HIGH RESOLUTION SPECTROSCOPY OF SOME GASEOUSTRANSITION METAL CONTAINING DIATOMIC MOLECULESByGEJIAN HUANGB. Sc. (Chemistry), Zhongshan University (China), 1984M. Sc. (Chemistry), University of British Columbia, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CHEMISTRYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1993© GEJIAN HUANG, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of ChemistryThe University of British ColumbiaVancouver, CanadaDate^May 7, 199-1DE-6 (2/88)AbstractThis thesis reports studies of the electronic spectra of some gaseous diatomic moleculescontaining transition metals. The (0,0) bands of the C 3 11 — X3 0, OH — X 3/ andfl(1) — a 1 A electronic transitions of niobium nitride (NbN) have been recorded at sub-Doppler resolution by intermodulated laser-induced fluorescence. The C 311 and OHstates, which belong to the same electron configuration (4d8) 1 (4thr) 1 , interact stronglywith each other by a second-order spin-orbit/Fermi contact mechanism; the structure ofthe resulting 011/C311 complex has been treated by a full Hamiltonian matrix analysis,which resulted in a deperturbed set of spin and hyperfine parameters as well as interactionparameters. From the well resolved low J lines of the PO — al z (0, 0) band, the hyperfineparameters a were measured accurately for the 'A and 1 0 states; they were found tohave been modified by the spin-orbit/Fermi contact cross terms < alAli/IX 3A > andI< f1 iblii1B34) > respectively. Wavelength-resolved fluorescence studies of the emissionspectra from the laser-excited e 1 I1 andelectronic state at 5863 and 9919 cm -1 , respectively. The low energy of the 1 E+ statesuggests that it arises from the electron configuration (5sa-) 2 ; while the 1 F state can onlycome from the configuration (4d6) 2 . The vibrational sequence bands of the B — X andC — X systems, along with some weak sub-bands in the region 18200 —18500 cm -1 , havebeen rotationally analyzed from the laser excitation spectra of NbN. Two of the weak sub-bands were assigned as vibrational bands of the transition (58a - ) 1 (4d7r)" Y3 11 2 — X303 ,from preliminary analysis of the hyperfine structure; the other weak sub-bands werefound to originate from the X 30 state and to have charge transfer upper states of Hsymmetry with B constants of about 0.45 cm -1 .f 1 (1) states located a new 1 E+ and a new "FiiThe B4 11 — X4 E- (1, 0) band of VO has been investigated at sub-Doppler resolutionby intracavity laser-induced fluorescence spectroscopy and at Doppler-limited resolutionby emission spectroscopy. Values of the principal hyperfine parameters a and (b + c) forthe B4I1 state have been deduced from the resolved hyperfine structures of the low J linesof the B41512 — X and B4 I131 2 — X sub-bands: a = 0.00964 cm -1 and (b+ c) = —0.01295cm -1 . Estimates of the dipolar parameter c show that it is small compared to b, sothat the Fermi contact parameter b is sizeable and negative; this allows the electronconfiguration of the B 4I1 state to be assigned as (3d6) 2 (3dir) 1 . Rotational analysis ofthe emission spectra of the B 411 — X4 E- (1,0) band proved that the B 411 1 / 2 (v = 1)substate is perturbed by an orbitally non-degenerate E state. The perturbing state hasa fairly large rotational constant, B = 0.5405 cm -1 , and very large hyperfine structure;this suggests that it is the 2 E+ state from the electron configuration (3d8) 2 (4sa) 1 .A portion of the red system of CoO, from 6100 to 6450 A, has been studied byintracavity laser-induced fluorescence spectroscopy. Hyperfine analysis of two 40 71 2 —X40 71 2 sub-bands and one 40 5 / 2 — X 40 5/ 2 sub-band, with origins at 6338 A, 6411 Aand 6436 A, gave the principal hyperfine parameters a and (b + c) for the upper andlower states. From the derived Fermi contact parameters b, it was possible to assignthe X4 0 state to the configuration (4s0) 2 (3d8)3 (3thr) 2 , and the two excited 40 statesto (4sa) 1 (3d8) 3 (3dir) 2 (3da) 1 and (0, 27)703 (430) 2 (3c/8) 3 (3c/70 3 . The spin structure of theX40 ground state of Co0 was measured directly by wavelength-resolved fluorescencewith the observation of spin satellites from the excited 40 71 2 and 405 / 2 levels. Valuesof the first- and second-order spin-orbit coupling constants A and A for X 40 have beendeduced from the sub-state origins: A = 166.2 cm -1 and A = 7.32 cm -1 . Rotationalanalysis of 16 other sub-bands of the red system of Co0 revealed a very complex upperstate energy level structure with extensive global and local perturbations. Vibrationallevels belonging to various 4 ,A, 4 4) and 411 excited states have been partially identified.iiiTable of ContentsAbstract^ iiList of Tables^ xList of Figures^ xiiAcknowledgement^ xviChapter 1 Introduction^ 1Chapter 2 The Theoretical Background of Molecular Spectroscopy^42.i Introduction ^42.ii Angular momentum operators and their matrix elements ^62.ii.A Angular momentum operators and wavefunctions ^62.ii.B Coupling of angular momenta and coordinate transformation .^82.ii.0 Coordinate rotations ^132.ii.D Irreducible spherical tensor operators ^172.iii Molecular Hamiltonian ^ 232.iii.A The general molecular Hamiltonian and its eigenfunctions^232.iii.B Rotational Hamiltonian ^272.iii.0 Electron spin Hamiltonian ^282.iii.D Nuclear spin Hamiltonian ^362.iv Hund's coupling cases and the Hamiltonian matrix ^ 422.iv.A Hund's coupling cases and the corresponding basis sets ^ 42iv2.iv.B Matrix elements of the Hamiltonian operator in a case (a o) basis . 502.iv.0 A-doubling ^592.v Line strength and selection rules ^62Chapter 3 Laser Induced Fluorescence Spectroscopy^ 683.i Introduction ^683.ii Spectral linewidth and broadening effects ^693.iii Principles of saturation spectroscopy ^733.iv Doppler-free saturation fluorescence spectroscopy ^773.iv.A Intermodulated fluorescence ^773.iv.B Intracavity fluorescence ^803.v Resolved fluorescence spectroscopy ^ 83Chapter 4 Laser Spectroscopy of NbN^ 88Part I Unusual Electron Spin and Hyperfine Effects in the ElectronicStates of the (4(16) 1 (580) 1 and (4db) 1 (4(170 1 Configurations^884.i Introduction ^884.ii Experimental details ^904.iii The spin-orbit and hyperfine structures of the (4c/8) 1 (4d7)1 1 1-1/3 1-1—(4d6) 1 (5so- ) 1 X30 system ^ 924.iii.A Description of the spectra ^924.iii.B Hamiltonian matrix for the 1 11/311 complex. ^964.iii.0 Fitting the spectral data to the energy expressions ^ 1014.iii.D Discussion ^ 1044.iv Hyperfine analysis of the PO — OA system ^ 109v4.iv.A Spectral characteristics ^ 1094.iv.B Determination of molecular constants of the a 1 / and f 1 states ^ 1104.iv.0 The hyperfine structures of the PO and elA states ^ 1174.v Conclusion ^ 118Part II Resolved Fluorescence Study of Two Low-lying Singlet Elec-tronic States of NbN^ 1204.vi Introduction ^ 1204.vii Experimental details ^ 1224.viii Data analysis 1224.viii.A Transitions from the (4d8) 1 (4dir) 1 OH state ^ 1224.viii.B The PO — c1 F transition ^ 1264.ix Results and discussion ^ 128Part III Rotational Analysis of the B — X, C — X Vibrational SequenceBands and Some New Subbands of NbN^ 1344.x Introduction ^ 1344.xi Experimental 1364.xii Appearance of the low-resolution spectra ^ 1374.xiii Analysis of the vibrational sequence bands of the B — X and C — X systems 1394.xiii.A Sequence structures of the B — X and C — X systems 1394.xiii.B Term values and rotational constants for the vibrational levels ofthe X30, B3 43. and C 3II states ^ 1414.xiv The new (5so) 1 (4dir) 1 3H 2 — (4d8) 1 (5so) 1 X30 3 subbands at 18419 and18276 cm-1 ^ 147vi4.xiv.A Rotational analysis ^ 1474.xiv.B The electrostatic perturbation between the Y 3 11 2 and C3I1 2 states 1504.xiv.0 The rotational and hyperfine structure of the (580- ) 1 (4(170 1 Y3 11excited state. ^ 1564.xv The charge transfer transitions ^ 1604.xv.A Branch analysis 1604.xv.B Determination of rotational constants ^ 1644.xv.0 Discussion ^ 164Chapter 5 Intracavity Laser Spectroscopy of VO^ 169Part I Hyperfine Parameters and Electron Configuration of the B 4 11State^ 1695.i Introduction 1695.ii Experimental details ^ 1705.iii Results ^ 1715.iv Hyperfine parameters for the B 41I, v = 1 level ^ 1775.v Discussion ^ 1825.vi Conclusion 184Part II Analysis of Rotational Structure and Perturbations in the B 411—X4 E- (1,0) Band of VO^ 1855.vii Introduction ^ 1855.viii Spectral data 1865.ix Analysis of the branch structure of the B 4 11— X 4 E- (1, 0) Band ^ 1865.x The energy levels of the 4H/2E+ and 4 E - states ^ 188vii5.xi Least-squares fitting of the data ^ 1915.xii Results and discussion ^ 1955.xii.A Spin-orbit coupling constants ^ 1955.xii.B The 2E+ perturbing state 1965.xii.0 A-doubling parameters ^ 1985.xiii Conclusion ^ 199Chapter 6 Laser Intracavity Spectroscopy of CoO^ 200Part I Hyperfine and Spin-orbit Structure of the 4A i Ground State ofCoO^ 2006.i Introduction 2006.ii Experimental details ^ 2016.iii Hyperfine structure 2036.iii.A Appearance of the spectra ^ 2036.iii.B Energy level expressions 2096.iii.0 Results ^ 2106.iii.D Interpretation of the hyperfine parameters ^ 2126.iii.E Centrifugal distortion of the hyperfine h parameters ^ 2166.iv Spin-orbit structure of the X 4A i ground state of CoO 2186.v Conclusion ^ 224Part II Rotational Analysis of Some Sub-bands of the Red System ofGaseous CoO^ 2266.vi Introduction 2266.vii Experimental ^ 227viii6.viii Appearance of the spectrum ^ 2286.ix Rotational analysis of the Co0 sub-bands ^ 2286.ix.A The 40 7/ 2 — X 4 A 7/ 2 sub-bands 2286.ix.B The 405/2 - X4 A5/ 2 and the perpendicular sub-bands ^ 2406.x Results and discussion ^ 2526.xi Conclusion ^ 260Bibliography^ 261Appendix A Matrix Elements of the Nuclear Electric Quadrupole Hamil-tonian in Case (ao ) Coupling^ 266Appendix B Derivation of the Matrix Elements of the Electron Spin-orbitand Nuclear Magnetic Hyperfine Hamiltonian for the 1 11/3 11Complex from the Sr Configuration of NbN 268Appendix C Line Frequencies of the Vibrational Bands of the B 3 (1) — X30and C311 — X3 0 Transitions of NbN in cm -1^279Appendix D Line Frequencies of the Charge Transfer Bands of NbN in--1cm^ 309Appendix E Line Frequencies of the B 4 11 — X 4 E- (1, 0) Band of VO in cm -1 313Appendix F Assigned Hyperfine Lines of the 6338, 6436 and 6411 ASub-bands of Co0 in cm -1^318Appendix G Assigned Rotational Lines of the Red System of Co0 in cm -1 325ixList of Tables4.1 Observed hyperfine transitions of the e 1 II — X30 2 (0, 0) Q(11) line andcalculated energies of the 1 1-1, J = 11 hyperfine levels ^984.2 Rotational line positions of the e 1 II — X302 (0, 0) band of NbN ^ 994.3 The structure of the 1 H/311 matrix in the Hund's case (a) parity basis .^1024.4 Matrix element representations of the labels used in Table 4.3 ^ 1034.5 Deperturbed rotational and hyperfine constants for the OH and C 3Il statestogether with the interaction parameters ^ 1054.6 Molecular constants of the Pot. and alA states of NbN ^ 1144.7 Hyperfine transitions of the PO — a 1 / (0,0) band of NbN ^ 1144.8 Rotational lines of the f 1 (1) — a 1 /. (0, 0) band of NbN 1164.9 Spin-orbit constants AA and hyperfine constants aA for the electronicstates from the configuration (4d6) 1 (4d/r) 1 of NbN ^ 1194.10 Rotational constants and line positions for the e 1 H — 6,1 E+ transition ofNbN 1274.11 Rotational constants and line positions for the PO — OF transition of NbN1304.12 Expected low-lying electronic states of NbN 1354.13 Term values and effective rotational constants of the vibrational levels ofthe X30 1 , B3 10 2 and C3110 substates of NbN ^ 1424.14 Term values and effective rotational constants of the vibrational levels ofthe X302, B3 1.3 and C3111 substates of NbN ^ 1434.15 Term values and effective rotational constants of the vibrational levels ofthe X303 , B3 4:10 4 and C3II 2 substates of NbN^ 144x4.16 Comparisons of the observed and calculated term values of the 0 3 11 2 vi-brational levels ^ 1464.17 Rotational constants derived from the 18419 and 18276 cm -1 subbands ofNbN ^ 150^4.18 Line positions of the Y 3 11 2 — X303 (1,1) and (5,4) bands of NbN 1514.19 Calculation of the hyperfine widths of the Y 3 I12 levels ^ 1584.20 Rotational constants of the charge transfer subbands of NbN ^ 1655.1 Rotational and hyperfine constants for the B 411, v = 1 level of VO derivedfrom the 4115/2 and 4 113/2 substates^ 1805.2 Hyperfine energies of the v = 1 vibrational level of B4 I1 5/ 2 and B41131 2substates of VO ^ 1815.3 Matrix elements of the rotational Hamiltonian for a 411/ 2 E+ complex incase (a) coupling ^ 1925.4 Corrections applied to the observed F2 and F3 line positions to allow forthe internal hyperfine perturbation shifts ^ 1935.5 Spectroscopic parameters for the B 4 11/2 E+ complex of VO derived fromrotational analysis of the B4 11 — X4 E- (1,0) band ^ 1946.1 Rotational and hyperfine constants derived from the red bands of Co0 . 2116.2 Assigned rotational lines of the ST = 5/2 — SI" = 5/2 band at 15580 cm -1and the 5/2 — 7/2 band at 15884 cm -1 ^ 2216.3 Results from wavelength-resolved fluorescence measurements of the groundstate of Co0 ^ 2236.4 Constants derived for the upper states of the red system of Co0 ^ 2376.5 Approximate spectroscopic constants for the excited states of Co0 . . . ^ 258xiList of Figures2.1 Polar and Cartesian coordinates ^92.2 Euler angles a, 0, and -y ^142.3 Vector diagram for Hund's coupling case (a) ^ 442.4 Vector diagram for Hund's coupling case (b) 462.5 Vector diagram for Hund's coupling case (c) ^ 493.1 "Hole burning" in the lower level velocity population distribution andgeneration of a corresponding population peak in the upper level ^ 753.2 (a) Two Bennett holes burned symmetrically into the velocity populationdistribution (n(v z )). (b) Lamb dip formed at the center (v = v o ) of theprofile of intensity versus laser tuning frequency 763.3 Schematic drawing of the intermodulated fluorescence experiment . . .