gives more lines than can be explained by the case (b) selection rule A J = AN = ± 1. These branches fall off very rapidly in intensity with increasing N . The intensities of satellite branches are small compared to those of main branches as long as case (b) applies to both states. The AN = ± 3, ± 5 satellite lines at the short wavelength end of the band arise because the B 6 Z + state has a spin-spin coupling parameter X which is considerably larger than that of the ground state, so that it is nowhere near pure case (b) coupling; the difference in coupling case between the B 6 L + and X 6 L + states is the reason for the breakdown of the AN = ± 1 selection rule (which holds strictly only for pure case (b) coupling). We have used the case (b) notation ANp, for the main branches since they show no discontinuities when followed down from the high N region. If one state belongs to case (a) and the other belongs to case (b), the strict selection rule is A J = 0, ±1 only, since N has no meaning in case (a) coupling; all transitions following this rule are allowed in principle (though some may be very weak in practice). The branches are then named with symbols indicating the formal change in the quantum number N , as if both states were in case (b) coupling, as a left superscript: e.g. ^ A J F F " . where F' and F" indicate the electron spin components of the upper and the lower states in agreement with A N = A J + F - F". Q branches (A J = 0) are normally very weak in parallel transitions, and their appearance here is explained partly by the large value of the electron spin quantum number S and partly by the breakdown of the pure case (b)-case (b) selection rules. Figure 3.6 shows the satellite branches we were able to assign following excitation of the R6 and P6 main branch lines; a total of four has been identified, namely 5 0 N 6 (+) 5 H H H ( - ) TRM(3) R6(5) P6(7) RP64(5) TQ 6 3 ( 3 ) F6 F5 J 7/2 7/2 A 6XH F3 F4 F2 F5 F1 19/2 F6 9/2 F3 F4 F2 F5 F1 F6 11/2 9/2 13/2 7/2 15/2 5/2 x 6 r F3 F4 F2 F5 F1 F6 7/2 5/2 9/2 3/2 11/2 1/2 Figure 3.6. Energy level diagram indicating the satellite branches induced by excitai.on of the F6spin component of the upper state. 5 1 TR64, RP64» RQ65> TQ63 (where the notation T R ^ means a spin satellite branch with AN = 3, AJ = 1 which involves the same set of upper state levels as the main branch but comes instead from the F4 electron spin levels of the ground state). Similarly T Rs3 , RP53, RQ54 satellite branches, which have a common upper state F5 component, have been assigned. The disappearance of these satellite branches around N=l 1 shows that they are becoming forbidden as spin uncoupling sets in and the upper state approaches case (b) coupling. The satellites coming from the F3 and F4 upper components are TR42, RP42» T R 3 i and RP3i, which appear to the blue of the main branches, and PR46, NP46, PR35, NP35 which appear to the red of the main branches.( The notation NP35 means AN = -3 , AJ = - 1 ) 3.6 Hyperfine structure analysis 5 5 M n has a nuclear spin I = 5/2, which interacts with the angular momentum J and splits each rotational level into six hyperfine components. High multiplet Z states almost invariably belong to the hyperfine case bpj coupling, where the hyperfine structures of the electron spin components are very characteristic. The magnetic hyperfine energy in case bpj coupling can be written as [24]: Ehfs - [ b(J-N) + JJ] [ F(F+1) -1(1+1) -J(J+1)]/2J (3.4) Ignoring the c term, eq. (3.4) can be rewritten as: Ehfs - [ b(J-N) ] [F(F+1) -1(1+1) -J(J+1)]/2J (3.5) The hyperfine width of a spectral line is given by the difference in the hyperfine splitting of the upper and the lower state energy levels. The hyperfine splitting, E(J, F = J +1) - E(J,F = J -1), of the ground state can be written by simplifying eq. (3.5): E(J, F = J + I) - E(J,F = J -1) = Ib"(J"-N") (2J"+1)/J" (3.6) where b" is the ground state Fermi contact hyperfine constant, which is equal to 433 ± 1 MHz, as determined by Baumann et al [23]. The hyperfine width of an MnO line, AVhfW, is expressed as: 52 A v h f w = Ib"(J"-N") (2J"+1)/J" - Ib'(J'-N') (2J*+1)/J* (3.7) where b' and J' represent the upper state hyperfine constant and rotational quantum number respectively. For the main branches (J"-N") equals (J'-N'), so that AVhfw can be rewritten as: A v h f w = 2I(J"-N")[(l-^r) b"- (l+~r)b'] (3.8) For high J values eq.(3.8) can be approximated to A v h f w = 2I(J"-N")(b"-b,) (3.9) where b' and b" are the upper and lower state hyperfine constants, respectively. Therefore, at high rotational quantum numbers, the hyperfine width is proportional to (b' - b") (J - N). This quantity is a constant within a particular Fj component, and it can be seen that the hyperfine widths for a 6 L + (b) state will be in the ratio 5: 3: 1: -1: -3: -5 for the six spin components (see Figure 3.7) as long as both states are in case (b) coupling. This hyperfine pattern carries much useful information for identifying the electron spin components. Unfortunately we cannot apply eq.(3.9) to the satellite branches because these have inconsistent hyperfine widths. The factor [F(F+1) - J(J+1) -1(1+1)] in eq.(3.4) shows that the hyperfine energy is proportional to F(F+1) for a particular N J rotational level. Therefore the relative hyperfine line positions can be approximated as E M s = T(N,J) + kF(F+l) (3.10) where k is proportional to (b'-b") (J-N). Since the factor (b'-b") is negative for the A - X and B - X transitions, the higher F quantum number components of the F i , F 2 and F 3 electron spin components will appear at the low-frequency (red) side of the rotational lines. The opposite holds true in the F4, Fs,and F$ lines. In addition, the assignment of the quantum number F is easy for the F i , F 2 , F5 and F6 electron spin components since there are no internal hyperfine perturbations and the hyperfine structure follows the Lande interval [25] pattern familiar in the spin multiplet structure A(36) /M38) P3(37) JV36) UUUUL~ A (38) A(37) L JtLLL Figure 3.7. Hyperfine patterns in the six electron spin components of the (0,0) band of the A«2> - X « E * transition of MnO. II = 24 25 26 27 28 29 Rx 24 23 22 21 20 19 Re V V Figure 3.8. Hyperfine patterns in the Rf(24) and R6(24) lines of the A«2> - X«L* (0, 0) band of MnO. of atomic spectra. The hyperfine patterns open out at the high F side, and the higher F lines have greater intensity as shown in Figure 3.8. The relative intensities of the hyperfine components of a rotational line are given as: Essentially, the higher the value of F the stronger the hyperfine component is, allowing for the Boltzmann factor of the initial state. Without fitting the results, we cannot give accurate values for the upper state hyperfine constants. However, we can calculate the b' values approximately for the A 6 L + and B 6 I + state from equation (3.9). The average b' value of the A 6 I + state at high rotational quantum numbers is negative in sign and its magnitude is almost as large as that of the ground state. Table (3.1) shows the variation of the b' value with the N quantum number for the Fi and F6 electron spin components. In Figures (3.9 )and (3.10) the upper term values and the hyperfine constant are plotted as a function of N(N+1). These graphs clearly indicate that whenever there is any rotational perturbation, there is a deviation from the normal hyperfine behaviour. (3.11) Table 3.1: F 6 (A- X transition) = Quantum Hyperfine Upper state hyperfine number N width (in cnr 1) constant, b (in MHz) 20 0.1921 -15.2 21 0.1310 125.2 22 0.2453 -141.2 23 0.2778 -218.1 24 0.2899 -247.2 56 25 0.2973 -265.3 26 0.3027 -278.6 27 0.3083 -292.4 28 0.3085 -293.4 29 0.3108 -299.3 30 0.3087 -294.8 32 0.2858 -241.6 33" 0.3130 -306.2 34 0.3064 -291.0 35 0.3091 -297.7 36 0.3172 -317.2 37 0.3167 -316.3 38 0.2888 -250.6 39 0.3088 -298.2 40 0.3153 -313.8 41 0.3127 -307.9 43 0.3099 -301.8 44 0.3229 -332.8 Table 3.2: Ft (A- X transition) Quantum Hyperfine Upper state hyperfine number N width (in cm-1) constant, b (in MHz) 20 21 22 0.2353 0.2688 -119.5 -199.4 23 0.2908 -251.6 24 0.3039 -283.0 25 0.3114 -301.2 26 0.3157 -311.8 27 0.3190 -320.0 28 0.3219 -327.3 29 0.3244 -333.5 30 0.3247 -334.6 31 0.3251 -335.9 32 0.3259 -338.1 33 0.3015 -280.6 34 0.3238 -333.7 35 0.3256 -338.3 36 0.3250 -337.1 37 0.2742 -216.9 38 0.3202 -326.2 39 0.3070 -295.1 42 0.3060 -293.3 43 0.3189 -324.1 According to eq.