^783.4 a) The formation of crossover resonances. b)Stick diagram of a spec-trum with OF AJ transitions and crossover resonances accompanyinga OF = AJ Q transition ^ 813.5 Schematic drawing of an intracavity fluorescence experiment ^ 823.6 The Doppler-limited and sub-Doppler spectra of the rR3 (16) line of H 2 CS,recorded by a fluorescence-based intracavity laser spectrometer ^ 844.1 The Q heads of the three C3 11 — X30 (0, 0) subbands of NbN ^ 934.2 The Q head of the e l 11 — X30 2 (0, 0) band of NbN at Doppler-limitedresolution ^954.3 Hyperfine structure of the Q(11) line of the eill — X 30 2 (0, 0) band of NbN 97xii4.4 The Q head of the PO — alA (0, 0) band of NbN at sub-Doppler resolution 1114.5 The hyperfine structure of the R(3) line of the PO — ct 1 0 (0,0) band ofNbN ^ 1124.6 The OH — a 1 A and A3 E;1_ transitions near 7300 A from laser excitationof the Q(22) line of the 011 — X 302 transition ^ 1234.7 State-selected elll — blE+ emission near 7700 A from laser excitation ofthe OH — X302 transition ^ 1254.8 Fluorescence patterns of the PO — c1 1" transition of NbN: emissions from(a) J' = 7, (b) J' = 21 and (c) J' = 34 levels ^ 1294.9 Electronic states and transitions of NbN known to this date ^ 1314.10 The relative energies of the molecular orbitals of NbN, formed from linearcombinations of the atomic orbitals of Nb and N ^ 1354.11 Broadband laser-induced fluorescence spectrum of NbN in the region 18200— 18500 cm-1 ^ 1384.12 A portion of the B 304 — X303 (3, 3) subband of NbN at Doppler-limitedresolution ^ 1404.13 Laser excitation spectrum of NbN in the region 18384 — 18433 cm -1 . . ^ 1484.14 Laser-induced resonance fluorescence of the Y 3 11 2 — X303 (1, 2) band, ac-companied by the collisionally induced C 311 2 — X303 (2, 2) emission (near5770 A) ^ 1544.15 Laser excitation spectrum of NbN in the region 18249 — 18303 cm -1 . ^ 1614.16 Energy levels of the charge transfer states and transitions of NbN . . . ^ 1665.1 Energy levels of the B 411 state of VO, as presently known, plotted against.1)2 ^ 1735.2 Hyperfine structure of the RP31(15) line of the B411 — X4 E- (1, 0) band ofVO ^ 1745.3 Part of the hyperfine structure of the T Q41 (1) rotational line of the B 4 1-1 —X4 E- (1,0) band of VO ^ 1755.4 The overlapping uR41 (0) and TQ41 (2) lines of the B 4 11 — X 4 E- (1, 0) bandof VO ^ 1765.5 The SR32 (1) line of the B 4 1131 2 — X4 E- (1, 0) band of VO^ 1785.6 Calculated energy levels of the B 4H, v = 1 and 2 E4- states of VO, plottedagainst (J + 2)2 ^ 1896.1 Hyperfine structure of the P(92) line of the SY = 7/2 — X 407/ 2 band ofCo0 at 6436 A ^ 2056.2 Hyperfine structure of the Q(32) line of the CY = 7/2 — X 4071 2 band ofCo0 at 6338 A 2076.3 Hyperfine structure of the Q(31) line of the CY = 5/2 — X 40 5 / 2 band ofCo0 at 6411 A ^ 2086.4 Resolved fluorescence from selective excitation of the R(192) line of thelt = 7/2 — X 40 7/ 2 (6338 A) band of Co0^ 2196.5 Resolved-fluorescence patterns from the (a) perturbed and (b) unper-turbed R(302) lines of the 6338 A band of Co0 2226.6 The low J structure of the 4 A7/ 2 — X46.7/2 subband of Co0 near 6127 A 2306.7 The low J structure of the 40 7/ 2 — X4 071 2 subband of Co0 near 6151 A 2316.8 The upper state energies of the 6127 A band of CoO, plotted against J(J+1)2326.9 The upper state energies of the 6151 A band of CoO, plotted against J(J+1)2326.10 The upper state energies of the 6221 A band of CoO, plotted against J(J+1)2346.11 The upper state energies of the 6388 A band of CoO, plotted against J(J+1)234xiv6.12 Comparisons of the widths of the (a) perturbed and (b) unperturbed ro-tational lines of the 6221 A band of Co0 ^ 2366.13 Partially resolved hyperfine structures of the P(5.5) line of the 6127 Aband (top) and the Q(4.5) line of the 6151 A band (bottom) ^ 2396.14 The R head region of the 4051 2 — X4051 2 subband near 6305 A ^ 2416.15 The upper state energies of the 6294, 6295, and 6418 A bands of CoO,plotted against J(J + 1) ^ 2446.16 The upper state energies of the 6370 and 6373 A bands of CoO, plottedagainst J(J + 1) ^ 2456.17 A magnified plot of the upper state energies of the 6373 A band ^ 2476.18 The upper state energies of the 6186 and 6305 A bands of CoO, plottedagainst J(J + 1) ^ 2486.19 The upper state energies of the 6152 and 6154 A bands of CoO, plottedagainst J(J + 1) 2496.20 The upper state energies of the 6314 A band of CoO, plotted against J(J+1)2516.21 The upper state energies of the 6195 A band of CoO, plotted against J(J+1)2536.22 The upper state energies of the 6142 A band of CoO, plotted against J(J+1)2536.23 Energy level structure of the red system of Co0 254AcknowledgementI sincerely wish to thank Prof. A. J. Merer for giving me the opportunity to do thisPh.D. project and allowing me to study spectroscopy and science in general under hisguidance. His constant advice, instructions, encouragement and considerateness giventhroughout my time here are deeply appreciated. I am very much grateful to Dr. D. J.Clouthier for his assistance with the vanadium oxide and cobalt oxide projects, to Dr. Y.Azuma and Dr. V. Srdanov for their assistance with the niobium nitride experiments. Iwould like to thank Mr. C. Chan for his technical assistance with the usage of electronicdevices, Dr. M.C.L. Gerry for allowing me to use some instruments in his laboratoryand Dr. M. Barnes for his help in preparing some thesis material. Thanks also go to Dr.A. Adam, Dr. P. Hajigeorgiou, Mr. B. Berno and all my colleagues in the spectroscopicgroup at U.B.C. for their useful and interesting discussions. Finally, I wish to thank myparents for their love, understanding and support.xviChapter 1IntroductionTransition metals have a common and distinctive feature in their atomic structures,that is their electron configurations all contain partially filled d subshells. This structuralcharacter has a significant influence on the chemical bonding and properties of transitionmetal-containing molecules. In this thesis, we study the electronic structures of somefirst and second row transition metal monoxides and nitrides by the method of opticalspectroscopy.The interest in transition metal-containing compounds stems from their importancein astrophysics, quantum chemistry and material science. In astrophysics, high concen-trations of the early transition metal monoxides exist in the atmospheres of cool (M-type)stars as a result of the high dissociation energies of these molecules. So far astrophysistshave identified most of the 3-d transition metal monoxides and some 4-d monoxides in thespectra of the M-type and S-type stars [1]. In quantum chemistry, the spectroscopy oftransition metal containing molecules presents unique and challenging features: the manyunpaired d electrons produce large numbers of low-lying electronic states whose energyorders are very difficult to calculate accurately because of complicated electron correla-tion effects; also the large magnetic moments of some transition metal nuclei give rise toimportant nuclear hyperfine effects which further complicate the energy level patterns.In the solid state, many transition metal containing alloys are good superconductive ma-terials. Although most pure "diatomic" crystals of transition metal compounds such as1Chapter 1. Introduction^ 2oxides are rather poor superconductors, some nitride crystals are known to exhibit excel-lent superconductivity. For example, the NbN crystal has a comparatively high criticaltemperature, 7 1, r-:.,- 16 K [2].Many gaseous diatomic molecules containing transition metals have very complicatedspectra; this is attributed mostly to the presence of large electron spin and orbital angularmomenta resulting from the many unpaired electrons. Although the interactions betweenthe angular momenta within the molecules complicate the molecular spectra tremen-dously, their analyses ultimately yield valuable information on the electronic structuresand chemical bondings of the molecules. Intra-molecular interactions can be measuredfrom molecular spectra in the form of parameters; these parameters are the expectationvalues of electron coordinates and can be used to test the quality of electronic wavefunc-tions calculated by ab initio methods. The magnetic hyperfine parameters which describethe electron spin-nuclear spin interactions are particularly sensitive to the distribution ofelectrons around the nuclei and provide vigorous tests for the proposed wavefunctions.Determination of such weak interactions as the magnetic hyperfine interactions requiresspectroscopic techniques with the utmost resolution and sensitivity; laser spectroscopyis capable of fulfilling these requirements.The aim of this thesis is to study the high resolution spectroscopy of two 3-d transitionmetal monoxides, VO and CoO, and a 4-d metal nitride, NbN, to obtain information onthe electronic structures of these molecules. The experiments described here employthe techniques of laser-induced fluorescence spectroscopy to acquire molecular spectraat both Doppler-limited and sub-Doppler resolution in the region from the visible tothe near infrared. The molecular parameters are derived from the spectra through aneffective Hamiltonian approach, where the operators act within a particular vibronic stateor states from which the spectra arise.Chapter 2 gives a summary of the theoretical background of molecular spectroscopy,Chapter I. Introduction^ 3including the couplings of molecular angular momenta, the construction of effectivemolecular Hamiltonians and the derivation of their matrix elements using the methods ofspherical tensors. Chapter 3 describes briefly the principles of laser induced-fluorescencespectroscopy and the different aspects of this technique. Chapters 4, 5 and 6 describerespectively the spectroscopic studies of gaseous niobium nitride (NbN), vanadium ox-ide (VO) and cobalt oxide (CoO). Chapter 4 consists of three Parts: Part I describesan intermodulated fluorescence study of the spin-orbit and hyperfine interactions in theelectronic states arising from the electron configurations (4d8) 1 (46-) 1 and (4d6) 1 (5so-) 1 ,Part II presents two new singlet states of NbN, blE+ and ciT, discovered by wavelength-resolved fluorescence, and Part III gives an analysis of some new subbands from thelaser excitation spectra, which include two (5sa) 1 (4dr) 1 Y3 11 2 — X303 subbands and anumber of charge transfer subbands of NbN. Chapter 5 is divided into two Parts; PartI studies the hyperfine structure of the B 4 H state of VO using fluorescence-based laserintracavity spectroscopy, and Part II gives a rotational analysis of the B 4H — X4 E - (1, 0)band, including a quantitative analysis of the interaction between the B 4 H, v = 1 leveland the v = 3 level of a 2 E+ state. Finally Chapter 6, on CoO, describes an intracavitylaser-induced fluorescence study of the spin-orbit and hyperfine structures of the X40state (Part I), and a rotational analysis of some subbands of the red system (Part II).Chapter 2The Theoretical Background of Molecular Spectroscopy2.i IntroductionMolecular spectra measure radiative transitions between the various energy levels of amolecule; the energy levels are the eigenvalues of the quantum mechanical Hamiltonianoperator H. When ft operates on an eigenfunction we have the eigenvalue equationHT = ET.^ (2.1)Eq. (2.1) is known as the time-independent Schr&linger equation, whose solution givesthe energy levels, E, and the wavefunctions, T, that represent the stationary states of amolecule.The usual procedure for solving Eq. (2.1) is to expand the wavefunction T as a linearcombination of a complete set of basis functions 74bi,T = E aoki .^ (2.2)This reduces the solution of the SchrOdinger equation to the problem of finding the rootsof the secular determinantI Hii — E Sii I = 0.^ (2.3)The matrix elements Hii and the overlap integrals S2; are defined asHi; < I fl I > = f 7 Hj dT,^(2.4)Sii = < 71,i10 .; > =^0: j dr. (2.5)4Chapter 2. The Theoretical Background of Molecular Spectroscopy^5One can almost always choose an orthogonal and normalized basis set zi) such that theoverlap integrals Sii become the Kronecker delta S ij ; this is defined as6t.7 = <^I^> =^ifij^(2.6)1 ifi=j.Thus solving the determinant (2.3) to obtain the eigenvalues, E, and wavefunctions,111, becomes a matter of diagonalizing the Hamiltonian matrix H calculated using thebasis functions 0. In principle, one can choose any complete basis set to carry out thecalculation, but a wise choice of is one that factorizes the Hamiltonian matrix intodiagonal blocks (sub-matrices) which are as close as possible to being diagonal.Since the present work is concerned with the interpretation of molecular spectra usingthe principles of quantum mechanics, a complete derivation of the molecular energy levelsand wavefunctions from first principles is not necessary. Our approach is to derive aneffective Hamiltonian, which is related to the exact Hamiltonian through some adjustableparameters, and to set up the Hamiltonian matrix with a limited basis set. The experi-mental data are fitted to the energy levels obtained from diagonalizing the Hamiltonianmatrix by the method of least-squares to produce a set of molecular constants. Since theconstants obtained are related to the actual Hamiltonian, they carry direct informationon the electronic and geometric structures of the molecule.In rotational problems, the Hamiltonian operator is generally expressed in terms ofthe angular momentum operators; therefore calculation of the Hamiltonian matrix re-quires a knowledge of how to manipulate the angular momentum operators. Section 2.iiwill deal with the principles of angular momenta and the spherical tensor methods usedfor the derivation of their matrix elements. In Section 2.iii, we will derive the Hamil-tonian expression for a diatomic molecule and the expressions for the related molecularparameters. In Section 2.iv the coupling of the angular momenta will be discussed forChapter 2. The Theoretical Background of Molecular Spectroscopy^6the five limiting situations known as Hund's cases (a) to (e); Hamiltonian matrices forhyperfine structures, calculated in an extended case (a) basis set, will also be presented.Finally, a brief discussion of the selection rules and line strengths for an electric dipoletransition will be given in Section 2.v.2.ii Angular momentum operators and their matrix elementsThe concepts of angular momentum play important roles in molecular spectroscopy,thus it is important to start this chapter by introducing the principles of angular mo-menta.2.ii.A Angular momentum operators and wavefunctionsIn quantum mechanics, a general angular momentum operator J is defined as anoperator whose components obey the following commutation rules,[jx , fit] = i jz, [jv , jz] = i jx, [iz , jx] = i 4, (2.7)ix , iv and 4 are the X, Y and Z components of the operator J in a Cartesian co-ordinate system. Although ix , Jy and Jz do not commute with each other, they docommute with the operator :T 2 , i.e.[Jx , :12] = 0, [iy , j2] = 0 , [Jz ,:i2] = 0. (2.8)This means one can choose an angular momentum wavefunction to be simultaneously aneigenfunction of :T 2 and any one of the Cartesian components of :T; traditionally the Jzoperator is chosen because it has a very simple differential form in a polar coordinatesystem. An eigenfunction of J2 and 4 is denoted in Dirac notation as 1JM > withJ and M being the respective quantum numbers of the two operators. The significanceof the operator J2 is that it commutes with the Hamiltonian operator H. ThereforeChapter 2. The Theoretical Background of Molecular Spectroscopy^7••an eigenfunction of 11 is also an eigenfunction of J2 and can be expressed as a linearcombination of angular momentum wavefunctions M >. In other words, the eigen-functions of the operators ./^2 and iz can be used as the basis functions to construct theHamiltonian matrix. The eigenvalues of .I 2 and Jz are given by [3]IJM >^J (J + 1) h 2 I J M >,^(2.9)jz IJM> = MhIJM>^ (2.10)where h is the Planck constant, h, divided by 27r. The quantum numbers J and M maytake integer or half-integer values; for each J value, M can take (2J +1) different values:J, J — 1,...—J. For convenience, we introduce a unit system in which h equals 1, andtherefore can be dropped from the eigenvalue expressions. The effects of the operatorsix and Jy on a wavefunction IJM > are most easily treated in terms of the raising andlowering (or ladder) operators f f. and , defined by3, = ix ±^ (2.11)The ladder operators i± have the property of transforming a wavefunction IJ M > intomultiples of IJM + 1 >; to be exact, the ladder operations are written [3]i± I J M > = VJ(J + 1) — M(M +1) I J M +1 > .^(2.12)From Eqs. (2.9), (2.10) and (2.12) we obtain the matrix elements of the angular momen-t um operators J2 , Jz and 3, as follows:< J M I:1 2 If^> = J(J^1).5.1J , 8mm , , (2.13)< J M I 41^> = M 6jj , Smmi, (2.14)<JMIi±IJ'M'> = J(J + 1) — M(M + 1) &LP 6m,m ,±1. (2.15)In the above equations, (5, i , is the Kronecker delta defined by Eq. (2.6).Chapter 2. The Theoretical Background of Molecular Spectroscopy^8The explicit forms of the angular momentum wavefunctions can be derived most easily••in a polar coordinate system. The angular momentum operators L2 and Lz , expressedin terms of polar coordinates 0 and 0 (see Fig. 2.1), are written as. aLz = —z -a6.^(2.16)L2 = ^1 82^1 a ( sin (2.17){sin' 0 802 + sin 0 80 ( 0—ao )1.The eigenfunctions of L 2 and Lz are the spherical harmonics, which can be shown tohave the form [3]1yim (9, 0) , (-1)m (21 + 1)(l — nil 2Pr (cos 0) ern° .[ 47r(l + m)!(2.18)The function Pim(cos 0) is called the associated Legendre function, and is defined by1^1 . ,^1 +'m171 (COS2 0 — 1 ) 1 .^= 2111 sin u d(cos 0)Pri (cos 0)For the special case when in = 0, the spherical harmonic becomesYi0 (0 , q) = ( 2/ + 1 )47r^/3/(cos 0),where Pi (cos 0) is the ordinary Legendre polynomial.(2.19)(2.20)2.ii.B Coupling of angular momenta and coordinate transformationVarious angular momenta arise in a molecule as a result of the circular motions of elec-trons and nuclei. Each angular momentum is associated with a magnetic moment sincea moving charged particle is known to produce a magnetic moment. Naturally, differentmagnetic moments will interact with each other, giving rise to additional energy terms.These magnetic interactions can be looked upon as couplings between the correspondingangular momenta. The logical order for the angular momentum coupling depends on theChapter 2. The Theoretical Background of Molecular Spectroscopy^9 YFigure 2.1: Polar and Cartesian coordinates, in which X = r sin 0 cos 0, Y = r sin 9 sin (0,Z = r cos 9 [4].Chapter 2. The Theoretical Background of Molecular Spectroscopy^10molecules, being governed by the relative magnitudes of the different magnetic interac-tions. Five limiting situations for angular momentum couplings in molecules have beenproposed [5], and will be discussed in Section 2.iv.A. In the present section, some impor-tant relationships describing the couplings of two or three general angular momenta, andtheir transformation from one axis system to another, are examined.The addition of two angular momenta 3 1 and 3 2 to form a resultant angular mo-mentum J can be treated from two points of view. First, the compound system canbe described by wavefunctions 13 1 32 J M >, normally abbreviated as IJ M >; these aresimultaneously eigenfunctions of the operators 312 , 322 , j2 (3 1 +3 2 ) 2 and 3, = 31z +32z)since all these operators commute with each other. This is called the coupled representa-tion. Alternatively, the same system can be described by wavefunctions I jimi; j2m2 >=jlml > Ii2m2 >, with I j i m i > and I j2m2 > being eigenfunctions of the operators 3 1 ,3 1z and 3 2 , 3 2z respectively; this is known as the uncoupled representation. These tworepresentations are equivalent, and are related to each other by a unitary transformation[3]IJA4-^E 14wirni; j2m2 ><jimi; j2m2 I JAI > •7711 7712(2.21)Complicated algebraic expressions for the vector coupling coefficients, <iimi; i2m2IJM>(also known as Clebsch-Gordan coefficients), have been derived, and are available inthe literature for calculations. However, it is more practical and useful to redefine thecoupling coefficient in terms of the Wigner 3-j symbol, that is:i 2^J ) .< 71 1, hrn2 I JM > = (-1).ii-h+m 1/2j + 1^ii (2.22)m l m2 -MThe advantage of using the Wigner 3-j symbols is that they have higher symmetry proper-ties than the Clebsch-Gordan coefficients and are much easier to manipulate. Combining^(E^(-0-i1-j2-Fm12^2i i2 + 1M12 ,m3,m1 ,m2" > =jig)m1 m2 — m12()il2 i3^Jx (-1)312-h+m ^2J + 1M12 7723j1m1; i2m2; i3n3 >^(2.25)Chapter 2. The Theoretical Background of Molecular Spectroscopy^11Eqs. (2.21) and (2.22), we haveIJM>= E H1)ii-J21-mv2 j + 1^j2 :111^i2n2 > .^(2.23)^M1 M2^ ml m2Eq. (2.23) provides a means to transform wavefunctions between coupled and uncoupledbases, and is a very important relation for the calculation of Hamiltonian matrices.The coupling of three or more angular momenta can be taken as two or more successivecouplings of two angular momenta. Accordingly the reverse process, uncoupling thetotal angular momentum into the individual angular momenta, can be accomplishedby applying Eq. (2.23) twice or more. For example, for the coupling of three angularmomenta, we have^I JAI >^i < j12rn12; i3n3 I JAI > I jurni2; i3n3 >m12 m3< ^32m2 I il2M12 >m12,m3,m1 ,m2< i12M12; .33rn3 I JM > 71n1; si2rn2; .73rn3 >^(2.24)Substituting for the coupling coefficients with Eq. (2.22) we findObviously Eq. (2.25) represents just one of the two possible mechanisms for the couplingof three angular momenta, that is the addition of j 1 and j 2 to give j 12 , followed byaddition of j 3 to give the resultant J. Alternatively, j 2 and j3 can couple first to givej 23 , then j 1 is added to produce J. The two coupling schemes are physically equivalentand are connected by a unitary transformation (just like the coupled and uncoupled/ji j2 j12 =^-1)31 +32 +33+Y(2ii2 + 1)(2j23 + 1)j3 J 323( il j2 j3 j4ml m2 —m3 rn4 J5^)m5 7123Chapter 2. The Theoretical Background of Molecular Spectroscopy^12representations for the addition of two angular momenta). Therefore we may write1j23 j1 JM > = i <^> ii12 j3^> .^(2.26)312The coefficients < ji2j3J1j1j23,/ >, called the recoupling coefficients, can be shown tocontain the products of four vector coupling coefficients, or the products of four Wigner3-j symbols; the most symmetric forms for these are the Wigner 6-j symbols, which aredefined asx < j12 i3 J I j1 i23 J > (2.27)A 6-j symbol can be written as a sum of products of four 3-j symbols; alternatively,products of 3-j symbols can be expressed in terms of a 6-j symbol as in the followingequation [6]= E (2j6 + 1) (- 1 )71+72-.73+74+75+76-m1-m4a61 j1 j2 j314 35 36 j5 ii i6 j2 34 j6 (2.28 )n/5 m1 n/6 m2 rn4 -nuLike 3-j symbols, 6-j symbols have very high symmetry and can be easily calculated withthe aid of computers.The coupling of four angular momenta can be treated in the same way as the couplingof three angular momenta. In this case, there are several possible coupling schemes, allof them connected to each other by unitary transformations. The recoupling of fourangular momenta is best described using Wigner's 9-j symbols, which are defined in aChapter 2. The Theoretical Background of Molecular Spectroscopy^13similar manner to 6-j symbols. A 9-j symbol may be written as products of three 6-jsymbols or as products of six 3-j symbols. The symmetry properties of 9-j symbols, aswell as those of 3-j and 6-j symbols, can be found in the literature [3, 6] and will not begiven here.2.ii.0 Coordinate rotationsIn spectroscopy, quantities defined in different coordinate systems are sometimestreated simultaneously. For instance, the electric interaction between a photon and amolecule involves the electromagnetic field of the photon, which is conveniently definedin the space-fixed axis system, and the electric dipole moment of the molecule which isusually defined in a molecule-fixed axis system. Proper description of such an interactionrequires the quantities involved to be transferred to the same frame of reference; this isdone by a coordinate rotation.The operator corresponding to a rotation operation about an arbitrary axis, say theZ axis, by an angle a is defined asiz(a) = (2.29)where jz , the Z component of the angular momentum operator :/, is the "generator of aninfinitesimal rotation about the Z axis". A space-fixed coordinate system can be broughtinto coincidence with a molecule-fixed coordinate system by three successive rotationsrepresented by the operator [3]:h(cei3-y) . e -joi, (2.30)In Eq. (2.30), Jy and Jz are the Y and Z components of the angular momentum operatordefined in the space-fixed axes; angles a, Q and -y are the Euler angles shown in Fig 2.2.The Euler angles are defined to be positive going from the space to the molecule-fixedChapter 2. The Theoretical Background of Molecular Spectroscopy^14Figure 2.2: Euler angles a, #, and -y relating the space-fixed XYZ and molecule-fixed xyzframes [3].Chapter 2. The Theoretical Background of Molecular Spectroscopy^15axes; this leads to normal commutation relations in the space-fixed axes but anomalousones in the molecule-fixed axes [7]:jYjX = ZiZ^SPACE,^ (2.31)iy iy^iz MOLECULE. (2.32)The anomalous sign of i in Eq. (2.32) can be eliminated by setting up the molecularoperators in the space-fixed axis system and referring them back to the molecule-fixedaxis system by a coordinate rotation. The expression for the operator describing thetransformation will be given in the next section.When a rotation operator i? acts on an angular momentum wavefunction I J M >,the resultant can be written as a linear combination of all the M components:100'01 JM > = E DIpm(ceo-y) I JO >,^(2.33)MIwhere the expansion coefficients Dkvm (a0-y) are simply the matrix elements of the op-erator R:Dkvm (ce,3•7) =< JM'I RI JM > .^(2.34)The rotation operation on wavefunction IJM >, represented by the left side of Eq.(2.33), is equivalent to transforming the space-defined wavefunction to a molecule-definedwavefunction If K >. Therefore Eq. (2.33) becomesJIi > = E Di/K(co-y) I JM >,Mto which the reverse expression isJM > = E Dif;K(ao-y) I JK > •KThe complex conjugate of the rotation matrix in Eq. (2.36) is given by [3]DMK (a13 = (- 1 )m-K D'Ai_K (a /3 ) •(2.35)(2.36)(2.37)Chapter 2. The Theoretical Background of Molecular Spectroscopy^16It can be shown that for a symmetric-top molecule the angular momentum wavefunctionbecomes.I2J -I- 1^,,IJKM>=^DMjIC(07)-87r 2 (2.38)The elements of a rotation matrix (also called D-functions hereafter) frequently appear inthe process of evaluating the matrix elements of Hamiltonian operators. One particularlyuseful relation concerning the D-functions is the integral over a product of three D-functions, which is given by [3]1 pr fir f2w NI 1mi (a0-y)Divl Al2 (a07) N. Al,(ciO-y) da sin /3d/3dy87r 2 a=0 ./3.o ^-y=oVM .12 J3^J1 J2 J3=^ (2.39)I M Mi ( M1 M2 M3A D matrix element with one of its projections equal to zero collapses to a sphericalharmonic, which depends on only two angles:47rDp0(07) = (-1)Y2/ + 1 Ylp(a/3)(2.40) ploq (a/13-Y) =47r2/ + 1 Y1q (07)(2.41)The expression on the right side of Eq. (2.41) is often referred to as a modified sphericalharmonic, labelled Cin,(00) (= V47r/(2/ + 1) 1//,(00)). From Eqs. (2.39) and (2.41) onecan derive the important Spherical harmonic addition theorem [3]:VT,(0 , 0) yin., (0 , , 0,) = 2/ + 1^Pi (cos w).^(2.42)4rmwhere w is the angle between the two directions (0, 0) and (0', 0'), and /3/ (cos w) is aLegendre polynomial. Expressed in terms of modified spherical harmonics, the Sphericalharmonic addition theorem becomesPi (cos c...,) . E ( — 1 )m c,,, (9, 0) cin,(0', 0').^(2.43)Chapter 2. The Theoretical Background of Molecular Spectroscopy^172.ii.D Irreducible spherical tensor operatorsThe matrix elements of a Hamiltonian operator are usually calculated using the meth-ods of spherical tensor algebra since it can greatly simplify the computation process. Inthis section, some basic ideas about tensor operators are discussed.An irreducible spherical tensor operator of rank k is defined as a set of (2k + 1)functions TI: (q = — k, — k +1, ... k) which transform under a rotation of the coordinateframe according toR(a 137)Tqk R- 1 (0 -y) = E D:, q (cti37)Tql;. (2.44)q'This means that, under rotation, the operator n is transformed into multiples of acomplete set of (2k + 1) components Tiqc„ where the expansion coefficients are the Wignerrotation matrix elements Dqk, q (R). Comparison of Eqs. (2.44) with (2.33) shows thattensor operators Ti, are proportional to the spherical harmonics Ykg (0, 0).It can be verified from the definition of Eq. (2.44) that a tensor operator of rankzero (n) is unchanged by rotation. Hence an operator Tc? behaves just like a numberor time, i.e. a scalar; therefore zero-rank tensor operators are called scalar operators.All Hamiltonian operators are scalar operators as they are invariant to rotation. Atensor operator of rank one consists of three components which behave under rotationlike the components of a vector or dipole. Therefore, angular momentum operators(vector operators) are represented by first-rank tensors. Relations between the Cartesiancomponents of an angular momentum operator and the components of a first-rank tensorcan be derived from Racah's original definition of a spherical tensor operator [8]:[ii, T iq ] = q Tqk^(2.45)[4, Tql = Vk(k + 1) — q(q +1) Tqicli .^(2.46)Chapter 2. The Theoretical Background of Molecular Spectroscopy^18By setting k = 1 and q = 0, we obtain, after some rearrangements, the following relations:Tc1, - = iz ,1^1n^,^. ,, = T ---- J± = +-- (J's + z•-iy)•(2.47)(2.48 )A second-rank tensor consists of five components and transforms under rotation like thespherical harmonics Y2 m, (0, 0), or the quadrupole moments of a charge distribution. Thustensor operators of rank two arise in the molecular Hamiltonian operator for terms suchas the electric quadrupole interaction and the electron spin-spin interaction, as we shallsee in the next section.All Hamiltonian operators can be written as scalar products of two tensor operatorsof the same rank, and therefore are related to zero-rank compound tensor operators.A compound tensor operator can be constructed in the same way as a coupled angularmomentum vector, since tensor operators behave just like spherical harmonics. Therefore,referring to Eq. (2.23) for the addition of two angular momenta, we obtain (ki k2 k )Tqkil (1) T:22 (2).qi q2 -q[Tk2(1) ® Tk2 (2)]: = E (-1)k1—k2+q V2k + 1919'2(2.49)In Eq. (2.49), a compound tensor operator of rank k is constructed from tensor operatorsTkl (1) of rank k1 acting on system (1) and Tk2(2) of rank k 2 acting on system (2). Whentwo tensor operators of the same rank couple to form a zero-rank tensor, or scalar, Eq.(2.49) becomes[Tk ( 1 ) ® Tk( 2 )ig = (_i)k V21c-l- 1 Tk(1) . Tk (2),^(2.50)where Tk(1) • Tk(2) is the conventional scalar product given byTk(1) • Tk(2) . E (_i)q Tiqc(i)T%(2).^(2.51)qThe matrix elements of a tensor operator ric can be calculated in the basis of theangular momentum wavefunctions 1-yJM > by means of the Wigner-Eckart TheoremChapter 2. The Theoretical Background of Molecular Spectroscopy^19[6, 9]:' k J< -y7 M' IT: -yJM > =^J^< 7" Ji II Tk II yJ > . (2.52)—M' q MEq. (2.52) shows that a matrix element factorizes into: (1) a part containing the 3-j symbol, which gives information about the orientation of the angular momenta (thegeometry), specified by the projection quantum numbers M' and M; (2) a reduced matrixelement < y'J'IlTk117J >, where Tk is assumed to act on functions described by thequantum number J, and all other quantum numbers are collapsed into the symbol -y.Reduced matrix elements can be worked out quite easily for an operator Tk acting onan uncoupled wavefunction, but can become rather difficult to calculate if J is part ofa coupled angular momentum scheme. In what follows, some reduced matrix elementsfrequently used in molecular spectroscopy will be discussed.