(3.8), there will be essentially zero hyperfine splitting if b'~ b"; the hyperfine transitions coincide at the position of the rotational transition, irrespective of the F quantum number. Because all the hyperfine transitions lie on top of each other the rotational line has a very high intensity. In the MnO spectrum, we were able to follow some intense branches of this type. An example is given in Figure 3.11, where the Ps(19) line at the right hand side of the figure has essentially zero hyperfine splitting and its six components collapse into one sharp "spike". By contrast the Ps(29) line 58 -100 -200 -300' 17928 17927 -17926 17925 • (b) • • • • + (a) + + + + — i i i — | i i i "i 600 800 1000 i — | — i — p -1400 1600 400 1200 N(N+1) Figure 3.9 (a) Upper state term values and (b) effective Fermi contact parameters of the Fi electron spin component of the v=0 level of the A 6 I* state of MnO. The term values have been scaled by subtraction of a quantity 0.47N(N+1) cm-1 to magnify the details. 59 l_ S D. 2 w > c | I w 8 | 17940 -V. u I 55 1 7 9 3 0 • 400 600 — 1 — r 800 1000 1400 1600 1200 N(N+1) Figure 3.10 (a) Upper state term values and (b) effective Fermi contact parameters of the F6 electron spin component of the v=0 level of the A 6 L + state of MnO. The term values have been scaled by subtraction of a quantity 0.46N(N+1) cm-1 to magnify the details. Figure 3.11. A portion of the A65> - X«2> (0,0) band of MnO; among the six components of the P(29) line is a line with zero hyperfine splitting, which has been identified as a P5(19) transition. 61 next to it has the "normal" hyperfine pattern for the A-X transition. With increasing rotation the hyperfine splitting of the "spike" branch changes slightly and it is found to split into six hyperfine components with very narrow hyperfine width, and correspondingly low intensity. Combination differences show that the lower state of this branch is the F5 electron spin component. Since its upper state does not match with any electron spin components of the A61+ state or the B6Z+ state this must be another perturbing electronic state which has the correct symmetry to interact with the A and B states. We can offer no electronic assignment for it, since no other branches of this type have been identified as yet. 3.7 Internal Hyperfine Perturbations In contrast to the patterns of Fj, F 2 , F5 and F6 the most intense hyperfine lines of the F3 and F4 branches are clustered together. Since doubled lines are not found at low N, the F quantum numbers of the F3 and F4 branches in the low N region can be assigned without much trouble. However in the range N = 15-50 strong internal hyperfine perturbations distort the expected pattems(see Figures 3.3, 3.7 and 3.11). An interna] hyperfine perturbation is a remarkable occurrence, where the F3 and F4 electron spin components of the ground state of MnO (J = N+l/2 and J = N-l/2 respectively) should cross at N = 26 in zero order because of the particular values of the rotational and electron spin parameters, as shown in Figure 3.12. This crossing cannot actually occur because the levels differ by one unit in J for given N, and therefore interact through matrix elements of the hyperfine Hamiltonian of the type AN = AF = 0, AJ = ± 1. Extra lines are induced by the internal hyperfine perturbation. Since the detailed course of the ground state is known, their positions tell whether a T 1 1 1 1 1 I 0 10 20 „ 30 N Figure 3.12. Electron spin structure of the v=0 level of MnO, X*2>. The quantity plotted is the electr on spin contribution to the total energy as a function of the rotational quantum number N. 63 branch containing them has F3 or F4 and also gives its N numbering by the ground state combination differences. The hyperfine matrix elements between F3 and F4 are F-dependent and of the order of about 0.05 cnr 1 . This reverses the Lande-type pattern for the spacings, which comes from the factor F(F+1) in the diagonal hyperfine matrix elements (eq.1.59). The patterns of hyperfine lines become very complicated because the F order of the hyperfine components inverts so that an internal hyperfine perturbation produces a kind of band head in the hyperfine structure. Even though Fi and F6, and F 2 and F5 cross in zero order they do not interact because they do not differ by one unit in J. The operator responsible for the internal hyperfine perturbations has contributions from the Fermi contact, dipolar and quadrupole parts of the Hamiltonian. Because the F3 and F4 electron spin components lie very close over a considerable range of N, internal hyperfine perturbations are found throughout this range, which is essentially all N values above N ~ 10. The largest effects occur where the electron spin components cross in zero order, near N=26. Only five of the six hyperfine components of each level actually avoid each other (see Figure 3.13). The reason is that the range of F values is different in the two levels. The F3 spin component (J=N+l/2) has the range F=N-2 to N+3 while F4(J=N-l/2) has F=N-3 to N+2. Hyperfine components with F=N-2 to N+2 occur in both F3 and F4 levels, and therefore perturb each other, but the F=N+3 component of F3 and the F=N-3 component of F 4 pass through the avoided crossing region unaffected. They give rise to the isolated strong lines which are very characteristic (Figure 3.14 ). The energy matrix for the interaction of the F3 and F 4 electron spin components of the X 6 Z + state of MnO can be written as: 64 A J = ± 1 , AN = AF=0 matrix elements of b I-S + d z S z Figure 3.13. Mechanism of an internal hyperfine perturbation. R3 (26), J" = 26.5 R4 (26), J" = 25.5 F" r ~ . •i i T ,1 T . .1 i i iJ ! i I V i i i i . Ii 11 1 / V i i ...i-i II i!i ,i 111 1 I L it in I .... I i MI 29 27 25 23 28 26 24 Figure 3.14. Internal hyperfine perturbation in the R(26) lines of MnOA 6 5> - X 6 I + 66 IF 3 , X > IF 4 , X > E. 3.F <bI.S> H' = <bI.S> E 4 ,F where <bFS> is the AJ=0, ±1 matrix element calculated from eq.(1.60). The zero order energies of F3 and F 4 are given by E3,F and E 4 j respectively. The Hamiltonian is diagonal in N and F, but not in J . Diagonalisation of the H' matrix leads to mixing of F3 and F 4 levels with the same N , and extra branches are induced. Extra lines obeying the selection rules AJ=0, ± 2 are induced. These extra lines give the energy separations of the F3 and F4 electron spin components of the ground state and upper state. In formal case(b) notation ^ A J F T " , the extra branches are classified as P Q 3 4 , R Q43 , p043 and RS34. The Q branches are permitted by the case(b) selection rules (though their intensities should be vanishingly small except at low N ) . But the other branches violate the A J =0, ± 1 selection rule for electric dipole transitions meaning that the hyperfine perturbations have destroyed the goodness of J as a quantum number. 67 CHAPTER 4 4.1 Electron configuration The Molecular orbital method, where molecular orbitals are constructed by linear combination of atomic orbitals (LCAO), will be assumed for discussing the electron configurations of MnO. The ground state electron configurations of the Mn and O o c 2 4 atoms are K L M 4s 3d and K 2s 2p respectively. Neglecting the filled sub-shells and the oxygen 2s orbital there are three atomic orbitals left with which to construct the molecular orbitals of MnO. The symbols o, rc, 8,... refer to the magnitude of the projection of the angular momentum of the individual electrons on the space fixed Z-axis in the atoms, and the intemuclear axis in the molecules. On a scale where the energy zero is the ionization energy, the metal 3d and 4s orbitals lie above the oxygen 2p by the difference in the ionization energies. The two degenerate orbitals are split by the electric field of the other atom into components where the lowest value of A, (i.e.,3da or 2po) is the highest in energy. Molecular orbitals can then be formed by linear combinations of atomic orbitals with the same A, values. Since there are no 8 orbitals from the oxygen , the metal 3d8 atomic orbital becomes a 8 molecular orbital without shifting its energy. The 2pa, 3da and 4sa atomic orbitals are involved in LCAO. The 2po atomic orbital is combined with the 3do rather than 4so which is energetically more favorable. This is 2 because there is greater overlap achieved by combining the localized lobes of the 3d z and 2p z orbitals than by combining the large, diffuse spherical 4s with 2pz > The 3d7t and 2p;t atomic orbitals form two p molecular orbitals, the upper of 68 Mn orbitals MnO orbitals O orbitals 3d 3d o 3d7t 4s 4s o • 3d 5 a •6 tjr-i o > 2p Fig 4.1. The relative energies of the molecular orbitals of MnO, formed from the linear combinations of the atomic orbitals of Mn and O atoms. 