(i). <^(J) II^>Since Tj, (J) = Jz from Eq. (2.47), we have< JM 'TO(J)^> = < JM iz IJM >= M.^(2.53)Applying the Wigner-Eckart theorem to the left side of Eq. (2.53), we obtain(-1 )J-m^1- -114. 0 MSubstitution for the 3-j symbol [9] produces< J II T i (J) J > = M.^(2.54)(-1)2(J-m) ,^ < J T1(J) J > M.VJ(J + 1)(2J + 1)(2.55)Chapter 2. The Theoretical Background of Molecular Spectroscopy^20Now J and M are both either integers or half-integers, so that (-1) 2 (J-m) = +1. Thuswe have<^> = J(J + 1)(2J + 1). (2.56)(ii). < J'K' II DT* (w) II JK >Rotation matrix elements VI,kg )*(w) are often used to project internal angular momentawritten in a space-fixed axis system back to a molecule-fixed axis system. Taking the7),9)*(c..,) as operators acting on symmetric top wavefunctions IJKM > and applyingthe Wigner-Eckart theorem to project out the space-dependence, we obtainJ' K J< JK'M IDpq ) * (w) JKM > =^ < JK IID.Q)* JK > .—M' p M(2.57)Recall that a symmetric top wavefunction is I JKM >. ^(2J 1)/87r2it);(w), fromEq. (2.38); then the left side of Eq. (2.57) may be written as< J'K'M' I pl,g ) *(w) I JKM >J(2J + 1)(2J' + 1)87 2x f DM,h,(w) /71,9 )*(w)^(w) dw (2.58)where w represents the Euler angles a, and 7, and dw = da sin Oc/13d-y. Substitutionfor the integral of 3 D functions with Eq. (2.39) produces<^f 'M' I 1.31,9 ) * (w) I JKM >^(-1)Ki-mV(2J + 1)(2J' + 1)J' K J J' k J. (2.59)—M' p M —K' q KChapter 2. The Theoretical Background of Molecular Spectroscopy^21Setting the right sides of Eqs. (2.57) and (2.59) equal gives' k J^< If ' II Vq )* GO K > = (-1)J-KV ^J(2J + 1)(2J' + 1) . (2.60)—K q K(iii). < jIAJ 1 II X l‘ (1, 2)11 :h. j2J > (Reduced matrix elements of compound tensoroperators in a coupled basis.)For a compound tensor operator Xel(1, 2) = [Tkl (1) 0 Tk2(2)]-14', whose componentsact on different parts of a coupled wavefunction Ijij2 JM >, the Wigner-Eckart theoremreadsJ' K J<^I XQ ( ,2)I ji j2 JM > =—M' Q Mx < AjW II XK (1,2)11 ji j 2 J > . (2.61)Derivation of the reduced matrix element < XK ( 1 1 2 ) 11j1 j2 J > is quite involvedand will not be given here. The result involves a 9-j symbol and is written as [3]< ilAJ/ II Xl‘ ( 1 , 2 )11 iii2.1 > =< ..71 II T ic' MU 1 > < A II T k2 ( 2 ) II i2 >A ji lc].X V(2J + 1)(2J' + 1)(2k + 1) A j 2 k_2J'^(2.62) J KThis formula is completely general, meaning that K, the rank of X, can take any value.In practice, it is rarely used because all the tensor operators appearing in the Hamiltonianhave to be scalar products, and if K = 0 the 9-j symbol collapses to a 6-j symbol.In Eq. (2.63), it is assumed that Tk acts on j 1 , Uk acts on j 2 and j1 + j 2<^ITk • Uk I Yj1j2JM > =6 j, 6m,m 1J A j1k j1 J2x E < -r'il II Tk II 7 " > < -y""A II uk 11-ri2 > .Chapter 2. The Theoretical Background of Molecular Spectroscopy^22For the scalar product of two commuting tensor operators, where k 1 = k2 = k andK = 0, the matrix element may be written in a coupled basis as(2.63)= J. Twoother special cases of particular importance are (a) a single operator acting on part of acoupled basis; and (b) two parts of a compound operator XK acting on the same system.The reduced matrix elements for both cases can be derived from the general formula ofEq. (2.62); they are given byx(a)^< KV' II Tki ( 1 )II^><^II Uk2 (2 ) I I^>1 +:72 +.1+ kl /(2J + 1)(2J' + 1){..11J j1 k1(_1).ii+:72-FY-Fk2V(2J + 1)(2J' + 1)< j; II T k1 ( 1 )^>,< j2 II Uk2 (2 ) II j2 >, (2.64)J j2 k2< I II X K -ri > = (-0K+3+7',/2K +1 E-Yuin^j^j' j"x < -rT II Tki^j " >< 7 " iiTk2 "Yi > • (2.65)Finally we need to establish a mechanism to refer an operator defined in the molecule-fixed coordinate system to the space-fixed axis system. If a, Q and 7 represent the Eulerangles required to bring the space-fixed axes into coincidence with the molecule-fixed axesk 1 k2 KChapter 2. The Theoretical Background of Molecular Spectroscopy^23by rotations, the molecule-fixed components of the tensor operator T 1:(A) are obtainedby rotation of the space-fixed components Tpk(A),Tqk(A) . E 131V (a 137) T I (A).^ (2.66)PMultiplying both sides of Eq. (2.66) by TIk)* and summing over q givesTrk, ( A ) = > vg).(a,37)Tqk(A).^ (2.67)qNote that the property of a unitary matrix,EE Tv. . pq = 1 ,P qis used in the above derivation. When evaluating the matrix elements of an internaloperator written in a space-fixed aixs system, Eq. (2.67) is used to project the operatorinto the molecule-fixed axis system. The reduced matrix elements of Dp*q (c037) are givenby Eq. (2.60).2.iii Molecular Hamiltonian2.iii.A The general molecular Hamiltonian and its eigenfunctionsA molecule can be regarded as a collection of charged nuclei and electrons. TheHamiltonian corresponding to the total energy of the molecule therefore consists of thekinetic energy terms for each particle and the potential energy terms representing all theelectrostatic interactions between the particles. In addition, relativistic effects resultingfrom the spinning motions of the electrons and the nuclei add small corrections to thetotal energy. For the sake of simplicity, the small relativistic corrections are droppedfrom the present discussion and will be added later as perturbations. Then the molecularHamiltonian may be written as1^14, Ho =^E /3 + E^+ v(q , Q)2m„^n 2mn (2.68)Chapter 2. The Theoretical Background of Molecular Spectroscopy^24where the potential energy operator V(q,Q) (in which q, Q represent the positions of theelectrons and nuclei respectively) is given byZn e2^e 2 ZnZnie2V(q , Q) —EE ^ +EE +EE ^(2.69)n e^• en^e e '>e^n n1>n^rnniIn Eq. (2.68), p e and pn refer to the linear momenta of electron e (mass me ) and nucleusn (mass inn ); in Eq. (2.69), e and Zn e denote the charges of an electron and of nucleusn respectively, while r ii is the distance from particle i to particle j.In theory, the energy levels of a molecule are obtained from the solutions of theSchrOdinger equation containing the molecular Hamiltonian operator H 0 given by Eq.(2.68). But since any molecule must contain at least three particles, one is up againstthe "three body problem". It is well known in classical mechanics that the equationfor the relative motions of three bodies cannot be solved exactly; similarly the quantummechanical analogue is equally insoluble. Therefore one must approximate in order toobtain solutions to the SchrOdinger equation of a molecular system. Born and Oppen-heimer assumed that electrons are so much lighter than nuclei ( and therefore so muchfaster-moving) that they adapt themselves instantaneously to one of their allowed energystates for the nuclear configuration at that moment. This means that the motions ofelectrons and nuclei can be described separately; in other words the total eigenfunction41 can be expanded into a complete set of functions which are products of an electronicpart V, ei (q , Q) and a nuclear part ik„(Q), i.e.klf = E oei(q,Q)oivr(Q).^(2.70)The electronic wavefunction 71.), i (q,Q) is an eigenfunction of the electronic Hamiltonian,according to1He = ^Epe2 V(q,Q),2m e (2.71)11,11,,,(q,Q) = Eei (Q) 4 ei(q , Q).^ (2.72)Chapter 2. The Theoretical Background of Molecular Spectroscopy^25Consequently, the total energy of a molecule is taken as the sum of the electron energyEel (for a particular nuclear configuration) and the nuclear energy E„:E = Ee i^Evr•^ (2.73)Inserting the molecular Hamiltonian of Eq. (2.68) and the total wavefunction of Eq.(2.70) into the SchrOdinger equation, we obtain E 1) + V (q ' (2) + E 2P4, E^Q) °vr (Q)= E E 0, i (q,Q)0: r (Q) (2.74)or{Cr(Q) [ 2m1 e^ EP! + V (q, Q)] 4 ei (q, Q)i-"Pei ( q'Q ) E 2rn °:'r (Q)^'Cbvr (C2) E 2:; „^Q)+ E^ PnOei(q, Q)PnOt,r(Q)} = E E oei(q, 7iiiivr(0•Mn(2.75)On multiplying both sides of Eq. (2.75) by V,: k and integrating over the electron coordi-nate q (which essentially picks out the state k from the electronic manifold), the equationbecomes{-Eek(Q) E 2Pn?tin + J z4k (q, Q) E 2P,,72in Oek(q, Q) dq} O lvcr(Q)+ E {I E1 pn 0:,,(Q)1): k (q,Q)pokei(q, Q) dq} E 0 1,,c r (Q).(2 .7 6)n " 6 nThere are two terms in Eq. (2.76). The first term has the form of a SchrOdinger equationin which the Hamiltonian consists of the nuclear kinetic energy, the electronic energyof state k as an effective potential energy and a small mass dependent term which con-tributes a small electronic isotope correction. Thus the first term essentially defines theChapter 2. The Theoretical Background of Molecular Spectroscopy^26vibrational and rotational functions of the electronic state k. The second term representsthe coupling between electronic motion and vib-rotational motion in different electronicstates, and is responsible for the failures of the Born-Oppenheimer approximation. Se-vere breakdown of the Born-Oppenheimer approximation occurs when the energy levelsof two electronic states lie close to each other. Such a situation results in strong interac-tions called vibronic (meaning vibrational and electronic) couplings between the levels ofthe two states. Vibronic coupling plays an important role in the spectra of polyatomicmolecules and active research has been carried out in this area. Since no such coupling ef-fects are observed in the molecules studied in this thesis, this subject will not be pursuedfurther.In the absence of strong vibration-rotation interactions, the nuclear wavefunction vrmay be taken as a product of vibrational and rotational factors; accordingly, the nuclearenergy is taken as the sum of the vibrational and rotational energies. As a result, thetotal energy of a molecule in the electronic state k may be written asE = Eek Evib Erot. (2.77)The electronic energy Eek or, more precisely, the electronic energy at the equilibrium nu-clear configuration is treated as a parameter in spectral analysis. The vibrational energyEvib is usually taken as that for a harmonic oscillator with anharmonicity corrections [20]1^1Evib^(v —2)We — (v —2)2we x (2.78)The symbols we and we x e are the determinable vibrational frequency and the anharmonic-ity constant, while v is the vibrational quantum number for the level of interest. Theexpression for the rotational energy within a particular vibronic state is derived next.Chapter 2. The Theoretical Background of Molecular Spectroscopy^272.iii.B Rotational HamiltonianThe classical rotational Hamiltonian is given by [10]R2^R2^R2Hrot = x + y + z ,2/s^24^2/,^ (2.79)where Rs , Ry and Rz are the components of the rotational angular momentum R alongthe principal axes of a molecule, and IS, Iy and I, are the principal moments of inertia ofthe molecule. For diatomic molecules, the angular momentum vector R is perpendicularto the molecular axis (which is normally chosen to be the z-axis); therefore we haveR, = 0^ (2.80)andIx = 4 = I = pr2 ,^ (2.81)with p being the reduced mass of the molecule given bymim2 II =^ (2.82)m1 + m2In Eq. (2.81), r is the distance between the two nuclei of masses m l and m 2 . As a result,Eq. (2.79) reduces toHrot =Rx + Ry2 = R2(2.83)21^2pr2.If the nuclear separation r is assumed to be constant during the rotational motion, Eq.(2.83) represents the Hamiltonian for a rigid rotor, and the factor 1/(2pr 2 ) before R2 iscalled the rotational constant denoted by the letter B:B = 1^.2pr 2(2.84)But since the internuclear distance does increase with rotation due to the effects ofcentrifugal forces, a centrifugal correction term is usually added to the rigid rotor Hamil-tonian to give a more complete rotational Hamiltonian which may be written, to secondChapter 2. The Theoretical Background of Molecular Spectroscopy^28order, in the formHrot = B112 — D R4 ,^ (2.85)where D is the centrifugal distortion constant. Centrifugal corrections are also requiredfor the other terms in the molecular Hamiltonian that are functions of the internuclearseparation r. The form of the second order centrifugal correction term for any term inthe Hamiltonian H(r) is given byHD = —12CD [H(r), ill= —1 CD [H(r)R2 + ii2 i-1(r)}^(2.86)2where CD is a proportionality constant and the anticommutator [x, Y]+ is required tomaintain the Hermitian form of the operator [11].The rotational angular momentum R is just one of many angular momenta that canarise in a molecule. Other important angular momenta include the total electron orbitalangular momentum, L, and the total electron spin angular momentum, S. The totalangular momentum J is what interacts with the external beam of photons; this is thesum of R, L and S, i.e.J = R + L - I- S.^ (2.87)The rotational angular momentum then has to be taken asR = J — L — S.^ (2.88)As a result, the rotational Hamiltonian given by Eq. (2.85) and the centrifugal correctionHamiltonian of Eq. (2.86) have to be rewritten accordingly.2.iii.0 Electron spin HamiltonianThe motion of a charged particle in space is known to produce a magnetic moment.The spin motion of an electron, although it has no classical anologue, also gives rise to aChapter 2. The Theoretical Background of Molecular Spectroscopy^29magnetic moment, which is directly proportional to the spin angular momentum,ge tin s =^h •(2.89)In this equation, ge is the relativistic "g-factor", 2.0023, [03 is the Bohr magneton,eh/2m,, representing the unit of electron magnetic moment, and s is the spin angu-lar momentum, with magnitude Js(s + 1) h. If all the electrons in a molecule are paired,the net spin angular momentum S is zero, and all the electron spin effects cancel. How-ever, if a molecule has unpaired electron(s), S is greater than zero, and various effectsarise: the electron spin magnetic moments interact with 1) the electron orbital magneticmoments, 2) the rotational magnetic moment and 3) each other, provided there are atleast two unpaired electrons. We describe these three effects as electron spin-orbit inter-action, spin-rotation interaction and spin-spin interaction. Therefore the electron spinHamiltonian fivin may be written= H30^Iss,^ (2.90)where the energy operators on the right side correspond to the three electron spin inter-actions, as indicated by their subscripts.Spin-orbit interactionAn electron moving round a charged nucleus with linear velocity v in a static electricfield E experiences a magnetic field B given by [12]E x vB = ^ (2.91)c2Now the electric field resulting from the charged nucleus is given byE^— (rr ) (ddVr^ (2.92)Chapter 2. The Theoretical Background of Molecular Spectroscopy^30where dV/dr is the gradient of the Coulomb potential and r/r is the unit vector fromthe position of the nucleus to the electron. (Eq. (2.91) then becomesB^c21 r c1V(r x v).^ (2.93)dr)The interaction energy of a magnetic moment it with a magnetic field B is — it • B;therefore on substituting for B with Eq. (2.93) and for it with the electron spin magneticmoment given by Eq. (2.89), we obtain the Hamiltonian operator for the spin-orbitinteraction in a two-particle system,Hg = -# • .6 gefiB dV) (r x v) • s.hc2r dr(2.94)Before we proceed to a multi-particle system, we need to label the coordinates to showtheir exact meanings; r becomes rem , the distance from the electron to the nucleus, vbecomes v,, the relative velocity of the electron with respect to the nucleus, and sbecomes s e , the electron spin. Also at this point we introduce a relativistic effect calledthe "Thomas precession" [13]: time appears to slow down for the fast-moving electron,and it appears to the nucleus that the electron spins only half as fast as if it were static.Thus the effective magnetic moment of the electron appears to be halved, and this effectcan be accounted for by taking the relative velocity of the electron y en, as (2v e — vm ).As a result, Eq. (2.94) becomesEso =^getiBhc2ren dren(dV rLien x (2^— 'b.)] • :4e.^(2.95)The spin-orbit interaction is additive for the various electrons and nuclei, so that for amolecule we have-il SO gel-03 v. E^dV^-ho^dr, k 2e n renge,UB^1^dV ti.en^.4ee n2hC2 L'^drenren.4e,eg /AB^1^dVE Eho „ enr dr. • 8e(2.96) Chapter 2. The Theoretical Background of Molecular Spectroscopy^31where the expression is separated into two terms: an electron term and a nuclear term.The second term of Eq. (2.96) represents the interaction between the electron spin andthe overall motion of the molecule, indicated by v n , and is therefore the spin-rotationinteraction. We will drop this term for now and reconsider it in the next section. Thefirst term becomes the normal spin-orbit Hamiltonian upon substitution of ("p en x rni/e)by the orbital angular momentum operator ien,140 — ge"2h771C2 re\---, z 1 n drV( d ) jen . S.ene n^k,By setting len Ps.: le , the spin-orbit Hamiltonian operator is usually approximated asii„ = Z a, l e • e e ,^ (2.98)ewhere the spin-orbit parameter a, is given by(2.97)a, gettB z 1 ( di 7 )_2hmc2 ten. ^dren j(2.99)Eq. (2.98) is called the "microscopic spin-orbit Hamiltonian", and can be written intensor-operator form asiiso = Z a, 711 (1,) • T i (se ).^ (2.100)In spectral analysis, ii„ is sometimes simply expressed as the scalar product of the totalelectron spin and orbital angular momenta, S and L:-ft 8. = A i • ,§ .^ (2.101)A is the spin-orbit coupling constant, S and L are the vector sums of the individualelectron spin angular momenta s e and the orbital angular momenta l e , that isS = Z S eeL = > 1.e(2.102)(2.103)Chapter 2. The Theoretical Background of Molecular Spectroscopy^32Eq. (2.101) is commonly used in situations where electronic states are studied in iso-lation, while the microscopic spin-orbit operator Ee a, I, • 8 e is required in cases whereinteractions between electronic states of different spin multiplicities need to be considered.Spin-rotation interactionAs indicated in last section, the magnetic interaction of the electron spin with the mag-netic field that the moving electrons experience in a molecule gives rise to a term definedas the spin-rotation interaction. The microscopic form of the spin-rotation Hamiltonianoperator is given by the second term of Eq. (2.96), i.e.geilB^1 (dV ,ilsr =^EE — --) ( ten X 4) n ) • :4 e•he2 e , ren dren(2.104)Similar to the spin-orbit interaction, a phenomenological form of the spin-rotation Hamil-tonian is normally used in spectral analysis; this is written asfisr = "Y -1 t • ,§,^ (2.105)where -y is the spin-rotation coupling constant.Besides the direct coupling of electron spin and molecular rotation, there is a secondcontribution to the spin-rotation Hamiltonian which is actually larger than the directcontribution. This indirect contribution comes from the off-diagonal spin-orbit matrixelements, and can be shown by second-order perturbation theory to have the same formas the spin-rotation Hamiltonian [14] given by Eq. (2.104). As a result, the spin-rotationcoefficient -y measured from molecular spectra contains two indistinguishable contribu-tions and should be written7 ^,),sr + 7S0^ (2.106)Chapter 2. The Theoretical Background of Molecular Spectroscopy^33Spin-spin interactionThe interaction between the spin magnetic moment of an electron with the magneticfield generated by the spin motions of other electrons is called spin-spin interaction.According to Maxwell's equations [12], the magnetic field produced by electron j withmagnetic moment /Li at coordinate i is writtenBi = V x Ai,^ (2.107)where V is the directional derivative operator, and A, is the magnetic vector potential,given by [12]Ai = X3r(2.108)with r,, = r j — ri. The Hamiltonian for the interaction at the point i between themagnetic moment tt i of electron i and the magnetic field B i is=^• B i7...=^• [V x µj x 2 i)3 ,g ttB2 2^ re ^/7 X S • X k 3t^si.2 riiThe vector directional derivative of a vector product (A x B) is given by [15]ss j(2.109)V x (A x B) = A x (V • B) — (A • V) x B(B • V) xA—Bx (V • A).^(2.110)Since the V operator is a function of spatial coordinates only, differential operations withrespect to spin functions will result in zero, i.e.V x s = 0,^V • s = 0.^ (2.111)Chapter 2. The Theoretical Background of Molecular Spectroscopy^34According to relations (2.110) and (2.111), equation (2.109) may be rewritten as2ge" Vrh2 r" )]X^•3^( • Si2^2ge[(83 • V) x • Si. (2.112)h2 rjiThe first term of Eq. (2.112) is zero except when rji = 0, because Gauss' divergencetheorem statesV' • ---2-Hrr3ji77 • V-1rii = —471-6(rji ),^(2.113)where b(rii ) is the Dirac delta function, which is zero everywhere except when r ii iszero. The Dirac function is defined such that it picks out the square of the amplitude ofthe electron wavefunction for electron j at the coordinate origin, i.e., at the position ofelectron i;<^ii) > = 1 zk (0 ) 1 2Therefore, the first term of Eq. (2.112) becomes2 2gettBh2 4711030)1 2 8i • Si.It can be shown that the second term of Eq. (2.112) factorizes toght 2B f 47r IV, (0)128, 83 + {s i • Si^3(Si • rii)(Si • Vil}h2^3^3 3 5rii(2.114)(2.115)(2.116)Combining the two terms given above and summing over all electrons, we obtain thespin-spin interaction Hamiltonian operator as11.93 = e ^v-•eh:B 4_, -3 10g^j(0)123i • ki2 2i>32 2+ h2^.[(•"ei • :43)7-Ji —^.2> 311'3203 • 1.30] r3 , (2.117)The first term turns out to give a constant contribution to the energy, and is usuallyincluded with the Born-Oppenheimer potential; the second term has the same form asChapter 2. The Theoretical Background of Molecular Spectroscopy^35the classical Hamiltonian for the magnetic interactions between bar magnets, or dipoles;therefore it represents the dipolar electron spin-spin interaction.The spherical tensor form of the spin-spin Hamiltonian is written [9]2 2Hss = — V6 ge2" E T 2 (si,si) • T2(C),^(2.118)hwhere T2 (C) is related to the spherical harmonics Y2q (0, 0) byTq2 (C)yq2(0, 0)3rii(2.119)In the calculation of matrix elements, T 2 (s i , si ) carries the quantum number dependenceand the Tq2 (C)'s are taken as parameters. Considering only the q = 0 term of the scalarproduct of [T2 (si, si) • T 2 (C)] of Eq. (2.118), the spin-spin Hamiltonian reduces toNow, since [16]P-33 =2 0 2E T2(s• , s.)T2(c) •i>j 03E < slIT2(s, sJ)11s)-- < slIns,^)11,9>,i>j(2.120)(2.121)it is convenient to write the spin-spin operator in trems of the total spin quantum numberS, and at the same time to introduce a parameter for the diagonal matrix element of thescaled spherical harmonic, TAC):1183 = 2 NA- A TAS , S)3= 3(3:5? — S2 ).^ (2.122)Expression (2.122) is commonly used in spectral analysis to describe the electron spin-spin interaction.Chapter 2. The Theoretical Background of Molecular Spectroscopy^362.iii.D Nuclear spin HamiltonianLike electrons, many nuclei possess intrinsic spin angular momenta, and thus nu-clear spin magnetic moments. In addition, the spin motions of nuclei produce electricquadrupole moments for nuclei with spin I > 1. As a result, the nuclear spin Hamil-tonian includes a magnetic term representing the interactions of the nuclear magneticmoments with other magnetic moments in a molecule, and an electric quadrupole termrepresenting the interaction between the electric quadrupole moment of the nuclei withthe electric field gradient in the molecule. In other words, the nuclear spin Hamiltonianmay be writtenHmag.hfs -E HQ. (2.123)The subscript "h f s" is the abbreviation for "hyperfine structure" which originates fromthe fact that the nuclear spin interaction is usually so small that it can cause only minorchanges to the energy levels (usually known as hyperfine structure). In the followingdiscussion, the two terms in the nuclear spin Hamiltonian are considered in detail.Magnetic hyperfine interactionThe nuclear spin magnetic interaction originates from the same mechanism as theelectron spin interaction: it consists basically of interactions between the magnetic mo-ment of a spinning nucleus with (1) the spin magnetic moments of the electrons; this isknown as the nuclear spin-electron spin interaction, (2) the orbital magnetic moments ofthe electrons; this is called the nuclear spin-electron orbit interaction, and (3) the spinmagnetic moments of other nuclei; this is the nuclear spin-spin interaction. Because themagnetic moments of nuclei are much smaller than those of electrons, nuclear spin-spininteraction is usually negligible in most electronic spectra.The Hamiltonian for the nuclear spin-electron spin interaction is exactly the same asChapter 2. The Theoretical Background of Molecular Spectroscopy^37that for the electron spin-spin Hamiltonian given by Eq. (2.117). Simply replacing oneof the two electron spin magnetic moments with a nuclear spin magnetic moment pi ,=^ (2.124)we haveHra = gett"n E Egn {-8-71 10,0)12 in ..4,n eh 2^3— [(in • .4,) re2„ — 3(in • ?'en )(se • i.,n )] —r5^(2.125)nwhere gn is the nuclear g-factor for the nucleus n, itn is the nuclear magneton Gin =eh/27np , with nrp being the mass of a proton) and I is the nuclear spin angular mo-mentum. In this case, the first term of Eq. (2.125) does not vanish, it represents theFermi contact interaction between electrons and nuclei which arises when electrons are inorbitals with non-zero amplitude at the nuclei. It is found that only unpaired electronsoccupying molecular orbitals derived from atomic s orbitals have non-zero contributionto the Fermi contact interaction. Like the electron spin-spin Hamiltonian, the secondterm represents the dipolar interaction between the nuclear spin magnetic moments andthe magnetic field generated by the spinning electrons at the nuclei.The Hamiltonian for the nuclear spin-electron orbit interaction can be obtained fromthe electron spin-orbit Hamiltonian (Eq. (2.98)) by replacing the electron spin angularmomentum with the nuclear spin angular momentum and changing the parameter a, toa,. It is therefore written as11.1. 1^EE aen in • i,n e--, . gn lin 1 dV(2hc2me r) dr n.=^ i^le. (2.126).dri L—deThe total magnetic hyperfine Hamiltonian is obtained on combining Eqs. (2.125) and(2.126). For diatomic molecules with only one spinning nucleus such as those discussedChapter 2. The Theoretical Background of Molecular Spectroscopy^38in this thesis, the Hamiltonian operator flmag.hf, is written1Hmag.hfs^gegn t^in ^•^-871-^(0)I 2 • .4 eh "2^{71 le + 3 e—^.9e) re — 3(1 • ie)(:se l'e)] s} •^(2.127)reIf the dipolar term is treated along the lines of Eq. (2.120), retaining only operators thatare diagonal in the electronic state quantum number, we haveHmag.hfs =geg"itn^v-1 i• Ale + 87 10e( 0 )1 2 -Ih 2^71^3(3 cos 2— [1^— 3-iv;z(e) .1 (2.128)270 — 1)1To a good approximation, the sum over electrons can be taken as a vector sum over theelectron spin and orbital angular momenta, s e and le , to give the total electron spin andorbital angular momenta, S and L; this yields^gegniliBtin [^A A^81rh2^ r3 + -i-10,(0)1 234,§ (3 cost —^z) ^2r3^(2.129)From the above expression, we obtain the magnetic hyperfine Hamiltonian commonlyused for open shell diatomic molecules [17]Hmag.hfs = a 1 • I, + bi • A§ + clz&,^(2.130)or, in the form of spherical tensor operators,Hmag.hfs = a r(I) • T l (L)^bT 1 (I) • T i (S)^cTd(I)V(S),^(2.131)wherea gegnPB P^>^n ^1 (2.132)h2^< W a,^87rgegnitBlin 10(0 ) 12^ (2.133)3h 2c = 3gegntiBtin < 3 cos 2 0 —1 (2.134)2h 2^r3Hmag.hfsChapter 2. The Theoretical Background of Molecular Spectroscopy^39andb = a — —1c.c^3 (2.135)The coefficients a, b and c are the spectroscopically determinable hyperfine parameters; ais the nuclear spin-electron orbit coefficient, b is (a, — 3c) with a, being the Fermi contactparameter and c is the dipolar electron spin-nuclear spin interaction constant.Electric quadrupole interactionA nucleus can be taken in first approximation as a collection of protons and neutrons,or nucleons. Since protons and neutrons have 'spins of -12-', nuclei may have spins of 0, 2,1 ... etc. If the nuclear spin is 0 or 1, a nucleus has a spherical charge distribution; if thespin I > 1, the nucleus has a quadrupolar charge distribution. An important hyperfineinteraction occurs between the quadrupole moment of a nucleus and the electric fieldgradient at the nucleus caused by the electron charge distribution. The interaction isthe first non-vanishing term in the multipole expansion of the electrostatic interactionbetween a nucleus and the electrons.According to Coulomb's law, the potential energy of an electron in an electric field isE = e V, (2.136)where the electrostatic potential at the electron, V, is produced by the n nucleons, withcharges qn :q.V = E R- .^ (2.137)nThe distance R„ is the separation of the electron and the nth nucleon. Let r n be thecoordinate of the nucleon relative to the center of the nucleus, and R be the distance ofthe electron from the nucleus; then R, is given byRn = Ir n — RI.^ (2.138)Chapter 2. The Theoretical Background of Molecular Spectroscopy^40From the law of cosines [18], Rn can be expressed in terms of the distances r n and R andthe angle On between the vectors r n and R according toRn = (R2 + rn2 — 2 R rn cos On) 1= R[1 + (L2 ) 2 — 2(--Lr ) cos On] 2^(2.139)HenceR = [1 — 2(.7 ) cos On + (L-2 ) 21 2 .^ (2.140)Rn L^R^RThe right side of Eq. (2.140) resembles the "generating function" F(x, y) = (1 — 2xyy 2 ) - 2 , so that it can be expanded as a series of Legendre polynomials [6] for y = rn /R ti 0:rns—D = E Pi (cos en ) ( 74- ) i .1(2.141)From this, Eq. (2.137) becomesV^E Pl(cos n ) r in qn^ (2.142)R1+ 11=0 nAt this point we use the Spherical harmonic addition theorem of Eq. (2.43) to replaceeach Legendre polynomial with a scalar product of two modified spherical harmonics.Upon summing over all electrons, we obtain the multipole expansion of the electrostaticHamiltonian asftmultipole^eVE E(-1)- [E eRi +^001 [E qnrs_i m on , cbrol . (2.143)1=0 m^e^eThe result is a sum of scalar products of tensor operators of rank 1, one referring to theelectron coordinates, and the other to the nuclear coordinates.Now let us consider the different expansion terms. For 1 = 0 we have the coulombicmonopole interaction between the nuclear charge and the electrons; this has been includedin the electronic Hamiltonian. The 1 = 1 term represents interactions involving theChapter 2. The Theoretical Background of Molecular Spectroscopy^41nuclear electric dipole moment, which by symmetry arguments, can be shown to be zero,as are the higher electric multipoles of odd order. Finally, the 1 = 2 term is the firstnon-vanishing term in the multipole expansion; it is the interaction of the nuclear electricquadrupole moment (Q) with the electric field gradient (VE) experienced by the nucleusdue to the charge distribution of the electrons. For nuclei with I > 1, i.e. those with aquadrupole moment, the quadrupole Hamiltonian is taken as [19]HQ = eT2 (Q) • T 2 (VE),^ (2.144)whereeT2 (Q) = E qt,r2 C 2 (07,,, On )nis the quadrupole tensor, andT 2 (VE) = E e C 2 (0,, (ke )//ieis the electric field gradient tensor. By convention, the nuclear electric quadrupole mo-ment operator is defined as1Q = — E go-7,2 (3 cos t Or, — 1).