6 9 which is still mostly associated with the metal and the lower of which is associated with the oxygen. An electron promotion from any of the lower molecular orbitals to a molecular orbital in the upper group is therefore a "charge transfer transition", in the sense that it takes an electron from the oxygen and gives it to the metal; such a promotion will cause a considerable lengthening of the bond. The energy separations are shown in Figure 4.1. The 3d5 orbital sinks below 4so* with increasing atomic number Z , while all the metal orbitals fall relative to 0(2p) because the metal ionization potentials rise with Z. Using the building-up principle the eleven valence electrons from the open atomic shells are loaded into the molecular orbitals. The first six electrons fill the lowest a and 7t orbitals, and the ground state of MnO will be determined by the configuration of the remaining five electrons. Since the pairing energy of the electrons is greater than the separation of the 4so and 3d8 energy levels, the electrons remain unpaired in the ground state. Table 4.1: Molecular constants for the X 6 Z + ground state of MnO [23] Parameter Present work ESR work [23] (in cm - 1) (in MHz) B 0.5012 X 0.57 y -0.003 b 433(1) c 46(12) 70 Table 4.1 contains the approximate B, A, and y values of the present work and the hyperfine constants of b and c as determined by Baumann £t al [23] for the ground state. The positive ground state value of b and its large magnitude indicates that there is an "unpaired" so" electron present in the ground state. Of all the possible electron configurations, the ground state corresponds to the configuration which has the maximum number of parallel spins, so that the most likely electron configuration for the ground state is (8c 2 37t4) 182 9a 1 4TI2 This configuration has an unpaired electron in the 9a orbital, derived from the 4s orbital of manganese, and the large size of the Fermi contact parameter b is consistent with this assignment. The possible electron configurations for the upper states A 6 Z + and B 6 L + can arise from two kinds of electron transfer. 1) Valence state transfer 9o --»> 10a. 2) Charge transfer 8a --*10a, 3TI --*4TI In a charge transfer transition, there is considerable lengthening of the bond, i.e. a reduction of the rotational constant; the very great decrease in B from 0.5012 cnr 1 to 0.39 cm - 1 is only consistent with the assignment of the B - X transition as a charge transfer transition. The possible electron configurations for the B state are (8a1 3TC4 ) 162 9a 2 47t2, and (8a 2 3TC3 ) 162 9a 1 4K3. A valence state transition does not produce much lengthening of the bond. The observed change in B from 0.5012 cnr 1 to 0.45 cm - 1 is consistent with the A 6 L + -B 6 Z + transition being the 9a->10a promotion. The most likely electron configuration 2 4 2 2 1 for the A 6 L + state is (8a 3rc ) 18 4n 10a , particularly since the Fermi contact parameter of the A 6 X + state is negative, which indicates that no unpaired electron is present in the 9a orbital. 7 1 Figure 4.2 Energy Level diagram of MnO [22]. Thick lines indicate the calculated states. Dotted line indicates the experimental A6I+state. 72 Table 4.2: Ground states of the 3d transition metal monoxides[26] Ground A G 1 / 2 Bo ro AA or A, electron state (cm-1) (cm-1) (A) (cm-1) configuration 4 5ScO 964.65 0.51343 1.668 c 4 8TiO 1000.02 0.53384 1.623 101.30 o5 51V0 4I " 1001.81 0.54638 1.592 2.03 o52 5 2CrO 5 n r 884.98 0.52443 1.621 63.22 o62n 55MnO 832.41 0.50122 1.648 0.57 56FeO % 871.15 0.51681 1.619 -189.89 o-sV 59CoO 851.7 0.5037o 1.631 (-240) a25 %2 58N i 0 828.5 0.5058 1.631 (26) o264n2 63CuO 2 n ; 629.39 0.44208 1.729 -277.04 C2647t3 The bond length in MnO is very similar to those in the other early transition metal oxides; a summary is given in Table 4.2. The quantities AA and A. are the first and second order spin-orbit splitting parameters; AA is tabulated for orbitally-degenerate states, A for L states. 4.2 Future work Since we already know the ground state constants, analyses of the (1,0) and (0,1) bands of the A6L+-X6X+ transition will show how the molecular parameters and the perturbations change with vibration. Theoretical calculations by Pinchemel et al [22]showed that there is a 6 n state with electron configuration 9a1 182 4ft110a1 lying about 22,000 cm-1 above the X6Z+ state (Figure 4.2). Since it has an unpaired electron in the 9a orbital, the contribution to the Fermi contact interaction in the 6 n state will be identical to that of the X6L+ state, b'~ b", and the 6 n -X6Z+ transition will 73 have very narrow hyperfine structure. Since the spin-orbit coupling constant for a molecule such as MnO is ~ 60 cm - 1 , each spin-orbit component 6n+7/2, 6n+5/2, 6n+3/2, 6n+1/2,6n_1/2,6n_3/2,will be separated by 60 cm" 1. Since the 6n -X6l+ transition will be a case(a) - case(b) transition one could expect very large numbers of satellite branches and the band structure will be extremely complicated. Since the ground state parameters of the X 6 1 + are known, the spectrum should be assignable using ground state combination differences. From the Figure 4.2, a 6 A state should exist near 30,000 cm - 1 . It might be possible to observe a 6A- 6F1 transition in the "infra red" region around 8000 cm - 1 in the MnO emission spectrum. Bibliography [I] A.Messiah, 1965, Quantum mechanics Vol 2 (Ncnth-Holland Publishing Co.) [2] T.A.Miller, MoLPhysics, 1& 105-120 (1969). [3] A.R.Edmonds, Angular momentum in Quantum mechanics., Princeton Univesity Press, Princeton (1974). [4] Richard N.Zare, Angular Momentum understanding spatial aspects in chemistry and physics, John Wiley & Sons, Inc. (1988). [5] H.Lefebvre-Brion and R.W.Field, Perturbations in the spectra of Diatomic Molecules, Academic Press , New York(1972). [6] J.T.Hougen, The calculation of rotational energy levels and line intensities in Diatomic Molecules, National Bureau of Standards Monograph 115,1970. [7] G.Herzberg, Spectra of diatomic molecules, 2 n d ed, Van Nostrand Co.Inc, New York(1950) [8] E.Hill and J.RVan Vleck, Phys.Review, 32, 250 -272(1928). [9] R.A.Frosch and RM.Foley, Phys.Review, 88, 1337-1349(1952). [10] A.S-CCheung and A.J.Merer, MoLPhysics, 4£,111-128(1982). [II] LKovacs, Rotational structure in the spectra of Diatomic Molecules, American Elsevier I?ubUshing Company Inc. New York (1969). [12] D.Richards and R.F.Barrow, Nature (London) 212., 1244-1245(1968). [13] A.J.Merer, Unpublished Notes. [14] J.M.Brown, A.S-C.Cheung and A.J.Merer, J.Mol.Spectroscopy, 124, 464-475 (1987). [15] A.S-C.Cheung, R.C.Hansen and AJ.Merer, J.Mol.Spectroscopy, 9_i, 165-208(1982). [16] C.RTownes and A.L.Schawlow" Microwave spectroscopy" McGraw - Hill, New York, 1955. 75 [17] P.H.Kasai, J.Chem.Physics., 49_, 4979-4984(1968). [18] M.S.Sorem and A.L.Schawlow, Opt.Comm.5_, 148-151(1972). [19] W.R.Bennet,Jr., Phys.Rev.126, 580-593(1962). [20] W.E.LambJr., Phvs.Rev.A134. 1429-1450(1964). [21] R.M.Gordon and A.J.Merer Can J.Phys.5_8, 642-56(1980). [22] B.Pinchemel and J.Schamps, Chem.Physics Hi, 481-489(1976). [23] C A Baumann, R.J Van Zee, and W.Weltner, J.Phys.Chem..8J_, 5084 -5093(1982). [24] J.L.Femenias, G.Cheval, A.J.Merer and U.Sassenberg, J.Mol.Spectrosc. 124 348-368(1987). [25] T.M Dunn, "Molecular spectroscopy, Modem Research", K.N.Rao, and C.W.Mathews, eds. 1972 Academic Press, Inc.New York, NY(Chapter 4.) [26] A.J.Merer, Annual Review of Physical Chemistry in Press. 