^ (2.145)e .The corresponding quantum mechanical observable, called the nuclear quadrupole mo-ment, is defined again by convention asQ =<I,rni=I1(51I,mi= I > .^(2.146)It can be easily seen that the quadrupole tensor operator is related to the conventionaldefinition bye n (Q) = 2 e 0^ (2.147)which givese < /,rni = I 1TO(Q) I I, m l = I > = e Q.^(2.148)Chapter 2. The Theoretical Background of Molecular Spectroscopy^42Similarly the field gradient coupling constant is defined asWI/qo = < J, Mj = J I ( a 2, 2 )0 1 J, Mj = J >wherea2v(-e—z2)0 = e R-3(3 cost oe — 1) ;therefore(2.149)(2. 150)To2 (VE) = 1 qo .^ (2.151)2The experimentally measurable parameter is the quadrupole coupling constant given aseQqo.2.iv Hund's coupling cases and the Hamiltonian matrix2.iv.A Hund's coupling cases and the corresponding basis setsIn principle, one has total freedom in choosing a set of basis functions to calculate theHamiltonian matrix provided that the basis functions are combinations of suitable eigen-functions of angular momentum operators. Therefore it may sometimes be advantageousto choose a basis set where the calculation of matrix elements is easy but the matrix itselfis far from diagonal, since diagonalization is routine nowadays with the aid of comput-ers. However, if the basis functions are almost the eigenfunctions for a given state, theHamiltonian matrix is nearly diagonal, and the diagonal matrix elements already almostreproduce the observed energy levels. The choice of basis functions obviously dependson the relative magnitudes of the various angular momentum couplings in a molecule.Hund was the first to investigate the different angular momentum coupling schemesin molecules; he proposed five limiting coupling cases which are now known as the Hund'scoupling cases (a) to (e) [5]. The original Hund's coupling schemes do not include thenuclear spin angular momentum, but can be extended to include nuclear spin effects.Chapter 2. The Theoretical Background of Molecular Spectroscopy^43Hund's coupling cases (a) and (b) are the most common cases for molecules with no veryheavy atoms, and will be considered in detail in this thesis for diatomic molecules. Theother three coupling cases are occasionally encountered and will be briefly discussed.Hund's coupling case (a) applies when the electron orbital and spin angular momentacouple fairly strongly to each other, but only weakly to the rotational angular momentumR. The orbital angular momentum L, representing the circular motion of the electronsaround the nuclei, couples strongly to the molecular axis as a result of the coulombic in-teraction between the electrons and electric field along the internuclear axis. The vectorL is said to `precess' rapidly about the internuclear axis (the z axis), with constant pro-jections Lz ; while L itself is not conserved. The spin angular momentum S is also weaklycoupled to the internuclear axis through the spin-orbit interaction, and has a constantprojection Sz . Fig. 2.3 is a vector diagram representation of case (a). The symbols Aand E represent the quantum numbers for the projections L, and 5', respectively, whileS2 is the sum of A and E. The total z-axis projection of the electron angular momenta,represented by quantum number ft, combines the rotational angular momentum R toform the total angular momentum excluding nuclear spin, J:= J (2.152)Since R is perpendicular to the molecular axis in diatomic molecules, the projection ofJ along the z axis must be the projection of the electron angular momenta, defined bySZ. In Hund's case (a) coupling, the angular momenta S and J are both well defined, asare the angular momentum z-axis projections J z , Sz and Lz .The basis functions for electronic states in case (a) coupling are simply taken as theproducts of the eigenfunctions of the well-defined angular momentum operators; they areusually written as^I //A; SE; J9 >,^ (2.153)Chapter 2. The Theoretical Background of Molecular Spectroscopy^44IttI.■^1A^II^sNL 1^I\ 00 I\^IL . 1^i^ k.,I\\\ X^/.N•AL,/Figure 2.3: Vector diagram for Hund's coupling case (a) [20].Chapter 2. The Theoretical Background of Molecular Spectroscopy^45where the symbol 9 refers to the other electronic coordinates needed to describe the elec-tronic state fully, and the semicolon separators indicate products of component wavefunc-tions. A case (a) electronic state is labelled with symbol 2s+ 1 A0 where the superscript(2S + 1) is called the spin multiplicity.Fig. 2.4 is a vector diagram representation of Hund's case (b) coupling. In this case,the spin-orbit interaction is zero or smaller than the rotational energy. Thus the spinangular momentum S is essentially not coupled to the molecular axis, but the orbitalangular momentum L remains coupled to the z-axis with a well-defined projection Lz .L, and the rotational angular momentum R form a resultant N, which is then coupledto the electron spin angular momentum S to form the total angular momentum J:N + S = J. (2.154)In case (b), projection quantum numbers E and CI are no longer defined; only A, theprojection quantum number of both the angular momenta L and N, is defined. Thebasis functions for a case (b) electronic state, labelled 2s-H-A, take the formIii; NASJ > . (2.155)Hund's cases (a) and (b) represent two idealized cases of angular momentum couplingin molecules. In reality, there are no pure case (a) or case (b) couplings; molecules indifferent electronic states may follow approximately case (a) or case (b), or somewherein between. In many molecules an electronic state changes from case (a) to case (b)coupling as a result of "spin-uncoupling". This happens because spin-rotation couplingincreases with rotational motion, and at large enough J (so that the rotational energy islarger than the spin-orbit interaction) the spin angular momentum S will couple morestrongly to the rotational angular momentum R than to the orbital angular momentumL. In this situation, S is said to be uncoupled from L, and therefore from the molecularNR5' 5'.1A^ >_ •M•Chapter 2. The Theoretical Background of Molecular Spectroscopy^46Figure 2.4: Vector diagram for Hund's coupling case (b) [20].Chapter 2. The Theoretical Background of Molecular Spectroscopy^47axis (case (b)). The relation between case (a) and case (b) basis functions can be derivedfrom the principles of angular momentum coupling discussed in Section 2.ii; it is givenby Brown and Howard [21]^ (ly; NASJ > = E(-1) N-s+ f1V2N +1 J S NE,SI^ It —E —A7/A; SE; ./S-2 > .^(2.156)When nuclear spin is included in the coupling scheme, the Hund's coupling casesmust be subdivided according to the relative magnitude of the couplings between thenuclear spin and the other angular momenta. The extended coupling cases are based onthe Hund's scheme, but with the subscript a indicating that the nuclear spin is stronglycoupled to the molecular axis, or a subscript # indicating that the nuclear spin is notstrongly coupled to the molecular axis, but to other angular momenta. The a-typecoupling schemes, such as cases (a c ) and (ba ), are never seen because the small size ofthe nuclear spin magnetic moment makes it unlikely that the nuclear spin would couplestrongly to the molecular axis by magnetic interaction with the molecular field. Thereforeonly 0-type coupling cases need to be considered. Four such coupling cases can be derivedfrom the Hund's (a) and (b) cases; they are: (an ), (130j), (bas) and (b0N). Cases (an)and (bnj ) are the most frequently encountered cases. In both cases, the nuclear spin I(assuming only one spinning nucleus is present) is coupled to the total angular momentumexcluding nuclear spin, J, to form a resultant F:J + I = F. (2.157)The corresponding basis functions for coupling cases (a n ) and (bpJ) arelyA; SE; JilIF >and I 77; NASJIF > respectively.In the (b0N ) and (bps ) coupling schemes, I is coupled to the angular momenta Nand S respectively. However case (b0N) is not expected to occur because the angularmomentum coupling between I and S or between I and J is so much stronger than thatChapter 2. The Theoretical Background of Molecular Spectroscopy^48between I and N that cases (bps) and (boj) are the most probable extended Hund's (b)coupling schemes. Case (bps) applies when the Fermi contact interaction a cI • S is largerthan any other electron spin interaction. In this case, I couples to S to form the angularmomentum G, which then couples to N to give the total angular momentum F:I + S = G (2.158)G + N = F. (2.159)An example of case (3,3s ) coupling has been reported in the ground state of scandiumoxide [22].Hund's coupling case (c) is observed in molecules containing one or more heavy atoms.The presence of a heavy atom (or atoms) results in extremely large spin-orbit interaction,which causes the electron spin and orbital angular momenta of each electron, s and 1, tocouple with each other to form a resultant j. The vector sum of j's for all the electronsis called Ja , which is equivalent to the sum of S and L. J a is then coupled to therotational angular momentum R to form the total angular momentum excluding nuclearspin, J. The vector diagram for case (c) coupling is shown in Fig. 2.5. One of thecharacteristics of this coupling scheme is that the spin angular momentum S is no longerdefined, therefore spin multiplicity is no longer defined either. The basis functions for acase (c) state, labelled with the quantum number ft, are written as I iJa ; JS1>, with 52being the only defined projection quantum number.Case (d) coupling occurs in molecules where an electron has been promoted to aRydberg orbital with a higher principal quantum number n. The long distance betweenthe electron and the nuclei weakens the coupling of the orbital angular momentum L tothe internuclear axis, so that L is quantized. Case (d) is equivalent to case (b) exceptfor the difference that L is now coupled to the resultant angular momentum N (or moreRChapter 2. The Theoretical Background of Molecular Spectroscopy^49Figure 2.5: Vector diagram for Hund's coupling case (c) [20].Chapter 2. The Theoretical Background of Molecular Spectroscopy^50precisely Ncore in this case) instead of S. In other words, Eq. (2.154) now becomesNcore + L = N.^ (2.160)The coupling of angular momenta N and S to form the total angular momentum J isvery small and usually not included in the coupling scheme. The basis functions in case(d) are given as I Ncore A L N >.Hund's coupling case (e) is of little practical significance since no examples of case(e) coupling have been well characterized. Detailed discussion of this coupling case canbe found in the literature [23] and will not be given here.2.iv.B Matrix elements of the Hamiltonian operator in a case (a o ) basisIn this section, we will derive the Hamiltonian matrix elements in a case (an) basis,since all the relevant electronic states discussed in this thesis follow the case (a n ) couplingscheme quite closely. The Hamiltonian operators employed are those discussed in Section2.iii. For isolated electronic states, it is sufficient to express the Hamiltonian operatorsin terms of the total electron spin and orbital angular momenta. Only in the presenceof perturbations between different electronic states is the microscopic form of the spin-orbit Hamiltonian required. Thus from Section 2.iii, the total Hamiltonian operatorappropriate for case (an ) coupling may be written"total^flrot^Hso^Hss^"sr^flmag.hfs^HQ^(2.161)where"rot = BR2 — D il4^(2.162)Hso = AtLiss —2 ),(3,523^zor E aii, • .42— :§ 2 )(2.163)(2.164)Chapter 2. The Theoretical Background of Molecular Spectroscopy^51-11 sr = 71? • .§^(2.165)14nag.hfs = ai • i + bI • .§ + clz,§z^(2.166)HQI Q --= — T 2 (V E) • T 2 (Q) . (2.167)In Eq. (2.161), a A-doubling operator /hp has been added to the molecular Hamiltonian.The origin of A-doubling, and the form of the operator will be discussed later in thissection. In what follows, the matrix elements of the terms given above will be derivedindividually.As can be seen in Eq. (2.152), case (a) coupling is described by (R+L+S)=J.Therefore it follows that the rotational Hamiltonian of Eq. (2.162) can be rewritten asfiret = B(.-I — i — ,§) 2 — D(:I — i — ,§) 4 .^(2.168)Neglecting the centrifugal term for the time being, expansion of the B term gives'rot = 02 + L 2 + :§ 2 — 2:i • ,§ + 2i . :§ — 2i - :/).^(2.169)Because the x and y components of the operator L connect electronic states of differentA values and their effects are incorporated into the A-doubling operator, they can beomitted from the rotational Hamiltonian. Therefore Eq. (2.169) becomes'rot = B (:1 2 + L!d- ,§ 2 — 2i,§z —2:ix & — 2.iy S's y +2L,..§, — Az :1z) (2.170)= B rI2 + :§ 2 — ,.‘I — ST! — (i+ ,§_ -I- i_,§+ )]^(2.171)where the off-diagonal term —(i+ ,,§_ + i_,§+ ) is the spin-uncoupling operator. Since case(a) basis functions I //A; SE; J52 > are simultaneously eigenfunctions of the angular mo-A^A^Amentum operators J2 , J,, S2 , S, and L, matrix elements of the rotational Hamiltoniancan be obtained by applying Eqs. (2.13) — (2.15). The results are< qA; SE; J12 I i/rot 177A; SE; JS2 > = BRJ(J + 1) + S(S + 1) — C22 — E 2 1^(2.172)Chapter 2. The Theoretical Background of Molecular Spectroscopy^52and< 9A; SE; JS/ I firot 17/A; SE ± 1; JS/ ± 1 >— BVRAJ + 1) — 52(52 1)][S(S + 1) — E(E 1)].^(2.173)In the derivation of the off-diagonal matrix element, the ladder operators for the totalangular momentum, Jt , are applied with the understanding that they behave likebecause of the anomalous commutation relations for J in the molecule-fixed axis system(see Eq. (2.32)). The matrix element corresponding to the — D term of the rotationalHamiltonian is obtained by squaring the matrix of the coefficients of the B term andchanging the sign.The matrix elements of the spin-orbit Hamiltonian can be derived using spherical ten-sor methods. First, consider the microscopic spin-orbit Hamiltonian; its matrix elementsare given in a case (a) basis as< iA; SE; All E aiii • :s i I 9 1A'; S'E'; J52 >EE(-1)q < SE I T:(si) I S'E' >< 9A I 71,(ai/i)^> .^(2.174)qApplying the Wigner-Eckart theorem to the matrix element of T:(s i ), we obtain< 9A; SE; J1 E a ii i • :si I i'A'; S'E'; JC2 > = E(-1)q(-1)s-ES 1 S'E <^II Ti(si) II s' ><^71-1q(aiii)111A/ >,^(2.175)—E q E'where the product of the two matrix elements is left as the experimental parameter. Thereis no need to use the powerful machinery of spherical tensor algebra for the alternativeform of the spin-orbit operator, A L • ,§, but the derivation has been done this way forconsistency. Corresponding to Eq. (2.175), we have< yA; SE; JC2 I AL • .§19'A'; S'E'; JS/ > = E(- 1)q (- 1)s-EChapter 2. The Theoretical Background of Molecular Spectroscopy^53S 1 S'^< S II T 1 (S) II S ' >< iA1111q(AL) 11'A' > .^(2.176)—E qSubstitution for the reduced matrix element <^> with Eq. (2.56) gives< VA; SE; JSZ I A • ,§177'A'; S'E'; J52 > = E(-1)q(-1) s-ES 1 S'—E q ^S(S 1)(25 + 1) bss , < 77A I Tl q (AL) 177'A' > . (2.177)Thus the only non-vanishing matrix elements of the operator A L • S are those diago-nal in S. The matrix element of Ti q (AL) may be rewritten in terms of the Cartesiancomponents of the operator L following Eqs (2.47) and (2.48):<^Tlq(AL)171' > l < 01 A Lz I y'A' >, AA SAA, for q = 0< 7/A1A L± 1 'V A' >= Ai 8A,A ,±1 for q = +1 .