76 APPENDIX Transitions of the (0,0) band A 6 Z + - X6S+ system B - X transition Fi electron spin component N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 J"-5/2 -1 Rl(-l) 1.5 17932.0570 .0947 1 PKD 3.5 931.3118 .3629 .4074 0 R1(0) 2.5 933.5401 2 Pl(2) 4.5 930.6863 .7338 .7751 .8100 .8385 .8603 1 Rl(l) 3.5 934.6791 .7330 .7771 .8124 .8387 .8544 3 Pl(3) 5.5 929.7698 .8149 .8548 2 Rl(2) 4.5 935.5431 .5933 .6360 .6712 .6997 .7207 4 Pl(4) 6.5 928.6028 .6469 .6865 .7212 .7517 .7765 3 Rl(3) 5.5 936.1890 .2356 .2786 .3153 .3463 .3698 5 PK5) 7.5 927.2241 .2688 .3096 .3441 .3810 .4051 4 Rl(4) 6.5 936.6996 .7511 .7971 .8364 .8710 .8994 6 Pl(6) 8.5 925.7226 .7710 .8229 .8536 .8882 .9191 5 Rl(5) 7.5 937.2907 .3492 .4106 .4489 .4895 .5251 7 Pl(7) 9.5 924.3008 .3561 .4085 .4605 .4975 .5347 5 Rl(5) 7.5 932.5790 .6412 .6961 .7453 .7887 .8257 7 Pl(7) 9.5 919.5891 .6483 .7019 .7516 .8336 6 Rl(6) 8.5 933.8277 .8728 .9164 .9551 .9894 34.0194 8 Pl(8) 10.5 918.8236 .8692 .9116 .9498 .9856 19.0169 7 Rl(7) 9.5 934.0927 .1324 .1695 .2039 .2326 .2597 9 Pl(9) 11.5 917.0858 .1246 .1604 .1933 .2242 .2518 8 Rl(8) 10.5 933.7533 .7919 .8277 .8603 .8898 .9164 10 Pl(10) 12.5 914.7405 .7777 .8449 9 Rl(9) 7.5 933.0669 .1057 .1418 .1754 .2060 .2337 11 Pl(ll) 13.5 912.0470 .0850 .1204 .1535 .1846 .2132 10 Rl(10) 12.5 932.1516 .1899 .2279 .2626 .2949 .3245 12 Pl(12) 14.5 909.1254 .1649 .2015 .2329* .2686 .2991 11 Rl(ll) 13.5 931.0792 .1221 .1622 .1985 .2334 .2652 13 Pl(13) 15.5 906.0515 .0934 .1326 .1697 .2044 .2371 12 Rl(12) 14.5 929.9478 .9940 930.0369 0776 .1158 .1507 77 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 J"-5/2 14 Pl(14) 16.5 17902.9179 .9633 903.0036 .0462 .0845 .1200 13 Rl(13) 15.5 925.7132 .7663 .8160 .8628 .9073 .9482 15 Pl(15) 17.5 896.6811 .7327 .7820 .8290 .8730 .9148 14 Rl(14) 16.5 924.8717 .9226 .9701 925.0146 .0572 .0969 16 Pl(16) 18.5 893.8365 .8860 .9336 .9781 894.0214 .0610 15 Rl(15) 17.5 923.5290 .5787 .6260 .6706 .7129 .7529 17 Pl(17) 19.5 890.4939 .5430 .5899 .6345 .6770 .7176 16 Rl(16) 18.5 921.9256 .9762 922.0246 .0711 .1148 .1564 18 Pl(18) 20.5 886.8897 .9403 .9883 887.0344 .0784 .1201 17 Rl(17) 19.5 920.2009 .2533 .3033 .3514 19 Pl(19) 21.5 883.1643 .2167 .2665 .3143 .3597 .4033 18 Rl(18) 20.5 918.4227 .4767 .5284 .5778 .6246 .6693 20 PI (20) 22.5 879.3891 .4425 .4940 .5433 .5899 .6350 19 Rl(19) 21.5 916.6222 .6764 .7275 .7784 .8259 .8711 21 Pl(21) 23.5 875.5890 .6429 .6943 .7441 .7916 .8374 20 Rl(20) 22.5 914.7737 .8274 .8783 .9263 .9662 15.0090 22 PI (22) 24.5 871.7427 .7961 .8465 .8945 .9344 .9776 21 Rl(21) 23.5 23 Pl(23) 24.5 22 Rl(22) 24.5 913.4804 .5378 .5940 .6477 .6995 .7492 24 Pl(24) 26.5 866.4569 .5139 .5692 .6231 .6750 .7251 23 Rl(23) 25.5 911.2573 .3196 .3800 .4385 .4946 .5481 25 Pl(25) 27.5 862.2400 .3018 .3623 .4201 .4766 .5301 24 Rl(24) 26.5 909.4803 .5454 .6085 .6691 .7279 .7842 26 Pl(26) 28.5 858.4682 .5331 .5958 .6565 .7158 .7728 25 Rl(25) 27.5 907.8640 .9305 .9949 908.0575 .1177 .1754 27 Pl(27) 29.5 854.8561 .9222 .9865 855.0492 .1090 .1673 26 Rl(26) 28.5 906.3032 .3705 .4354 .4988 .5600 .6189 28 Pl(28) 30.5 851.3035 .3705 .4357 .4988 .5600 .6194 27 Rl(27) 29.5 904.7583 .8258 .8918 .9552 905.0172 .0773 29 PI (29) 31.5 847.7687 .8362 .9018 .9656 848.0278 .0874 78 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 Jn-l/2 J"-3/2 J"-5/2 28 Rl(28) 30.5 17903.2089 .2776 .3436 .4079 .4703 .5308 30 Pl(30) 32.5 844.2289 .2968 .3628 .4269 .4894 .5502 29 Rl(29) 31.5 901.6322 .7007 .7671 .8319 .8952 .9566 31 Pl(31) 33.5 840.6613 .7292 .7956 .8603 .9234 .9848 30 Rl(30) 32.5 900.0703 .1384 .2055 .2703 .3332 .3950 32 PI (32) 34.5 837.1132 .1811 .2473 .3125 .3759 .4376 31 Rl(31) 33.5 898.4349 .5035 .5699 .6348 .6980 .7600 33 Pl(33) 35.5 833.4899 .5584 .6247 .6898 .7535 .8154 32 Rl(32) 34.5 896.7768 .8450 .9117 .9763 897.0401 .1027 34 Pl(34) 36.5 829.8448 .9129 .9795 830.0438 .1074 .1692 33 Rl(33) 35.5 895.2400 .3027 .3641 .4247 .4841 .5415 35 PI (35) 37.5 826.3239 .3865 .4481 .5082 .5677 .6254 34 Rl(34) 36.5 893.1186 .1867 .2519 .3174 .3804 .4424 36 Pl(36) 38.5 822.2181 .2856 .3513 .4159 .4792 .5416 35 Rl(35) 37.5 891.3033 .3716 .4383 .5029 .5670 .6289 37 Pl(37) 39.5 818.4210 .4888 .5548 .6194 .6831 .7457 36 Rl(36) 38.5 889.3779 .4450 .5119 .5769 .6403 .7029 38 PI (38) 40.5 814.5826 .6481 .7130 .7775 .8398 37 Rl(37) 39.5 887.4583 .5119 .5657 .6210 .6850 .7325 39 PI (39) 41.5 810.6129 .6662 .7206 .7756 .8302 .8843 38 Rl(38) 40.5 885.1314 .1983 .2636 .3277 .3902 .4516 40 Pl(40) 42.5 806.3109 .3771 .4424 .5066 .5693 .6303 39 Rl(39) 41.5 882.6602 .7241 .7866 .8480 .9082 .9672 41 Pl(41) 43.5 802.0444 .1046 .1640 40 Rl(40) 42.5 42 Pl(42) 44.5 41 Rl(41) 43.5 43 Pl(43) 45.5 42 Rl(42) 44.5 878.5174 .5810 .6439 .7051 .7649 .8234 44 Pl(44) 46.5 791.8577 .9196 .9808 92.0410 .0995 43 Rl(43) 45.5 875.6603 .7259 .7916 .8544 .9172 .9792 Appendix, continued 79 N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"--3/2 J"-5/245 45 Pl(45) 47.5 17786.9646 787.0304 .0950 .1587 .2222 .2842 Extra lines of Ri component observed by perturbatioi n 21 Rl(21) 23.5 916.3032 .3594 .4142 .4664 .5168 .5650 23 Pl(23) 25.5 871.2748 .3307 .3850 .4379 .4883 .5369 F 2 electron spin component 6 R2(6) 7.5 17945.8999 .9200 .9368 .9507 8 P2(8) 9.5 930.9319 .9488 7 R2(7) 8.5 945.6625 .6813 .6976 .7113 .7231 .7330 9 P2(9) 10.5 928.6786 .6977 .7100 .7230 .7350 .7456 8 R2(8) 9.5 945.0887 .1067 .1204 .1360 .1477 .1576 10 P2(10) 11.5 926.0929 .1093 .1232 .1367 9 R2(9) 10.5 944.2048 .2222 .2378 .2512 .2630 .2730 11 P2(ll) 12.5 923.2001 .2157 .2306 .2434 .2554 10 R2(10) 11.5 943.0389 .0525 .0682 .0821 .0938 .1044 12 P2(12) 13.5 920.0224 .0392 .0541 .0673 .0792 .0900 11 R2(ll) 12.5 941.6008 .6180 .6334 .6466 .6589 .6694 13 P2(13) 14.5 916.5860 .6026 .6173 .6303 .6417 .6528 12 R2(12) 13.5 939.9273 .9445 .9595 .9727 .9852 .9961 14 P2(14) 15.5 912.9036 .9197 .9345 13 R2(13) 14.5 938.0232 .0406 .0558 .0700 .0820 .0936 15 P2(15) 16.5 908.9952 909.0120 .0264 .0399 .0496 14 R2(14) 15.5 935.9087 .9258 .9413 .9555 .9678 .9796 16 P2(16) 17.5 904.8778 .5842 .9352 .9471 15 R2(15) 16.5 933.6008 .6183 .6344 .6484 .6613 .6709 17 P2(17) 18.5 900.5671 .5842 .5993 .6133 .6278 .6381 16 R2(16) 17.5 931.1221 .1400 .1541 .1710 .1847 .1971 80 Appendix, N Branch J" F"=J"+5/2 J"+3/2 18 P2(18) 19.5 17896.0874 .1047 17 R2(17) 18.5 928.5015 .5204 19 P2(19) 20.5 891.4665 .4847 18 R2(18) 19.5 925.7979 .8160 20 P2(20) 21.5 886.7643 .7839 19 R2(19) 20.5 923.1049 .1287 21 P2(21) 22.5 882.0732 .0965 20 R2(20) 21.5 920.4850 .5135 22 P2(22) 23.5 877.4542 .4827 21 R2(21) 22.5 917.8889 .9219 23 P2(23) 24.5 872.8620 .8948 22 R2(22) 23.5 915.3419 .3719 24 P2(24) 25.5 868.3108 .3463 23 R2(23) 24.5 912.9102 .9481 25 P2(25) 26.5 863.8915 .9295 24 R2(24) 25.5 910.6579 .6978 26 P2(26) 27.5 859.6449 .6844 25 R2(25) 26.5 908.5637 .6048 27 P2(27) 28.5 855.5550 .5956 26 R2(26) 27.5 906.5854 .6270 28 P2(28) 29.5 851.6665 .7050 27 R2(27) 28.5 904.7309 .7712 29 P2(29) 30.5 847.7393 .7799 28 R2(28) 29.5 902.8424 .8850 30 P2(30) 31.5 843.8599 .9025 29 R2(29) 30.5 901.0229 .0660 31 P2(31) 32.5 840.0-1 y8 .0927 30 R2(30) 31.5 899.2115 .2546 32 P2(32) 33.5 836.2518 .2946 31 R2(31) 32.5 897.3941 .4370 33 P2(33) 34.5 832.4467 .4894 continued J"+l/2 J"-l/2 J"-3/2 J"-5/2 .1202 .1355 .1490 .1613 .5369 .5527 .5669 .5800 .5029 .5168 .5310 .5447 .8368 .8536 .8690 .8839 .8018 .8191 .8347 .8493 .1508 .1714 .1907 .2086 .1182 .1389 .1581 .1761 .5410 .5667 .5908 .6137 .5098 .5350 .5591 .5820 .9529 .9257 .9507 .9728 .9958 .4060 .4383 .4691 .4985 .3800 .4122 .4429 .4719 .9846 913.0192 .0519 .0833 .9653 .9997 864.0332 .0644 .7359 .7725 .8068 .8399 .7227 .7586 .7932 .8262 .6441 .6814 .7175 .7520 .6348 .6723 .7083 .7427 .6675 .7064 .7431 .7784 .7419 .7774 .8253 .8561 .8103 .8481 .8842 .9228 .8187 .8568 .8929 .9343 .9256 .9657 903.0036 .0406 .9437 .9832 844.0217 .0583 .1074 .1476 .1864 .2239 .1344 .1743 .2129 .2502 .2964 .3365 .3758 .4136 .3360 .3764 .4157 .4532 .4791 .5193 .5586 .5967 .5310 .5715 .6104 .6488 81 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 r-i/2 J"-3/2 J"-5/2 32 R2(32) 33.5 17895.5553 .5986 .6402 .6807 .7200 .7586 34 P2(34) 35.5 828.6232 .6657 .7072 .7477 .7869 .8248 33 R2(33) 34.5 893.6815 .7242 .7658 .8068 .8463 .8846 35 P2(35) 36.5 824.7625 .8050 .8466 .8873 .9266 .9651 34 R2(34) 35.5 891.7589 .8027 .8451 .8855 .9252 .9626 36 P2(36) 37.5 820.8564 .8998 .9417 .9823 821.0213 .0596 35 R2(35) 36.5 889.7596 .8030 .8441 .8845 .9240 .9624 37 P2(37) 38.5 816.8759 .9182 .9602 817.0005 .0393 .0776 36 R2(36) 37.5 887.6025 .6446 .7265 .7647 .8013 38 P2(38) 39.5 812.7362 .7780 .8187 .8578 .8961 .9329 37 R2(37) 38.5 884.5727 .6008 .6277 .6542 .6797 .7046 39 P2(39) 40.5 807.7282 .7564 .7832 .8092 .8348 .8597 38 R2(38) 39.5 884.5557 .5918 .6372 .6751 .7118 .7468 40 P2(40) 41.5 805.7324 .7739 .8139 .8519 .8883 .9230 39 R2(39) 40.5 881.9336 .9756 882.0159 .0557 .0947 .1319 41 P2(41) 42.5 801.1294 .1710 .2111 .2508 .2891 .3267 40 R2(40) 41.5 879.4857 .5259 .5651 .6046 .6426 .6802 42 P2(42) 43.5 796.7068 .7463 .7852 .8247 .8628 .9012 41 R2(41) 42.5 876.8314 .8705 .9092 .9470 .9837 877.0202 43 P2(43) 44.5 792.0788 .1182 .1566 .1938 .2311 .2668 42 R2(42) 43.5 872.4699 .4945 .5175 .5409 .5632 .5845 44 P2(44) 45.5 785.0504 .0922 .1323 .1708 .2080 .2434 43 R2(43) 44.5 873.7488 ..7907 .8304 .8692 .9065 .9416 45 P2(45) 46.5 785.0504 .0922 44 R2(44) 45.5 870.8602 .9017 .9427 .9825 871.0209 .0582 46 P2(46) 47.5 780.1909 .2332 .2743 .3140 .3900 Extra lines of F 2 component observed by perturbation 37 R2(37) 38.5 17882.2186 .2633 .3029 39 P2(39) 40.5 805.3792 .4219 .4630 .5018 .5388 .5746 42 R2(42) 43.5 878.1177 .1442 .1696 .1949 44 P2(44) 45.5 791.3922 .4181 .4436 .4685 .4923 .5162 82 Appendix, continued F 2 electron spin component N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 JM-5/2 2 R2(2) 3.5 17937.9607 .9936 938.0200 4 P2(4) 5.5 931.1541 3 R2(3) 4.5 938.6848 .7106 .7321 .7493 .7618 .7710 5 P2(5) 6.5 929.8046 .8248 .8416 4 R2(4) 5.5 939.0096 .0312 .0500 .0646 .0770 .0857 6 P2(6) 7.5 928.0880 .1058 .1215 .1349 .1479 .1564 5 R2(5) 6.5 939.0214 .0410 .0575 .0716 .0828 .0919 7 P2(7) 8.5 926.0718 .0884 .1029 .1153 .1271 .1367 6 R2(6) 7.5 938.7672 .7854 .8012 .8142 .8251 .8341 8 P2(8) 9.5 923.7974 .8131 .8267 .8501 .8594 7 R2(7) 8.5 938.2735 .2915 .3064 .3193 .3302 .3390 9 P2(9) 10.5 921.2900 .3050 .3187 .3307 .3415 .3513 8 R2(8) 9.5 937.5556 .5728 .5875 .6006 .6120 .6217 10 P2(10) 11.5 918.5594 .5778 .5881 .5976 .6088 .6246 9 R2(9) 10.5 936.6274 .6436 .6589 .6721 .6841 .6932 11 P2(ll) 12.5 .6384 .6522 .6646 .6761 .6866 10 R2(10) 11.5 935.4970 .5140 .5290 .5431 .5550 .5657 12 P2(12) 13.5 912.4841 .4996 .5139 .5271 .5392 .5502 11 R2(ll) 12.5 934.1756 .1928 .2091 .2228 .2352 .2476 13 P2(13) 14.5 909.1567 .1730 .1880 .2015 .2140 .2263 12 R2(12) 13.5 932.6749 .6929 .7090 .7240 .7379 .7499 14 P2(14) 15.5 905.6511 .6679 .6839 .6985 .7121 .7247 13 R2(13) 14.5 931.0088 .0271 .0445 .0611 .0750 .0887 15 P2(15) 16.5 901.9796 .9980 902.0151 .0298 .0454 .0594 83 Appendix, continued F5 electron spin component (Main branches) N Branch J" F"=J"+5/2 JM+3/2 r+1/2 J"-l/2 J"-3/2 J"-5/2 4 R5(4) 2.5 17963.9480 .9158 .8894 .8689 6 P5(6) 4.5 952.5601 .5162 .4788 .4479 5 R5(5) 3.5 962.1693 .1381 .1112 .0893 .0724 .0605 7 P5(7) 5.5 948.9338 .8969 .8647 .8371 .8144 .7964 6 R5(6) 4.5 960.2922 .2625 .2375 .2160 .1986 .1848 8 " P5(8) 6.5 945.1221 .0882 .0593 .0343 .0127 944.9948 7 R5(7) 5.5 958.2536 .2288 .2051 .1844 .1670 .1531 9 P5(9) 7.5 941.1180 .0898 .0639 .0409 .0204 .0027 8 R5(8) 6.5 956.0485 .0238 .0018 955.9821 .9655 .9516 10 P5(10) 8.5 936.9308 .9048 .8813 .8605 .8410 .8241 9 R5(9) 7.5 953.6756 .6528 .6323 .6136 .5976 .5831 11 P5(ll) 9.5 932.5705 .5472 .5253 .5054 .4875 .4709 10 R5(10) 8.5 951.1444 .1237 .1041 .0864 .0709 .0568 12 P5(12) 10.5 928.0476 .0264 .0058 927.9870 .9697 .9539 11 R5(ll) 9.5 948.4709 .4515 .4332 .4168 .4014 .3875 13 P5(13) 11.5 923.3787 .3587 .3399 12 R5(12) 10.5 945.6725 .6543 .6371 .6208 .6057 .5921 14 P5(14) 12.5 918.5836 .5651 .5475 .5284 .5146 .5000 13 R5(13) 11.5 942.7609 .7403 .7230 .7084 .6968 .6891 15 P5(15) 13.5 913.6764 .6560 .6381 .6229 .6102 .6018 14 R5(14) 12.5 939.7744 .7570 .7431 .7269 .7130 .7002 16 P5(16) 14.5 908.6929 .6759 .6594 .6441 .6299 .6164 15 R5(15) 13.5 936.7159 .6996 .6841 .6696 .6560 .6436 17 P5(17) 15.5 903.6361 .6201 .6043 .5893 .5748 .5611 16 R5(16) 14.5 933.6068 .5909 .5757 .5613 .5471 .5342 18 P5(18) 16.5 898.5285 .5131 .4968 .4819 .4678 .4538 17 R5(17) 15.5 930.4616 .4453 .4296 .4148 .4009 .3873 19 P5(19) 17.5 893.3866 .3697 .3543 .3386 .3249 .3101 18 R5(18) 16.5 927.2990 .2817 .2652 .2499 .2346 .2198 20 P5(20) 18.5 888.2105 .1943 .1779 .1627 .1481 .1313 84 Appendix, N Branch J" F"=J"+5/2 J"+3/2 19 R5(19) 17.5 17924.1463 .1280 21 P5(21) 19.5 883.0791 .0604 20 R5(20) 18.5 921.0574 .0364 22 P5(22) 20.5 877.9930 .9721 21 R5(21) 19.5 918.1026 .0784 23 P5(23) 21.5 873.0437 .0196 22 R5(22) 20.5 915.3604 .3324 24 P5(24) 22.5 868.3045 .2764 23 R5(23) 21.5 912.8543 .8238 25 P5(25) 23.5 863.8069 .7760 24 R5(24) 22.5 910.5548 .5217 26 P5(26) 24.5 859.5136 .4807 25 R5(25) 23.5 908.4005 .3663 27 P5(27) 25.5 855.3648 .3304 26 R5(26) 24.5 906.3183 .2824 28 P5(28) 26.5 851.2911 .2555 27 R5(27) 25.5 904.1950 .1587 29 P5(29) 27.5 847.1783 .1420 28 R5(28) 26.5 901.5617 .5236 30 P5(30) 28.5 842.5540 .5156 29 R5(29) 27.5 902.0298 901.9910 31 P5(31) 29.5 841.0324 840.9939 30 R5(30) 28.5 32 P5(32) 30.5 31 R5(31) 29.5 897.3082 .2559 33 P5(33) 31.5 832.3376 .2854 32 R5(32) 30.5 894.9113 .8764 34 P5(34) 32.5 827.9569 .9219 33 R5(33) 31.5 890.6996 .6589 35 P5(35) 33.5 821.7585 .7172 34 R5(34) 32.5 892.8673 .8303 continued J"+l/2 J"-l/2 J"-3/2 r-5/2 .1100 .0927 .0762 .0607 .0423 .0251 .0081 882.9917 .0159 920.9780 .9601 .9516 .9316 .9125 .8942 .0546 .0321 .0104 917.9888 872.9958 .9728 .9507 .9297 .3055 .2798 .2548 .2305 .2493 .2231 .1735 .7942 .7653 .7376 .7110 .7463 .7172 .6891 .6625 .4900 .4594 .4295 .4010 .4484 .4180 .3877 .3589 .3330 .3006 .2693 .2396 .2969 .2646 .2332 .2027 .2478 .2143 .1823 .1512 .2209 .1874 .1550 .1234 .1230 .0887 .0555 .0228 .1066 .0720 .0385 .0059 .4858 .4484 .4121 .3766 .4777 .4404 .4039 .3680 .9527 .9149 .8761 .8377 .9554 .9170 .8785 .8401 .2095 .1666 .1267 .0960 .2388 .1959 .1555 1262 .8394 .8024 .7659 .7291 .8854 .8484 .8113 .7743 .6186 .5795 .5430 .5042 .6769 .6376 .5992 .5618 .7928 .7550 .7167 .6784 85 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 J"-5/2 36 P5(36) 34.5 17821.9431 .9061 .8687 .8305 .7920 .7541 35 R5(35) 33.5 890.4173 .3793 .3413 .3033 .2646 .2250 37 P5(37) 35.5 817.5114 .4734 .4356 .3965 .3579 .3192 36 R5(36) 34.5 887.9631 .9256 .8885 .8517 .8153 38 P5(38) 36.5 813.1121 .0740 .0369 812.9994 .9622 .9257 37 R5(37) 35.5 884.2533 .2154 .1782 .1423 .1073 .0724 39 P5(39) 37.5 807.3880 .3501 .3140 .2769 .2414 .2070 38 R5(38) 36.5 884.9650 .9284 .8914 .8541 .8172 .7799 40 P5(40) 38.5 806.1210 .0839 .0472 .0102 805.9729 .9354 39 R5(39) 37.5 882.4681 .4301 .3925 .3540 .3153 .2753 41 P5(41) 39.5 .6054 .5672 .5286 .4901 .4501 40 R5(40) 38.5 880.1043 .0693 .0334 879.9951 .9558 .9154 42 P5(42) 40.5 797.3050 .2702 .2336 .1957 .1561 .1154 41 R5(41) 39.5 877.4389 .4010 .3649 .3289 .2946 .2602 43 P5(43) 41.5 792.6654 .6274 .5914 .5558 .5211 .4873 42 R5(42) 40.5 875.5532 .5011 .4497 .4055 .3512 .3033 44 P5(44) 42.5 788.8014 .7490 .6977 .6479 .5992 .5513 43 R5(43) 41.5 873.0608 .0223 872.9834 .9459 .9074 .8692 45 P5(45) 43.5 784.3426 .3043 .2665 .2283 .1895 .1512 44 R5(44) 42.5 870.5088 .4697 .4318 .3940 .3568 .3205 46 P5(46) 44.5 779.8205 .7813 .7434 .7054 .6680 .6319 Extra lines of F5 component observed by perturbation 33 R5(33) 31.5 17896.7025 .6620 .6219 .5829 .5451 .5078 35 P5(35) 33.5 827.7632 .7227 .6829 .6438 .6059 .5683 37 R5(37) 35.5 889.2045 .1710 .1365 .1023 .0699 .0375 39 P5(39) 37.5 812.3360 .3016 .2678 .2340 .2012 .1691 86 Appendix, continued F6 electron spin component (Main branches) N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 JT-5/2 7 R6(7) 4.5 17955.5106 .4451 .3840 .3314 .2869 .2446 9 P6(9) 6.5 8 R6(8) 5.5 952.5026 .4479 .3998 .3583 10 P6(10) 7.5 9 R6(9) 6.5 949.4467 .3974 .3539 .3157 .2828 .2549 11 P6(H) 8.5 10 R6(10) 7.5 946.3454 .3022 .2631 .2284 .1985 .1726 12 P6(12) 9.5 11 R6(ll) 8.5 943.3006 .2643 .2313 .2017 .1761 .1528 13 P6(13) 10.5 918.2152 .1801 .1472 12 R6(12) 9.5 940.4304 .4020 .3757 .3522 .3310 .3116 14 P6(14) 11.5 913.3469 .3194 .2937 13 R6(13) 10.5 937.7649 .7432 .7230 .7050 .6882 .6748 15 P6(15) 12.5 908.6635 14 R6(14) 11.5 929.4922 .4590 .4258 .3984 .3711 .2768 16 P6(16) 13.5 15 R6(15) 12.5 926.6662 .6285 .5925 .5596 .5289 .3466 17 P6(17) 14.5 893.5877 .5500 .4812 .4497 .4205 16 R6(16) 13.5 924.0547 .0129 923.9734 .9369 .9028 .5008 18 P6(18) 15.5 888.9774 .9363 .8972 .8603 .8254 .7932 17 R6(17) 14.5 921.6836 .6390 .5967 .5568 .5201 .8712 19 P6(19) 16.5 884.6096 .5652 .5233 .4837 .4457 .4103 18 R6(18) 15.5 919.5063 .4598 .4153 .3719 .3339 .4840 20 P6(20) 17.5 880.4350 .3889 .3449 .3025 .2626 .2254 19 R6(19) 16.5 917.4475 .4011 .3567 .3146 .2750 .2971 21 P6(21) 18.5 876.3779 .3321 .2878 .2455 .2052 .1673 20 R6(20) 17.5 915.3842 .3419 .3018 .2634 .2268 .2375 22 P6(22) 19.5 872.3185 .2765 .2367 .1979 .1609 .1256 21 R6(21) 18.5 913.0442 .0162 912.9890 .9624 .9370 .9124 23 P6(23) 20.5 867.981 .9540 .9268 .9007 .8746 .8497 87 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 JM+l/2 J"-l/2 J"-3/2 r-5/2 22 R6(22) 19.5 17913.3844 .3298 .2783 .2293 .1830 .1391 24 P6(24) 21.5 866.3270 .2732 .2220 .1730 .1267 .0822 23 R6(23) 20.5 911.4799 .4190 .3606 .3054 .2525 .2021 25 P6(25) 22.5 862.4298 .3694 .3110 .2554 .2026 .1519 24 R6(24) 21.5 909.7387 .6754 .6146 .5566 .5012 .4488 26 P6(26) 23.5 858.6950 .6318 .5710 .5129 .4575 .4043 25 R6(25) 22.5 908.0325 907.9678 .9059 .8463 .7897 .7352 27 P6(27) 24.5 854.9935 .9289 .8667 .8073 .7500 .6957 26 R6(26) 23.5 906.0873 .0214 905.9583 .8978 .8399 .7846 28 P6(28) 25.5 851.0570 850.9916 .9285 .8680 .8101 .7544 27 R6(27) 24.5 905.3748 .3077 .2433 .1818 .1229 .0665 29 P6(29) 26.5 848.3544 .2876 .2234 .1618 .1029 .0469 28 R6(28) 25.5 903.4703 .4032 .3390 .2776 .2180 .1617 30 P6(30) 27.5 844.4598 .3927 .3285 .2668 .2077 .1510 29 R6(29) 26.5 901.6829 .6162 .5511 .4892 .4292 .3721 31 P6(31) 28.5 840.6817 .6150 .5505 .4884 .4284 .3709 30 R6(30) 27.5 899.6563 .5902 .5262 .4643 .4049 .3476 32 P6(32) 29.5 836.6693 .6031 .5387 .4770 31 R6(31) 28.5 899.5072 .4423 .3234 .2704 .2214 33 P6(33) 30.5 834.5325 .4670 .4062 .3490 .2461 32 R6(32) 29.5 897.0155 896.9463 .8804 .8188 .7588 .7025 34 P6(34) 31.5 830.0536 829.9846 .9192 .8571 .7975 .7410 33 R6(33) 30.5 894.9341 .8673 .8017 .7380 .6771 .6183 35 P6(35) 32.5 825.9893 .9218 .8569 .7933 .7321 .6729 34 R6(34) 31.5 892.4420 .3768 .3138 .2519 .1930 .1356 36 P6(36) 33.5 821.5134 .4479 .3843 .3227 .2638 .2060 35 R6(35) 32.5 892.8465 .7818 .7183 .6568 .5966 .5374 88 Appendix, continued N Branch J" F W + 5 / 2 J-+3/2 r+1/2 J"-l/2 J"-3/2 J"-5/2 37 P6(37) 34.5 17819.9349 .8704 .8078 .7457 .6853 .6260 36 R6(36) 33.5 889.9933 .9240 .8589 .7950 .7349 .6761 38 P6(38) 35.5 815.1011 .0325 814.9669 .9038 .8433 .7849 37 R6(37) 34.5 887.5657 .4987 .4332 .3700 .3085 .2490 39 P6(39) 36.5 810.6922 .6255 .5602 .4966 .4355 .3756 38 R6(38) 35.5 884.3634 .3021 .2429 .1849 .1291 .0746 40 P6(40) 37.5 805.5139 .4528 .3942 .3362 .2803 .2254 39 R6(39) 36.5 884.8861 .8222 .7596 .6980 .6372 .5773 41 P6(41) 38.5 804.0594 803.9948 .9326 .8705 .8096 .7496 40 R6(40) 37.5 882.0463 881.9781 .9130 .8496 .7887 .7310 42 P6(42) 39.5 799.2427 .1731 .1076 .0450 798.9837 .9245 41 R6(41) 38.5 879.4043 .3378 .2732 .2108 .1502 .0916 43 P6(43) 40.5 794.6249 .5582 .4937 .4310 .3708 .3120 42 R6(42) 39.5 44 P6(44) 41.5 43 R6(43) 40.5 875.6518 .5890 .5243 .4626 .4017 .3419 45 P6(45) 42.5 786.9282 .8639 .8010 .7390 .6782 .6184 44 R6(44) 41.5 873.1509 .0845 .0191 872.9535 .8900 .8280 46 P6(46) 43.5 782.4584 .3910 .3259 .2610 .1976 .1352 F3 electron spin component (Main branches) 4 R3(4) 4.5 17947.6046 .6282 .6532 .6773 .6993 .7169 6 P3(6) 6.5 5 R3(5) 5.5 948.3278 .3442 .3634 .3841 .4046 .4233 7 P3(7) 7.5 6 R3(6) 6.5 948.6010 .6111 .6255 .6431 .6615 .6802 8 P3(8) 8.5 7 R3(7) 7.5 948.4462 .4519 .4621 .4765 .4934 .5119 9 P3(9) 9.5 8 R3(8) 8.5 9:7.9044 .9054 .9129 .9248 .9403 .9588 10 P3(10) 10.5 89 Appendix, continued N Branch J" F"=J"+5/2 r+3/2 J"+l/2 J "-1/2 J"-3/2 J"-5/2 9 R3(9) 9.5 17947.0202 .0181 .0225 .0323 .0466 .0648 11 P3(ll) 11.5 10 R3(10) 10.5 945.8291 .8239 .8262 .8342 .8472 .8653 12 P3(12) 12.5 11 R3(ll) 11.5 944.3610 .3542 .3542 .3610 .3722 .3897 13 _P3(13) 13.5 12 R3(12) 12.5 942.6434 .6313 .6287 .6334 .6434 .6632 14 P3(14) 14.5 13 R3(13) 13.5 940.6990 .6861 .6827 .6861 .6959 .7137 15 P3(15) 15.5 14 R3(14) 14.5 938.5455 .5307 .5257 .5282 .5376 .5545 16 P3(16) 16.5 Extra lines of F3 component observed by perturbation 2 R3(2) 2.5 17951.1646 .2095 .2487 .2838 4 P3(4) 4.5 944.3960 .4170 .4398 .4624 .4826 .4996 3 R3(3) 3.5 952.2028 .2384 .2700 .2993 .3218 .3417 5 P3(5) 5.5 943.3124 .3330 .3555 .3780 .3992 .4181 4 R3(4) 4.5 953.0382 .0692 .1000 .1272 .1512 .1701 6 P3(6) 6.5 942.0956 .1164 .1389 .1616 .1849 .2063 5 R3(5) 5.5 953.7696 .7993 .8280 .8557 .8806 .9018 7 P3(7) 7.5 940.7986 .8200 .8431 .8660 .8896 .9139 6 R3(6) 6.5 954.4374 .4675 .4958 .5224 8 P3(8) 8.5 939.4465 .4701 7 R3(7) 7.5 955.1200 Main lines and extra lines of F3 from Internal hyperfine perturbation 23 R3(23) 23.5 17914.0761 .0572 .0532 .0572 .0687 .0883 25 P3(25) 25.5 865.0493 .0272 .0219 .0248 .0349 .0537 24 R3(24) 24.5 911.5551 .5511 .5672 .5866 90 Appendix, continued N Branch J" F'=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 Jn-5/2 26 P3(26) 26.5 17860.5539 .5323 .5270 .5323 .5410 .5599 25 R3(25) 25.5 909.2329 .2140 .2105 .2140 .2263 .2459 27 P3(27) 27.5 856.2177 .1952 .1906 .1952 .2052 .2233 26 R3(26) 26.5 907.0299 .0103 .0066 .0103 .0228 .0421 28 P3(28) 28.5 852.0223 851.9988 .9946 .9988 .0086 .0270 27 R3(27) 27.5 904.9316 .9107 .9071 .9107 .9228 .9418 29 P3(29) 29.5 847.9343 .9096 .9050 .9096 .9193 .9343 28 R3(28) 28.5 902.9122 .8900 .8850 .8900 .9008 .9193 30 P3(30) 30.5 843.9025 .8975 .8924 .8964 .9062 .9237 29 R3(29) 29.5 900.9412 .9160 .9117 .9160 .9268 .9452 31 P3(31) 31.5 939.9619 .9335 .9281 .9335 .9415 .9590 30 R3(30) 30.5 32 P3(32) 32.5 836.0408 .0102 .0035 .0068 .0166 .0338 31 R3(31) 31.5 33 P3(33) 33.5 32 R3(32) 32.5 895.8347 .8323 .8152 .8073 .8073 .8152 34 P3(34) 34.5 829.8866 .8890 .8712 .8629 .8629 .8687 Branches labelled by RS34 and PQ34 are induced by internal hyperfine perturbations 23 RS34(23) 23.5 17914.1816 .1893 .1988 .1988 .1893 25 PQ34(25) 25.5 865.1466 .1600 .1654 .1654 .1576 24 RS34(24) 24.5 911.6762 .6902 .6964 .6887 26 PQ34(26) 26.5 860.6492 .6644 .6714 .6714 .6644 25 RS34(25) 25.5 909.3326 .3491 .3560 .3560 .3491 27 PQ34(27) 27.5 856.3103 .3268 .3346 .3358 .3299 26 RS34(26) 26.5 907.1271 .1433 .1518 .1532 .1464 28 PQ34(28) 28.5 852.1125 .1299 .1386 .1405 .1350 27 RS34(27) 27.5 905.0258 .0432 .0519 .0540 .0481 29 PQ34(29) 29.5 848.0215 .0400 .0495 .0519 .0469 28 RS34(28) 28.5 903.0036 .0205 .0302 .0327 .0274 30 PQ34(30) 30.5 844.0086 .0274 .0368 .0401 .0361 91 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 J"-5/2 29 RS34(29) 29.5 17901.0285 .0469 0565 .0598 .0558 31 PQ34(31) 31.5 840.0434 .0624 .0730 .0766 .0730 30 RS34(30) 30.5 .1502 32 PQ34(32) 32.5 836.1192 .1385 .1491 31 RS34(31) 31.5 33 PQ34(33) 33.5 32 RS34(32) 32.5 895.9414 .9495 .9495 .9435 .9314 34 PQ34(34) 34.5 828.9972 829.0062 .0062 .0010 828.9894 Main lines and extra lines of F4 from Internal hyperfine perturbation 8 R4(8) 7.5 17945.1221 10 P4(10 8.5 9 R4(9) 9.5 944.9736 .9754 .9736 .9689 .9603 .9475 11 P4(ll) 10.5 10 R4(10) 9.5 944.6072 .6072 .6028 .5953 .5838 .5663 12 P4(12) 11.5 11 R4(ll) 10.5 943.9953 .9932 .9874 .9783 .9645 .9432 13 P4(13) 12.5 12 R4(12) 11.5 943.0379 .0350 .0310 .0199 .0038 942.9764 14 P4(14) 13.5 13 R4(13) 12.5 941.6365 .6452 .6487 .6452 .6196 .5873 15 P4(15) 14.5 14 R4(14) 13.5 939.7898 .7898 .7857 .7764 .7603 .7279S 16 P4(16) 15.5 13 R4(13) 12.5 941.4643 .4657 .4717 .4827 .4987 15 P4(15) 14.5 14 R4(14) 13.5 939.6180 .6145 .6180 .6281 .6458 16 P4(16) 15.5 25 R4(25) 24.5 909.1880 .1957 .1950 .1918 .1813 .1145 27 P4(27) 26.5 856.1656 .1733 .1749 .1716 .1615 .0877 26 R4(26) 25.5 906.9518 .9589 .9589 .9554 .9448 .8749 92 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 J' "-5/2 28 P4(28) 27.5 17851.9374 .9451 .9451 .9451 .9334 .8561 27 R4(27) 26.5 904.8129 .8203 .8203 .8166 .8063 .7327 29 P4(29) 28.5 847.8090 .8165 .8165 .8144 .8050 .7241 28 R4(28) 27.5 902.7300 .7374 .7374 .7333 .7232 .6452 30 P4(30) 29.5 843.7354 .7432 .7432 .7402 .7310 .6462 29 R4(29) 28.5 900.6396 .6436 .6436 .6381 .6278 .5465 31 P4(31) 30.5 839.6548 .6595 .6595 .6548 .6458 .5578 Branches labelled by RQ43and P043 are induced by internal hyperfine perturbations 25 RQ43(25) 24.5 17909.0691 .0568 .0554 .0619 .0783 27 P043(27) 26.5 856.0504 .0373 .0346 .0404 .0552 26 RQ43(26) 25.5 906.8349 .8217 .8192 .8250 .8403 28 P043(28) 27.5 851.8242 .8098 .8061 .8112 .8253 27 RQ43(27) 26.5 904.6982 .6844 .6811 .6858 .7000 29 P043(29) 28.5 847.6970 .6819 .6774 .6819 .6952 28 RQ43(28) 27.5 902.6168 .6017 .5971 .6017 .6153 30 PO43(30) 29.5 843.6245 .6082 .6028 .6064 .6191 29 RQ43(29) 28.5 900.5270 .5086 .5021 .5046 .5174 31 P043(31) 30.5 839.5448 .5252 .5196 .5196 .5311 Upper state electron spin components induce the following satellite branches 3 TR64(3) 2.5 17966.2909 .2963 .2993 5 RP64(5) 4.5 957.1397 .4798 .4572 4 TR64(4) 3.5 965.6767 .6767 .6743 .6711 .6678 .6643 6 RP64(6) 5.5 954.5808 .1240 .1088 .0954 .0841 .0759 5 TR64(5) 4.5 964.9101 .9101 .9077 .9043 .8999 .8952 7 RP64(7) 6.5 951.8354 .5714 .5610 .5504 .5397 .5300 6 TR64(6) 5.5 963.9879 .9903 .9903 .9879 .9835 .9780 8 RP64(8) 7.5 948.9259 .8310 .8242 .8158 .8061 .7951 7 TR64(7) 6.5 962.9549 .9597 .9597 .9597 .9560 .9503 93 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 J"-5/2 9 RP64(9) 8.5 17945.8960 .9259 .9222 .9163 .9079 .8966 8 TR64(8) 7.5 961.8712 .8802 .8853 .8869 .8853 .8802 10 RP64(10) 9.5 .8999 .8999 .8960 .8888 .8769 9 TR64(9) 8.5 955.1563 .1829 .2034 .2182 .2260 .2260 11 RP64(11) 10.5 9 TR64(9) 8.5 960.8442 .8592 .8693 .8754 .8764 .8722 11 RP64(11) 10.5 10 TR64(10) 9.5 959.9870 860.0092 .0258 .0374 .0430 .0411 10 TR64(10) 9.5 954.7228 .7495 .7705 .7855 .7947 .7947 12 RP64(12) 11.5 11 TR64(11) 10.5 959.3335 .3622 .3850 .4015 .4114 .4114 13 RP64(13) 12.5 11 TR64(11) 10.5 953.9722 .9930 .0084 .0183 .0218 .0156 13 RP64(13) 12.5 12 TR64(12) 11.5 953.0747 .0911 .1018 .1070 .1070 .0936? 14 RP64(14) 13.5 13 TR64(13) 12.5 952.2581 .2704 .2774 .2790 .2742 .2554 15 RP64(15) 14.5 14 TR64(14) 13.5 951.6567 .6654 .6688 .6670 .6586 .6336 16 RP64(16) 15.5 2 TQ63(2) 2.5 965.7751 .7804 .7855 .8028 3 TQ63(3) 3.5 965.6243 .6298 .6381 .6466 .6547 4 TQ63(4) 4.5 965.1779 .1837 .1927 .2038 .2151 .2250 5 TQ63(5) 5.5 964.5029 .5087 .5190 .5321 .5457 .5592 6 TQ63(6) 6.5 963.6456 .6520 .6631 .6775 .6938 .7109 7 TQ63(7) 7.5 962.6547 .6620 .6749 .6910 .7101 .7297 4 RQ65(4) 2.5 961.2660 .2175 .1788 .1527 5 RQ65(5) 3.5 958.7038 .6634 .6300 6 RQ65(6) 4.5 7 RQ65(7) 5.5 953.2385 .2082 .1813 94 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 JM-3/2 J"-5/2 2 TR53(2) 2.5 17968.4588 .4787 .4990 .5194 4 RP53(4) 4.5 961.6907 .6873 .6873 .6907 .7023 .7094 3 TR53(3) 3.5 969.0917 .1059 .1214 .1358 .1541 .1710 5 RP53(5) 5.5 960.1986 .1986 .2056 .2130 .2244 .2336 4 TR53(4) 4.5 969.4259 .4326 .4512 .4662 .4795 .4915 6 RP53(6) 6.5 958.4836 .4836 .4901 .5004 .5136 .5274 TR53(5) 5.5 969.5236 .5328 .5463 .5619 .5778 .5924 7 RP53(7) 7.5 956.5519 .5537 .5613 .5727 .5874 .6040 6 TR53(6) 6.5 969.4117 .4199 .4326 .4485 .4662 .4844 8 RP53(8) 8.5 954.4217 .4235 .4314 .4444 .4608 .4803 7 TE153(7) 7.5 969.1099 .1175 .1298 .1464 .1650 .1849 9 RP53(9) 9.5 952.1068 .1089 ..1172 .1307 .1485 .1694 8 TR53(8) 8.5 968.6356 .6420 .6541 .6706 .6905 .7126 10 RP53(10) 10.5 949.6222 .6238 .6324 .6466 .6652 .6882 9 TR53(9) 9.5 968.0074 .0133 .0252 .0415 .0621 .0859 11 RP53(11) 11.5 946.9872 .9886 .9975 947.0120 .0323 .0551 10 TR53(10) 10.5 967.2457 .2510 .2629 .2792 .3000 .3248 12 RP53(12) 12.5 944.2193 .2193 .2296 .2439 .2628 .2887 11 TR53(11) 11.5 966.3707 .3715 .3818 .3991 .4236 .4560 13 RP53(13) 13.5 941.3391 .3342 .3414 .3571 .3826 .4125 12 TR53(12) 12.5 965.4080 .4096 .4202 .4403 .4619 .4899 14 RP53(14) 14.5 13 TR53(13) 13.5 964.3729 .3747 .3842 .3980 15 RP53(15) 15.5 3 RQ54(3) 2.5 963.5971 .5859 4 RQ54(4) 3.5 962.1890 .1793 .1693 .1611 .1528 .1481 5 RQ54(5) 4.5 960.6063 .5995 .5920 .5839 .5762 .5695 95 Appendix, continued N Branch J" F' ,,=J"+5/2 F+3/2 J"+l/2 J"-l/2 JT-3/2 J"-5/26 RQ54(6) 5.517958.8268 .8231 .8174 .8103 .8029 .7945 7 RQ54(7) 6.5 956.8522 .8522 .8474 .8421 .8342 .8249 8 RQ54(8) 7.5 0 TR42(0) 1.5 947.6046 .6282 2 RP42(2) 3.5 1 TR42(1) 2.5 950.3203 .3784 .4257 .4638 .4877 3 RP42(3) 4.5 2 TR42(2) 3.5 952.4580 .5130 .5578 .5928 .6189 .6352 4 RP42(4) 5.5 3 TR42(3) 4.5 954. .6169 .6590 .7228 .7422 5 RP42(5) 6.5 4 TR42(4) 5.5 956.7159 .7627 .8028 .8365 .8651 .8867 6 RP42(6) 7.5 5 TR42(5) 6.5 958.8350 .8932 .9299 .9483 .9756 7 RP42(7) 8.5 6 PR46(6) 3.5 937.7254 .6748 .6311 .5959 .5687 .5608 8 NP46(8) 5.5 922.5715 .5264 .4837 7 PR46(7) 4.5 935.6298 .5806 .5379 .5031 .4747 .4530 9 NP46(9) 6.5 918.5081? .4593 .4227 .3835 .3523 .3257 8 PR46(8) 5.5 933.6408 .5928 .5512 .5148 .4854 .4606 10 NP46(10) 7.5 914.5365 .4909 .4495 .4126 .3809 .3529 9 PR46(9) 6.5 931.6578 .6103 .5686 .5315 .4991 .4724 11 NP46(11) 8.5 910.5620 .5154 .4744 .4366 .4010 .3735 10 PR46(10) 7.5 929.5957 .5488 .5059 .4668 .4316 .4043 12 NP46(12) 9.5 906.5094 .4626 .4194 .3806 .3452 .3138 11 PR46(11) 8.5 927.4303 .3810 .3359 .2958 .2595 .2277 13 NP46(13) 10.5 902.3456 .2970 .2526 .2118 .1746 .1411 12 PR46(12) 9.5 925.0513 .0002 924.9533 .9112 .8743 .8383 14 NP46(14) 11.5 13 PR46(13) 10.5 922.4258 .3734 .3248 .2812 .2407 .2048 96 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 J"-3/2 J"-5/2 15 NP46(15) 12.5 14 PR46(14) 11.5 17919.4585 .4055 .3567 .3112 .2687 .2294 16 NP46(16) 13.5 1 TR31(1) 1.5 955.1244 .1873 .2446 3 RP31(3) 3.5 2 TR31(2) 2.5 4 RP31(4) 4.5 951.1625 .2095 .2487 .2935 3 TR31(3) 3.5 960.8556 .8947 .9292 .9600 .9865 961.0098 5 RP31(5) 5.5 952.1033 .1241 5 PR35(5) 3.5 939.7742 .7571 .7413 .7267 .7129 .7003 7 NP35(7) 5.5 6 PR35(6) 4.5 938.4717 .4551 .4405 .4280 .4191 .4114 8 NP35(8) 6.5 923.3112 .2998 .2888 .2788 .2694 .2608 7 PR35(7) 5.5 937.0599 .0400 .0222 .0072 936.9940 .9835 9 NP35(9) 7.5 919.9226 .9005 .8814 .8628 .8472 .8336 8 PR35(8) 6.5 .2191 .1948 .1764 10 NP35(10) 8.5 916.1212 .0973 .0760 .0550 .0382 .0221 9 PR35(9) 7.5 933.0124 932.9871 .9649 .9443 .9263 .9108 11 NP35(11) 9.5 911.9054 .8802 .8564 .8348 .8147 .7968 10 PR35(10) 8.5 930.3873 .3632 .3412 .3208 .3030 12 NP35(12) 10.5 907.3180 .2908 .2657 .2426 .2212 .2017 11 PR35(11) 9.5 927.4830 .4555 .4365 .4158 .3936 .3697 13 NP35(13) 11.5 902.3920 .3641 .3376 .3133 .2905 .2694 12 PR35(12) 10.5 14 NP35(14) 12.5 13 PR35(13) 11.5 15 NP35(15) 13.5 14 PR35(14) 12.5 917.0089 916.9774 .9481 .9213 .8968 16 NP35(16) 14.5 885.9271 .8954 .8655 .8379 .8131 .7898 15 PR35(15) 13.5 913.0384 .0079 912.9791 17 NP35(17) 15.5 97 A - X transition Appendix, continued N Branch J" F"=J"+5/2 JM+3/2 J"+l/2 r-i/2 J"-3/2 JM-5/2 5 Rl(5) 7.5 17881.9906 882.0485 .0965 .1407 .1802 .2137 7 Pl(7) 9.5 869.0019 .0548 .1039 .1477 .1876 .2234 6 Rl(6) 8.5 8 Pl(8) 10.5 7 Rl(7) 9.5 9 Pl(9) 11.5 8 Rl(8) 10.5 882.9759 883.1292 10 Pl(10) 12.5 863.9653 864.0189 .0697 .1175 .1634 .2061 9 Rl(9) 11.5 881.0621 11 Pl(ll) 13.5 860.0414 .0853 .1265 .1644 .19?? 10 Rl(10) 12.5 879.9983 880.0477 .0944 .1381 .1783 .2166 12 Pl(12) 14.5 856.9738 857.0222 .0682 .1116 .1523 .1907 11 Rl(ll) M3.5 878.7132 .7649 .8138 .8640 .8965 .9320 13 Pl(13) 15.5 853.6858 .7372 .7846 .8277 .8676 .9034 12 Rl(12) 14.5 877.0906 .1401 .1866 .2293 .2690 .3059 14 Pl(14) 16.5 850.0609 .1094 .1550 .1976 .2377 .2751 13 Rl(13) 15.5 875.1209 .1683 .2139 .2562 .2960 .3328 15 Pl(15) 17.5 846.0880 .1349 .1800 .2220 .2620 .2993 14 Rl(14) 16.5 872.7894 .8363 .8809 .9226 .9507 16 Pl(16) 18.5 841.7536 .8002 .8445 .8860 .9151 .9627 6 Rl(6) 8.5 900.3728 .4365 .4941 .5465 .5883 .6278 8 Pl(8) 10.5 885.3759 .4373 .4935 .5450 .5869 7 RK7) 9.5 900.8137 .8750 .9307 .9816 901.0229 .0660 9 Pl(9) 11.5 883.8078 .8668 .9218 .9725 884.0184 .0607 8 Rl(8) 10.5 900.9072 .9672 .0718 .1178 .1590 10 Pl(10) 12.5 881.8952 .9530 882.0557 98 Appendix, continued Fi electron spin component (Main branches) N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 r--3/2 J"-5/2 9 Rl(9) 11.5 17900.6908 .7500 .8039 .8541 .8997 .9412 11 Pl(ll) 13.5 879.6802 .7309 .8341 .8801 10 Rl(10) 12.5 900.1764 .2339 .2878 .3373 .3836 .4257 12 Pl(12) 14.5 877.1548 .2110 .3138 .3649 11 Rl(ll) 13.5 898.8757 .9307 13 Pl(13) 15.5 12 Rl(12) 14.5 898.2396 .2958 .3513 .3970 .4431 .4854 14 Pl(14) 16.5 871.2099 .2649 .3206 .3666 .4126 4563 13 Rl(13) 15.5 896.7920 .8456 .8986 .9463 .9933 897.0361 15 PK15) 17.5 867.7576 .8118 .8636 .9129 .9584 868.0017 14 Rl(14) 16.5 894.9997 895..0535 .1041 .1530 .1984 16 PK16) 18.5 863.9653 864.0189 .0697 .1175 .1634 .2061 15 Rl(15) 17.5 892.8929 .9908 .0357 .0738 .1186 17 PI (17) 19.5 859.8585? .9075? .9555 860.0008 .0853 16 Rl(16) 18.5 18 Pl(18) 20.5 17 Rl(17) 19.5 887.3498 .4006 .4485 .4947 .5383 .5795 19 Pl(19) 21.5 850.3141 .3644 .4125 .4596 .5019 .5440 18 Rl(18) 20.5 884.1291 20 PI (20) 22.5 846.0880 .1349 .1800 .2220 F 2 electron spin component (Main branches) 3 R2(3) 4.5 17891.1606 .2096 .2512 .2853 .3119 .3322 5 P2(5) 6.5 882.2812 .3247 .3623 .3947 .4301 .4561 4 R2(4) 5.5 891.4740 .50?? 6 P2(6) 7.5 5 R2(5) 6.5 891.2538 .2904 .3226 .3914 7 P2(7) 8.5 878.3052 .3385 .3681 .3944 .4169 .4368 6 R2(6) 7.5 890.6589 .6800 .7081 .7324 .7534 .7710 8 P2(8) 9.5 875.6797 .7091 .7354 .7587 .7793 .7977 7 R2(7) 8.5 889.6544 .6831 .7084 .7310 .7499 .7660 9 P2(9) 10.5 872.6717 .6982 .7220 .7432 .7624 .7795 99 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 r-1/2 r-3/2 r-5/2 8 R2(8) 9.5 17888.3323 .3586 .3818 .4026 .4209 .4367 10 P2(10) 11.5 869.3373 .3623 .3846 .4046 .4229 .4391 9 R2(9) 10.5 886.7520 .7774 .8008 .8210 .8397 .8560 11 P2(ll) 12.5 865.7468 .7713 .7932 .8132 .8313 .8487 10 R2(10) 11.5 884.9890 885.0152 .0392 .0607 .0801 .0977 12 P2(12) 13.5 861.9772 862.0024 .0257 .0465 .0657 .0831 11 R2(ll) 12.5 883.0944 .1225 .1483 13 P2(13) 14.5 858.0761 .1030 .1280 .1507 .1724 .1918 12 R2(12) 13.5 881.0675 .0977 .1243 .1498 .1730 .1944 14 P2(14) 15.5 854.0436 .0726 .0992 .1243 .1473 .1689 13 R2(13) 14.5 878.8582 .8893 .9186 .9455 .9705 .9933 15 P2(15) 16.5 849.8311 .8612 .8895 .9161 .9408 9639 14 R2(14) 15.5 876.4040 .4365 .4666 .4949 .5209 .5452 16 P2(16) 17.5 845.3747 .4062 .4362 .4638 .4896 .5138 15 R2(15) 16.5 873.6550 .6884 .7191 17 P2(17) 18.5 840.6213 .6540 .6841 .7128 .7399 .7645 16 R2(16) 17.5 870.6645 .6918 .7175 18 P2(18) 19.5 835.5349 .5680 .5991 .6284 .6556 .6813 17 R2(17) 18.5 19 P2(19) 20.5 830.1828 .2107 .2371 18 R2(18) 19.5 863.3017 .3356 .3676 .3993 20 P2(20) 21.5 824.2662 .2999 .3612 .3901 .4166 F5 electron spin component (Main branches) 8 R5(8) 6.5 17886.2374 .2261 .2150 .2046 .1947 .1855 10 P5(10) 8.5 867.0806 9 R5(9) 7.5 884.5918 .5673 .5456 .5258 .5087 .4940 11 P5(H) 9.5 863.4870 .4625 .4393 .4183 .3993 .3823 10 R5(10) 8.5 882.7340 .7110 .6903 .6847 .6547 .6399 100 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 J"-l/2 r-3/2 JM-5/2 12 P5(12) 10.5 17859.6366 .6135 .5920 .5719 .5536 .5371 11 R5(ll) 9.5 880.6426 .6214 .6016 .5826 .5675 .5492 13 P5(13) 11.5 855.5498 .5282 .5077 .4892 .4719 .4560 12 R5(12) 10.5 878.2913 .2717 .2536 .2366 .2186 .2064 14 P5(14) 12.5 851.2031 .1835 .1647 .1472 .1305 .1144 13 R5(13) 11.5 875.6552 .6370 .6203 .6042 .5890 .5736 15 P5(15) 13.5 846.5706 .5524 .5346 .5181 .5024 .4890 14 R5(14) 12.5 872.7138 .6982 .6810 .6662 .6518 .6389 16 P5(16) 14.5 841.6309 .6136 .5817 .5670 .5527 15 R5(15) 13.5 869.4365 .4204 .4046 .3905 .3757 .36?? 17 P5(17) 15.5 16 R5(16) 14.5 865.7567 .7378 .7213 .7053 .6912 .6748 18 P5(18) 16.5 830.6769 .6599 .6428 .6269 .6114 .5969 17 R5(17) 15.5 864.7384 .7174 19 P5(19) 17.5 827.6638 .6456 .6222 .6026 .5834 .5650 18 R5(18) 16.5 20 P5(20) 18.5 19 R5(19) 17.5 855.1240 .1090 .0985 .0864 .0746 .0632 21 P5(21) 19.5 814.0573 .0445 .0312 .0189 .0073 813.9951 20 R5(20) 18.5 850.1881 .1772 .1659 .1550 .1442 .1343 22 P5(22) 20.5 807.1203 .1146 .1036 .0929 F6 electron spin component (Main branches) .0816 .0706 8 R6(8) 5.5 17890.5635 .5098 10 P6(10) 7.5 871.4607 .4083 .3614 .2840 .2530 9 R6(9) 6.5 888.3948 .3432 .2976 .2581 .2234 .1943 11 P6(ll) 8.5 10 R6(10) 7.5 886.3173 .2489 .1811 .1161 .0514 885.9925 12 P6(12) 9.5 863. .1606 .0937 .0290 862.9657 11 R6(ll) 8.5 884.0503 .0034 883.9602 .9218 .8870 .8572 13 P6(13) 10.5 858.9652 .9192 .8765 .8375 .8022 .7728 12 R6(12) 9.5 881.7310 .6885 .6490 .6119 .5781 .5464 101 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 J"+l/2 JM-l/2 r--3/2 J"-5/2 14 P6(14) 11.5 17854.6482 .6063 .5666 .5292 .4939 .4612 13 R6(13) 10.5 877.5867 .5381 .4921 .4494 .4010 .3604 15 P6(15) 12.5 848.5026 .4544 .4073 .3614 .3171 .2739 14 R6(14) 11.5 876.5060 .4666 .4286 .3938 .3617 .3321 16 P6(16) 13.5 845.4270 .3873 .3499 .3150 .2827 .2524 15 R6(15) 12.5 873.5182 .4799 .4439 .4106 .3793 .3512 17 P6(17) 14.5 840.4383 .4002 .3643 .3308 .2992 .2703 16 R6(16) 13.5 870.2388 .2020 .1675 .1353 .1048 .0775 18 P6(18) 15.5 835.1607 .1244 .0899 .0575 .0272 834.9987 17 R6(17) 14.5 866.6434 .6079 .5741 .5428 .5139 .4867 19 P6(19) 16.5 829.5689 .5333 .5001 .4684 .4391 .4115 18 R6(18) 15.5 862.7043 .6709 .6363 .6048 .5751 .5479 20 P6(20) 17.5 823.6312 .5962 .5631 .5323 .5018 .4739 19 R6(19) 16.5 858.3336 .2943 .2576 .2227 21 P6(21) 18.5 20 R6(20) 17.5 22 P6(22) 19.5 21 R6(21) 18.5 848.9482 .9210 .8946 .8704 .8470 .8248 23 P6(23) 20.5 803.8883 .8609 .8351 .8096 .7864 .7643 22 R6(22) 19.5 843.3672 .3412 .3165 .2929 .2706 .2493 24 P6(24) 21.5 796.3102 .2841 .2600 .2364 .2127 .1920 13 R6(13) 10.5 879.2334 .1908 .1517 .1154 .0827 .0528 15 P6(15) 12.5 850.1518 .1094 .0705 .0342 .0005 849.9699 F3 electron spin component (Main branches) 3 R3(3) 3.5 17886.8008 .8191 .8493 .8836 .9092 .9288 5 P3(5) 5.5 877.8997 .9045 .9253 .9586 .9747 .9846 4 R3(4) 4.5 886.5914 .6170 .6439 .6704 .6946 .7141 6 P3(6) 6.5 875.6501 5 R3(5) 5.5 886.2004 .2090 .2282 .2489 .2734 .2963 7 P3(7) 7.5 873.2386 .2500 .2660 .2854 .3072 .3296 1 0 2 Appendix, continued N Branch J" F"=J"+5/2 r+3/2 J"+l/2 r-i/2 J"-3/2 J"-5/2 6 R3(6) 6.5 17885.6771 .6918 .7114 .7327 .7554 .7777 8 P3(8) 8.5 870.6875 .6966 .7105 .7294 .7507 .7747 7 R3(7) 7.5 885.1912 .2055 .2247 .2469 .2709 .2959 9 P3(9) 9.5 8 R3(8) 8.5 884.7698 .7840 .8036 .8268 .8804 10 P3(10) 10.5 865.7567 .7663 .7822 .8034 .8277 .8568 9 R3(9) 9.5 884.2343 .2481 .2675 11 P3(ll) 11.5 863.2147 .2240 .2402 10 R3(10) 10.5 883.4764 .4884 .5062 .5295 .5466 .5561 12 P3(12) 12.5 860.4503 .4583 .4737 .4942 .5202 11 R3(ll) 11.5 882.4459 .4561 .4733 .4957 .5227 .5541 13 P3(13) 13.5 857.4125 .4189 .4336 .4539 .4797 .5117 12 R3(12) 12.5 14 P3(14) 14.5 854.0789 .0843 .0992 .1173 .1429 .1750 13 R3(13) 13.5 879.4634 .4704 .4857 15 P3(15) 15.5 850.4238 .4278 .4405 .4598 .4850 .5172 14 R3(14) 14.5 877.4721 .4777 .4827 .4921 .5125 .5381 16 P3(16) 16.5 846.4314 .4314 .4334 .4557 .4641 .4890 15 R3(15) 15.5 875.1302 .1342 .1473 .1668 .1927 .2257 17 P3(17) 17.5 842.0867 .0860 .0980 .1159 .1407 .1729 16 R3(16) 16.5 872.4247 .4276 .4395 .4584 .4838 .5175 18 P3(18) 18.5 837.3810 .3810 .3909 .4084 17 R3(17) 17.5 19 P3(19) 19.5 18 R3(18) 18.5 865.8873 .8873 .8976 .9152 .9396 .9717 20 P3(20) 20.5 826.8464 .8433 .8516 .8680 .8912 .9230 19 R3(19) 19.5 860.1872 .1899 .1950 .1997 .2167 .2423 F4 electron spin component (Main branches) 4 R4(4) 3.5 6 P4(6) 5.5 103 Appendix, continued N Branch J" F"=J"+5/2 J"+3/2 r+1/2 J"-l/2 J"-3/2 J"-5/2 5 R4(5) 7.5 P4(7) 6.5 17874.8991 .8947 .8902 .8854 .8810 .8769 6 R4(6) 8.5 888.4641 .4803 .5009 .4926 .5062 .5087 8 P4(8) 7.5 873.3992 7 R4(7) 9.5 887.9235 .9175 .9097 .8998 .8885 .8765 9 P4(9) 8.5 870.8602 .8539 .8441 .8332 .8175 .7993 8 R4(8) 7.5 886.3226 .3143 .3038 .2907 .2764 .2599 10 P4(10) 9.5 9 R4(9) 8.5 884.4457 .4409 .4349 .4211 .4103 .3976 11 P4(ll) 10.5 10 R4(10) 9.5 12 P4(12) 11.5 859.0485 .0429 .0410 .0355 .0221 858.9979 11 R4(ll) 10.5 879.6505 .6485 .6457 .6426 13 P4(13) 12.5 854.5918 .5879 .5855 .5740 .5490 .5267 12 R4(12) 11.5 876.9631 .9642 .9092 .8903 14 P4(14) 13.5 849.8946 .8913 .8866 .8787 .8470 .8248 13 R4(13) 12.5 15 P4(15) 14.5 14 R4(14) 13.5 870.9017 .8968 .8870 .8801 .8684 16 P4(16) 15.5 839.8210 .8268 .8257 .8188 .8061 .7917 Upper state electron spin component F4 induce the following satellite branches 12 TR42(12) 13.5 17898.7992 .8400 ..8774 .9120 .9441 14 RP42(14 15.5 871.7775 .8166 .8536 .8872 .9199 .9492 13 TR42(13) 14.5 897.3995 .4411 .4791 .5153 .5492 .5793 15 RP42(15) 16.5 868.3715 .4122 .4429 .4861 .5186 .5496 14 TR42(14) 15.5 895.7275 .7694 .8073 .8456 .8798 .9115 16 RP42(16) 17.5 864.6971 .7384 .7775 .8136 .8482 .8708 15 TR42(15) 16.5 893.7592 .8020 .8393 17 RP42(17) 18.5 860.7278 .7697 .8092 .8461 .8809 .9133 104 Appendix, continued N Branch J" F"=J"+5/2 T+3/2 J"+l/2 r-i/2 J"-3/2 J"-5/2 16 TR42(16) 17.5 17891.4686 .5123 .5518 .5865 .6241 .6569 18 RP42(18) 19.5 856.4344 .4768 .5161 .5532 .5883 .6212 17 TR42(17) 18.5 888.8223 .8648 .9051 .9424 .9774 19 RP42(19) 20.5 851.7901 .8319 .8715 .9091 .9452 .9774 18 TR42(18) 19.5 885.7926 .8348 .8743 .9120 .9473 .9804 20 RP42(20) 21.5 846.7596 .8011 .8404 .8777 .9133 .9462 19 TR42(19) 20.5 882.3463 .3880 .4275 .4644 .4997 .5329 21 RP42(21) 22.5 841.3129 .3540 .3928 .4298 .4651 .4984 20 TR42(20) 21.5 878.5065 .5451 .5810 .6174 22 RP42(22) 23.5 835.4743 .5132 .5498 .4848 

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