^(2.178)As a result, the diagonal and off-diagonal spin-orbit matrix elements may be written as< 7/A; SE;n1AL= (-1)s-E• :SW;S^1—E 0SSE; J52 >VS(S + 1)(2S + 1) AAE= A AE, (2.179)<77A;SE;JS2lAi•:§10+1,SE+1;JS2>(-1)s- + 1S^1 S8E,E , +1)[= AiV(S(S + 1)(2S + 1)—E^1 E'S^1^S+ (-1)s-1-1^SE,D+1(—E^—1^E'=^iS(S + 1) — E(E^1) (2.180)Chapter 2. The Theoretical Background of Molecular Spectroscopy^54where A and Al are the measurable spin-orbit parameters. Eq. (2.180) gives the matrixelement between two interacting electronic states with the same spin multiplicity butdifferent A values. To describe the interactions between two electronic states of differentspin multiplicities, the more general spin-orbit matrix element of Eq. (2.175) is required.The electron spin-spin Hamiltonian operator of Eq. (2.164) has only diagonal matrixelements, given by< yA; SE; Ai I isi„17/A; SE; JS2 > = 3 A[3E 2 — S(S + 1)].^(2.181)Because of the rotational motion of a molecule, centrifugal corrections are requiredfor the electron spin-orbit and spin-spin interaction terms. The corresponding operators,characterized by the spin-orbit centrifugal correction parameter AD and the spin-spincentrifugal correction parameter AD, are, according to Eq. (2.86)2 AD [LA, /42] =_ 21^- - -^•• 2 J••z2- AD [L,S,, J2 + S —^— z2 — ( i + SA _ + J_S+ )]^(2.182)+ +z \ D [0.-3z9 L.,,2 _ ,62 i I.b2]^1= -2 AD [§! — :§ 2 , :/2 + :9 2 - j1 - ::5?, - (i+A§_ + j_i§+)]0'+ +(2.183)The matrix elements of the operators (2.182) and (2.183) can be obtained by multiplyingthe matrix of the coefficients of the B term by the matrix elements of the spin-orbit andspin-spin operators respectively. Like the rotational operator, the spin-rotation Hamil-tonian may be written in a form appropriate for case (a) coupling asIL = 7(:/ — i — ,§) • :§= 7 PA — LA — :92 + P+:9_ + j_,§+)]= -y [S, — ,§ 2 + 2(j+k- + j-,§4.)] .^(2.184)From this, the diagonal and off-diagonal matrix elements of _Hs, are calculated to be< 71A; SE; Jf/ I if's, I yA; SE; J9 > = 7[E 2 — S(S + 1)],^(2.185)Chapter 2. The Theoretical Background of Molecular Spectroscopy^55< qA; SE; JS1 I ii„ I qA; SE + 1; ./Sr2 + 1 >1 ^= —27 vi[J(J + 1) — S2(S2 + 1)] [S(S + 1) — E(E + 1)]. (2.186)The spherical tensor expression for the magnetic hyperfine Hamiltonian has beengiven as Eq. (2.131). The matrix elements of all three terms in the Hamiltonian can bederived individually using spherical tensor methods. As an example, we derive the matrixelement of the Fermi contact interaction term b i • .§ = b T 1 (I) • Tl(S). According to Eq.(2.63), which gives the matrix element of a scalar product of two commuting first-ranktensors, we have< qA; SE; JS1IF I bT 1 (I) • T i (S) I 77A.; S'E'; J'52' I F >= (-1) J 1+1-+F F I J11 J' Ix < qA; SE; Jf2 11 T 1 (S) II li'A i ; S'E'; J/9' >< IIIIn Eq. (2.187), the reduced matrix element < III TV) II I > equals V/(/ + 1)(2/ + 1),from Eq. (2.56), while the tensor operator T1 (S) needs to be projected from the space-fixed axes to the molecule-fixed axes withT1(S) . Ev:).(07 )Tql(s).^(2.188)qTherefore, Eq. (2.187) becomes< qA; SE; Alf F I bT 1 (I) • T i (S) 1 71A; S'E'; J'S -2' IF >I1 ^= (-1)P^F I J+I+F^V , I(I + 1)(21 + 1)1 J' Ix E < SE IT: (S) I S'E' >< A/ II Vq1)*(a /3-y) II J'S2' > b.q(2.189)After applying the Wigner-Eckart theorem, and Eq. (2.60) to project out the q-dependence,we obtain the desired result:< qA; SE; JOIF 1 bT 1 (I) • T 1 (S)1 77A; SE'; J'll' I F >T 1 (.1)11 I > b.^(2.187)Chapter 2. The Theoretical Background of Molecular Spectroscopy^56IF I J1= b (-1) .P+ I+F1 J' Ix VS(S + 1)(2S + 1)/(/ + 1)(2/ + 1)(2J + 1)(2J' + 1)(^ ' )^( S^S )x E(-1)./--°^J 1 J^(-1)s-E^1^.^(2.190)q^-42 q S2' —E q E'The matrix elements of the other two terms in the magnetic hyperfine Hamiltonian, ai•Land ciA, can be derived by a similar method. The results are< nA; SE; JIT I Fla T i (I) • T 1 (L) I yA; SE; XIII F >= a A (- 1e+/+FF II^J1^J'^IJ 1 J'(x (-1) J-f I V I(I + 1)(2I + 1)(2J + 1)(2J' + 1) ) (2.191)—12 0 9<0; SE; Alf F I cn(I)Tj,(S) I riA; SE; J/C2IF >= c E ( — le +/+FI1F I J1^J'^IJ 1x (-1) j-f 1 VI(I + 1)(21 + 1)(2J + 1)(2J' + 1) ( (2.192)—52 0 .llAfter substitution for the 3-j and 6-j symbols [6, 9], the non-vanishing matrix elementsof the magnetic hyperfine Hamiltonian are found to be< riA; SE; JQI F I --flinag.hf s 17/A; SE; Alf F >h ft R(J) (2.193)= 2J(J + 1)< 77A; SE; JS1 I Fliimag .hf 3 10; SE; J — 1, DI F >^—h 1/J 2 — Sr P(J)Q(J) ^—^ (2.194)2JA/4J 2 — 1Chapter 2. The Theoretical Background of Molecular Spectroscopy^57 < r/A; SE; Ali F I iimag .hfs I 9A; SE + 1; JC2 1I F >b (J R S1)(J f 52 + 1) R(J)V(S)—^4J(J + 1)< yA; SE; A-1_1F I Hmag .hfs I YA; SE + 1; J — 1, ft 1I F >^(J S2)(J S2 + 1) P(J)Q(J)V(S)(2.195)(2.196)where4JV4J 2 — 1R(J)^F(F 1) Aor -I- 1) I(I -*1)^(2.197)P(J) = V(F—I-FJ)(F+J+I+1) (2.198)Q(J)^"(Jr I F)(F^+ I + 1)^(2.199)V(S) = JS(S + 1) — E(1] + 1) (2.200)h^aA^(b+ c)E^ (2.201)The quadrupole Hamiltonian is written as the scalar product of the field gradienttensor, T2 (VE), which operates on the electron orbital functions, i.e. IA >, and thenuclear quadrupole tensor, T 2 (Q), which acts on I/ >, the nuclear spin. Thus the matrixelements of the quadrupole Hamiltonian operator can be written according to Eq. (2.63)as< 9A; SE; AZIF I HQ = e T 2 (VE) • T 2 (Q) I 70'; SE; J'S2' IF >= (-1)./14-1+F F I J2 J' Ix e < / T 2 (Q) II I >< 7111; JSZ II T 2 (VE) II 7/A';^>^(2.202)The reduced matrix element of the quadrupole tensor operator can be derived from Eq.(2.148) by applying the Wigner-Eckart theorem to the left side of the equation; the result-1( / 2 IeQ^E(-1).I-1)—I 0 /^q= (-1) J1+I+F F I J2 J' I 21Chapter 2. The Theoretical Background of Molecular Spectroscopy^58is -1e < / il T 2 (Q) ii / > = 1 e Q ( I 2 I2^—I 0 I(2.203)where Q is the conventional quadrupole moment. After the field gradient tensor T 2 (VE)is projected from the space-fixed to the molecule-fixed axes withT2(VE) . E V: )*(0-y)T72 (VE),^(2.204)qEq. (2.202) becomes< q A; SE; MI F I 1IQ I ii i A'; SE; J'S1' I F >—1{F I li ( 12 I= (-1) Ji+1+F^— eQ2 J' I 2^—I 0 Ix E < JC2 II iSq2)*(0-y) II J'52' >< 77 A IT: (VE) 177' A' > .qSubstituting for the matrix element of DV)* from Eq. (2.60), we obtain< 7/A; SE; JS/IF I HQ I ?I' A'; SE; J'S2' I F >(2.205) ( J 2 J'—It q C2 1x NA2J + 1)(2,/' + 1) < yA I Tq2 (VE) I q'A' > .^(2.206)For linear molecules, the only non-vanishing matrix elements < 7/A I T9(VE) I y'A' > arethe q = 0 and +2 components, and these are related to the field gradient constants, q,according to [7, 24]go = 2 < A I TAVE) I A >q±2 = 2.16 < A171 2 (VE) I A' > .(2.207)(2.208)Chapter 2. The Theoretical Background of Molecular Spectroscopy^59Thus Eq. (2.206) reduces to the desired forms:< 77A; SE; JSZIF I fIQ I riA; SE; fir F >I^ )= (-1)Ji+I+F F I J^I 2 I2 J' I ^0 /'x E(-1)J ^ J 2 J-iy(2J+ 1)(2J' + 1)^eQq0/4^(2.209)—SI q f2'and< 7/A; SE; JSZIF HQ ITO + 2; SE; J'n'IF >^I F I J^1 2 I )2 J' I ^0 I ^J 2^) 1 x E(-1)J-f2 V(2J + 1)(2J' +1)q cit^4V6 eQq±2 .^(2.210)Algebraic expressions for the diagonal and off-diagonal matrix elements of the quadrupoleHamiltonian can be obtained from Eqs. (2.209) and (2.210) after substitution for thespecific 3-j and 6-j symbols. The results are given in Appendix A.2.iv.0 A-doublingElectronic states with A^0 are orbitally degenerate, and should always remain sowithin the Born-Oppenheimer approximation which allows the separation of electronicand nuclear motions. But in reality, molecular rotation interferes with the electronicmotion and causes a breakdown of the Born-Oppenheimer approximation. When theonset of molecular rotation makes it impossible for the electronic orbital angular mo-mentum to remain strictly quantized along the molecular axis, the orbital degeneracyof an electronic state is lifted, giving rise to a phenomenon known as A-doubling. The=Chapter 2. The Theoretical Background of Molecular Spectroscopy^60Hamiltonian operators responsible for the A-doubling effect involve the electronic orbitalangular momentum operator L , and connect the +A and the —A components. To beexact, the A-doubling Hamiltonian 1/ contains the L-uncoupling operator —2BJ • L fromthe rotational Hamiltonian and the spin-orbit Hamiltonian E i a i i i • .S i [4], i.e.V = 14 + C/2 = —2B .7 • i E ai ii • Si. (2.211)The form of the A-doubling can be evaluated by degenerate perturbation theory [26];for H electronic states it is a second-order effect because the orbital operators need tobe applied twice to connect one component, say IA = 1 >, through various distant Estates (A = 0), to the other component IA = —1 >. Direct application of second-orderperturbation theory produces rather complicated terms. It is therefore an advantage touse the concept of effective Hamiltonian, in which the effects of the off-diagonal matrixelements between electronic and vibrational states are folded into a matrix representingthe state of interest only. The second-order effective Hamiltonian may be written as[4, 26]6-(2)^0, f; (Q0) Poileff = to v a (2.212)where 1/ is the A-doubling Hamiltonian given by Eq. (2.211), and the other two symbolsare^Po = E I10k >< 10k I,^(2.213)Q0(a )^E E lk >< lki .(2.214)1010 k^EO — ElIn Eqs. (2.213) and (2.214), / refers to any vibronic state with energy El , and /0 tothe particular vibronic state of interest with energy E 0 , while k labels all the variousrotational, spin and electronic quantum numbers needed to specify the components ofa vibronic state. Substitution of V, from Eq. (2.211), into Eq. (2.212) produces threeChapter 2. The Theoretical Background of Molecular Spectroscopy^61terms for the effective A-doubling Hamiltonian of a H state:HLD = EE E0 —1 E1 lok > (< lok 11-.41 lk >) 2 < lok Il to k +2 E E En 1—^ iok >< iok Ilk ><^>< lok1010 k -E0 1±^lok > (< 10k If/2 I lk >) 2 < lok+ E1010As can be seen from Eq. (2.215) the three terms differ in how many times the matrixelements of the operators —2B:I • L and E i aili • Si appear. The matrix elements of thespin-orbit operator E i a il i • 8i are given by Eq. (2.175), while the matrix elements of theoperator —2BJ • L are similar to those of the operator AL • S in Eq. (2.177), that is< 77A; SE; J52 — 2B :/ • 170'; SE; AY >= —2 E(_1)9(—i)J -c2 J 1 Jq^x ^J(J + 1)(2J + < qA Tq (BL) I y'A' > .^(2.216)When the three terms in Eq. (2.215) are evaluated it is found that the final expressioncontains factors that are the same as those resulting from the operators (, -/_2 + i2 ),+i_k_) and G§.1_ +,§), that is, we may write an effective operator that is assumedto act between the IA = 1 > and A = —1 > components of the H state:HLD = (0 + p + q)(:5_2f. ,§!) — (p 2q)(f+ S+^q(4 +^(2.217)The coefficients o, p and q are given by [27]-11^S S 2^ 1o = ^E EE^s(s+ 1)(2S + 1) 1 1 s^71 'A' .9 ,^Enll — EWEx (-1)s S S 2E < s Ti(si)^><^Ti(s) s >1 1 s ,(2.215)^Chapter 2. The Theoretical Background of Molecular Spectroscopy^62^x < 77, A = 11Til (ai/i)1//, A' = 0 >< 7/ 1 , A' = 0171 1 (ai li)^A = 1 > (2.218)P = 4^E E1 ^1 Vs(s + 1)(2S + 1) n'A' v' Enn — En /EX < 77, A = 1 I BTil (L) I 71' , A' = 0 > E < 77', A' = 01V i (aiii) I 7 , A = 1 ><^T i (si) S >^ (2.219)q = 4 17E E Linn 1^ o^77si < , A = 11BTil (L)177', A' = 0 > 1 2 .^(2.220)— rin/EThe matrix elements of HLD operating within a case (a) II state are< A = Tl;E+ 2;Alli-hp IA = +1;E;JS2 >= -2-1 (o p q)\AS(S + 1) — E(E 1)][S(S + 1) — (E 1)(E + 2)] (2.221)< A=T1;E±1;JfiTlIfiLD IA=±1;E;JS2>1=2(p + 201[S(S + 1) — E(^1)][J(J + 1) — 52(S2 1)]^(2.222)< A = +1;E;Jfi 2 AD 1A = +1;E;JSZ >= 2q vi[J(J + 1) — C2(S2 1)][J(J + 1) — (ft 1)(S2 2)].^(2.223)The phase choice employed here is such that the Wang combinations of case (a) functions1../f2+ > = {1A = 1; SE; JD > 1 A = —1;S, —E; J, (2.224)have parities +(-1) -1-S, respectively. By convention [28], the rotational levels with parity( —1)J-k are called e levels, while levels with parity —(-1)"-k are called f levels wherek = -12- for states with half-integer S and k = 0 otherwise.2.v Line strength and selection rulesA molecular spectrum is a collection of discrete spectral lines produced by absorptionsor emissions of electromagnetic radiation by molecules at certain frequencies. The photonChapter 2. The Theoretical Background of Molecular Spectroscopy^63frequency is given byErn — Env = ^ (2.225)h^'where Ern and En, are the upper and lower energy levels of the molecule connected by aradiative transition. The relative intensities of the spectral lines depend on the relativepopulations of the emitting or absorbing energy levels and the line strengths given bythe square of the matrix elements of the interaction operator. The molecular populationdistribution at equilibrium is described by the Boltzmann distribution, while the spectralline strengths can be calculated as follows. The Hamiltonian for an interaction betweenthe electric field of a photon beam (either emitted or absorbed) and the molecular dipolemoment (for electric dipole radiation) is written1^I = --#•.k= —T 1 (p) • T 1 (E)= —E(-1)P T1,(11)T1p (E),^(2.226)Pwhere p symbolizes the three components of the two first-rank tensors in a space-fixedaxis system. Assuming the two interacting energy levels follow case (a) coupling, thematrix elements of 11 are given by< 77'; fil'l F' M;-, 1 — T i (tt) • T1(E) Ill; Jai F MF >= —E(-1)Pr p (E) < 71'; J'Sr .1 - F' M 1F 17;1(1)111; J52. IF MF >,^(2.227)Pwhere 17/ > represents the electronic and vibrational parts of the wavefunction, namelyv; ASE >. Applying the Wigner-Eckart theorem to remove the MF-dependence fromthe matrix element of Tpl(p), we obtain< 77'; J /C2'/F 1/1PF I — 711 (11 ) • T1(E) I 11; AHFMF >= - E(- 1)P Vp (E) (- 1)J'-M.P^—114 P MFChapter 2. The Theoretical Background of Molecular Spectroscopy^64x < 77'; J' CI' I F' II T i (II) II 71; Ja I - F > .^ (2.228)Upon projection of T 1 (µ) from space-fixed axes to molecule-fixed axes with the rotationoperator DT* (a137), Eq. (2.228) becomes< 71'; f f2' I F' OF I — T i (it) • T 1 (E) 1 11; Jai F MF >(^ )= — E( -1)P vp(E) (-1)P-1V^J' 1 JP^ -114' P MFx E > < r1 ^IF' II 7:, q1)* (07) II ii" ; f Cr IF" >q IIx < li n ; J'' fi li _1 - F li II Ti)ql ( ^II 77; JC1IF >,^(2.229)where the unit operator E" kr; J" fr IF" >< 71"; J„ S2 „ IF”' has been inserted betweenVq1 )*(07) and T l (A). The rotation operator DT* (a,37) operates on only the IJQ >part of the wave function, thus it is diagonal in y, while T 1 (it), being the dipole momentoperator, acts on the ly > part only. As a result, the summations in Eq. (2.229) collapsetoE < A' IF' II Vq1)* (07) H A2I F >< 71 I Tql(A ) I Ti > .qSince the rotation operator acts on only part of a coupled basis, its matrix element isgiven by Eq. (2.64), i.e.< 71'; J'f2 ' - 1 -tii II Vq1) * (ci,(3-y) II y; Alf F > =(-1)J1+1+F+1 /(2F + 1)(2F' + 1)x{J' F' I 1< J'9 ' II Vq )*(a137) II Jf2 >F J 1Therefore Eq. (2.229) reduces to(2.230) < n'; JVIF'APF 1 — T1(11) • T1(E) 1 Ti; Ka' FMF >2( J' 1 J )= 1,E11/1 ' P MFF , , -114 p MF(2.233)Chapter 2. The Theoretical Background of Molecular Spectroscopy^65^, „„ ^J'^1 J )^= —E(-1)PT1p (E)(-1)'''- `v`F (-1).11+1+F+1\^A2F + 1)(2F + 1)P^—11/4 P MFx IF F' I E < J'n' II vp . (a,37) II Js/ >< 71'1 T:(it)17/ > •F J 1} q(2.231)Substitution for the reduced matrix element of DT*(a0-y) with Eq. (2.60) gives< 71 ' ; J'frIF'M'F 1 — T i (11 ) • 711 (E) I 11; Ali F MF >(^ )= —E(-1)PT_l_p (E)(-1) .P-mJ'^1 JP^ -A4." P MF^ {J' F' Ix (-1) Ji+1+F+ 1 V(2F + 1)(2F' -1- 1)F J 1^ ( J' 1 J^x E(-1)JI-°Y(2J + 1)(2J' + 1)^< li'IT:(A)17/ > . (2.232)q^ —ft' q ftIn the absence of an external field, the M-dependence is not resolved, and we need tosum over all the MF components to obtain the total line strength. Since the intensity isproportional to < —/./, • E > 2 [20] and the summation over the square of a 3-j sumbol isthe line strength for an electric dipole transition is given byI cx 1T 1 (E)1 2 (2J + 1)(2J' + 1)(2F + 1)(2F' + 1)x [1 J' F' I ^J' 1 J ) < y'1Tql (A)177 >1F J 1 }q(--Si'qfi2(2.234)where 1T 1 (E)1 2 is related to the strength of the light source, and < 77/17: 71 (p)177 > is calledthe transition moment, Rq . To describe the relative intensity in a real spectrum, onlyChapter 2. The Theoretical Background of Molecular Spectroscopy^66a Boltzmann factor needs to be added to the above expression. From Eq. (2.234), onecan calculate the line strength for a particular transition, or determine if a transitionis allowed or forbidden, from the symmetry properties of the 3-j and 6-j symbols. Fora transition to occur, the line strength must be non-zero, and the conditions for whichexpression (2.234) is non-zero are called the selection rules for electric dipole transitions.From the property of a 3-j symbol, we obtain the rotational selection rules as,AJ = IJ'—JI = 0, +1 (2.235)Al/ = ISY-1-21 = 0, +1. (2.236)The hyperfine selection rules can be derived from the the symmetry of the 6-j symbol inEq. (2.234); they areOF = IF'—F1 = 0, +1. (2.237)For transitions from the vibrational levels of an electronic state to the vibrational levelsof another electronic state, the wave functions ly > can be factorized into electronic partsand vibrational parts, i.e. ly >= le > Iv >. Since the dipole moment operator Tql(p)acts on only the electronic function, the transition moment Rq may be re-expressed asRq =< 71' IT: (A) I y > = < e' I Tel, q (A) I e >< v'lv >, (2.238)where < elTelq(it)Ie > is the electronic transition moment, and < v'lv > is called theFranck-Condon overlap integral. In principle, any spectral bands with non-vanishingFranck-Condon factors can be observed in an electronic transition; therefore there areno strict vibrational selection rules. However the number of vibrational bands observedand the intensity distribution depend on how much the geometry of the molecule changesin the electronic transition. For diatomic molecules, if the electronic transition causesonly a small change in the bond lengths, only a few bands are expected to be presentin either emission or absorption from a particular vibrational state; the reason is thatChapter 2. The Theoretical Background of Molecular Spectroscopy^67the vibrational functions in the two states are nearly identical, so that the integral <viv > decreases very rapidly with increasing A v, and produces only a short vibrationalprogression with the Av = 0 sequence bands being the strongest. A moderate bondlength change will give rise to a longer vibrational progression in which the strongestbands are usually those with Av 0. For the electronic transition moment Re to benon-zero, additional electronic selection rules are required. For case (a) electronic states,these selection rules areA S = 0 ( 2 .239 )DA = 0, ±1 (2.240)A E = 0, ±1. (2.241)There are other important selection rules with regard to the symmetry properties ofthe two combining energy levels. They can be derived from the symmetry property of thetransition moment Rmn = < y m I E g li:(µ)1On > for transition from state m to state n.For the transition to occur, Rm, must be totally symmetric. A rotational level is calledpositive or negative if the total wavefunction 0 remains unchanged or changes its signupon an inversion at the origin of the molecule. Since the dipole moment (p = E i eiri)changes sign for such an inversion, a positive rotational level can only combine with anegative level, and vice versa. This selection rule may be written symbolically:<^ +, >^ (2.242)Chapter 3Laser Induced Fluorescence Spectroscopy3.i IntroductionThe introduction of the laser as a light source into the field of spectroscopy in the1960's had profound and revolutionary effects. The laser has now become a powerfultool for spectroscopic studies in every region from the infrared to the vacuum ultraviolet[29, 30, 31]. Laser spectroscopy has achieved many orders of magnitude improvementsin resolution, accuracy and sensitivity over the traditional methods of spectroscopy.There is a variety of laser spectroscopies currently in use for the study of molecular oratomic structures, of which laser-induced fluorescence is the most widely used techniquefor studies in the visible region. "Laser induced fluorescence" here means a process wherea molecule absorbs laser radiation at one frequency and then emits some portion of theenergy as light at the same or different frequencies. It is a much more sensitive techniquethan laser absorption spectroscopy [32]. The monochromaticity and high power densityof laser light makes it an ideal light source for high resolution spectroscopy. Various non-linear spectroscopic techniques based on laser light have been developed to eliminate theDoppler-broadening of spectral lines so that the highest resolution can be acheived inoptical spectroscopy. Lamb-dip spectroscopy [33, 34] was the first sub-Doppler techniqueto be developed; it is based on the principle that the nonlinear interactions of two counter-propagating laser beams with a group of thermally populated gaseous molecules result ina dip (Lamb dip) at the center of the absorption or fluorescence Doppler profile. If just the68Chapter 3. Laser Induced Fluorescence Spectroscopy^ 69Lamb dip signals can be detected, the resulting spectra are free of Doppler-broadening.The introduction of the very sensitive intermodulated fluorescence (IMF) technique [35]in the 1970's greatly improved the applicability of non-linear sub-Doppler spectroscopyand now permits spectroscopists to study the hyperfine structures of molecules on aregular basis.In this chapter, we will consider the three laser spectroscopic techniques employed inthe present work; they are (i), intermodulated fluorescence, (ii), resolved fluorescence,and (iii), intracavity fluorescence. Before that, we shall discuss line broadening effects ingaseous spectroscopy and the laser saturation methods used for elimination of Doppler-broadening.3.ii Spectral linewidth and broadening effectsAn absorption or emission spectral line usually has a measurable width, which isdefined in terms of the full width at half maximum of the line (FWHM), which is theinterval between the frequencies at which the intensity of the line falls to half of itsmaximum value at the center frequency. Obviously the smaller the linewidth (AO is,the higher the spectral resolution, defined as v/Av, is. We now consider the variousreasons for the finite linewidth.1). Natural linewidthAn excited molecule can emit its excitation energy as spontaneous radiation, whichimplies that the excited state must have a finite lifetime. According to Heisenberg'suncertainty principle, a state with a lifetime 7- has a spread of energy given byAE > —h .T(3.1)The energy uncertainty of the excited state consequently gives a breadth to the spectrallines, which increases with decreasing lifetime. For a transition between state n withChapter 3. Laser Induced Fluorescence Spectroscopy^ 70lifetime -r,,, and state in with lifetime Tm , the natural linewidth may be taken as the sumof the energy spreads in the two states; expressed in frequency units, this is11^1 ,Av = --„^—).z Tn TynIf state n is the ground state, then Tr, = oo, and the natural linewidth measures thenatural lifetime Tm of the excited state. Since the natural lifetime is the reciprocal of theEinstein spontaneous emission coefficient A m,„ [20], which is proportional to the thirdpower of the transition frequency, (v r,3,,) [20], the natural linewidth increases rapidly withincreasing frequency. Even so, the lifetime broadening is generally much smaller thanother broadening effects. In the visible region, the lifetimes of most excited states are onthe order of 10-6 — 10 -7 seconds, which gives a natural linewidth of about 1 MHz.2). Doppler broadeningDoppler broadening is caused by the thermal motions of the absorbing or emittingmolecules. Consider a molecule with a velocity v relative to the rest frame of an observer;the proper absorption or emission frequency vo is shifted by the Doppler effect to (vok • v) [36] where k is the wave vector of the radiation. This means that the apparenttransition frequency v is increased if the molecule moves parallel to the light propagation(k • v > 0) and v is decreased when the molecule moves against the wave propagation(k • v < 0). If the z direction is chosen to coincide with the wave vector and lki = 2'r/a,the Doppler-shifted frequency becomesv zv = vo(1 —c) (3.3)At thermal equilibrium, the z-components of the molecular velocities obey the Maxwellpopulation distribution: ni(vz) = ^ e—(v z /v p ) 2Voirr(3.4)(3.2)Chapter 3. Laser Induced Fluorescence Spectroscopy^ 71where n 2 (vz ) is the number of molecules per unit volume in level E i with velocity com-ponent vz , Ni is the density of all molecules in level E, and vp = ^2kT/m is the mostprobable velocity with T and in being the temperature and mass of the molecules. Sub-stituting for vz in Eq. (3.4) with an expression obtained from Eq. (3.3), we get thenumber of molecules whose frequency has been shifted from vo to v:n i (v) = ^ e—[c(v — vo )/(vp v0 )} 2^vpvir(3.5)Since the intensity of the emitted or absorbed radiation, I(u), is proportional to thedensity n i (v), we have^ni( )^I(v)—^ (3.6)ni(vo )^/(vo) •Combining Eqs. (3.5) and (3.6) we obtain the intensity profile of a Doppler-broadenedspectral line:I(u) = I(v0) e —[c(v — vo)/(vpv0)1 2 .^(3.7)This is a Gaussian profile whose linewidth (FWHM) can be calculated to be110Ay =8kTln2m •(3.8)With c = 2.9979 x 108 m s" and k = 1.38 x 10 -23 JK - 1 , we obtain the Doppler widthin frequency units (s -1 ):Ov = 7.16 x 10 - 7 vo T^(3.9)InThe Doppler width therefore increases linearly with frequency vo , and is proportional toVT/m. In the visible region, The Doppler width is about 0.8 GHz for molecules with amass of ti 60 g mol' at temperature 300 K.3). Pressure broadeningPressure broadening is also known as collisional broadening. Qualitatively thiseffect can be considered from two points of view. First the intermolecular collisionsChapter 3. Laser Induced Fluorescence Spectroscopy^ 72shorten the lifetime of an excited state, thereby enlarging the energy uncertainty of theexcited state according to Heisenberg's principle and consequently broadening the spec-tral lines. Alternatively, the intermolecular forces and collisions perturb the stationarystate energies of the isolated molecule, and result in line broadening. Obviously thepressure-broadening linewidth depends on the collisional frequency, which in turn de-pends on the molecular density and the velocities of the molecules. Quantitatively, theprofile of a pressure-broadened line can be shown to be [36](Nlio-b 2I (v) = I (v0) (v — vo — N Yo + (N iTo-b) 2 '^(3.10)where N is the number of particles per unit volume, 15 is the mean relative velocity, ando and ab are the collision cross sections defined by the phase shift of the excited moleculedue to elastic collisions [36]. Eq. (3.10) corresponds to a Lorentzian profile with linewith(FWHM) given byAy = 2Nr)o-b,^ (3.11)and a shift of the line center,V I — vo = N 'I) a s .^ (3.12)For molecular systems near room temperature, the collision broadening is typically 10MHz/torr, which is considerably smaller than the Doppler width in the visible region.4). Saturation broadeningA sufficiently strong light source such as a laser can significantly change the pop-ulation densities of the molecular levels as a result of induced absorption or emission.Saturation of the population densities also causes additional line broadening.If a spectral line has a homogeneous linewidth, Ay, (meaning that all molecules con-tribute equally to the linewidth, through natural or pressure broadening), the saturation-broadened linewidth is given by(3.13)Chapter 3. Laser Induced Fluorescence Spectroscopy^ 73where So is the saturation parameter at the line center, defined as [36]B12P( 110) SO = R (3.14)In this equation, B12 is Einstein induced absorption or emission coefficient, p(v o ) is thepower density of the radiation and R is the mean value of the relaxation rate of levels 1and 2.For an inhomogeneous line profile (caused by Doppler-broadening ), the saturationeffect is different for molecules with different velocity components along the direction ofthe radiation. For a particular group of molecules with the same velocity component, andtherefore the same transition frequency, the saturation broadening manifests itself in ananalogous way to the homogeneous case. Thus the important factor in determining themagnitude of saturation broadening is the saturation parameter S o , given by Eq. (3.14).Since the dye laser used in the present work produces a relatively low power density, theadditional broadening caused by saturation effects is much smaller than the pressure andDoppler broadenings, and can be neglected.3.iii Principles of saturation spectroscopySaturation spectroscopy is based on selective saturation of a Doppler-broadened tran-sition by optical pumping with a monochromatic tunable laser. As mentioned in the lastsection, a coherent light wave of frequency v interacts only with a sub-group of particleswhose velocity components along the propagation direction satisfy the Doppler conditionof Eq. (3.3). This selective excitation of particles with a certain velocity componentchanges the equilibrium distributions of partial velocities in each level of the transition.There are too few particles with a particular velocity in the lower level, but too manywith that same velocity in the upper level. Schematically, this can be shown as a hole anda peak in the Maxwell-Boltzmann velocity distributions for the lower and upper levelsChapter 3. Laser Induced Fluorescence Spectroscopy^ 74respectively (see Fig. 3.1). This saturation effect is known as "Bennett hole burning"[37]. The depth of the Bennett hole depends on the degree of saturation by the lightfield, while the width corresponds to the homogeneous linewidth which can be severalorders of magnitude smaller than the Doppler width in the absence of other broadeningeffects.In order to detect the Bennett hole, which has been burned into the velocity distribu-tion by the so-called "pump" beam, a second light beam, called the "probe", is neededin order to measure the population depletion. This probe beam may either come fromthe same laser that provides the pump beam, or it may come from another laser. In asimple arrangement, the pump beam is reflected by a mirror back to the sample, such aswhen a sample is placed inside the resonator of a laser. If the pump beam interacts witha group of molecules with velocity v z , the reflected wave will interact with another groupof molecules with velocity —v, because of its opposite wave vector, —k. Therefore, twoholes are burned symmetrically into the profile of the velocity population distribution atv., = ±(vo — v)/k as shown in Fig. 3.2(a). In this case, the total fluorescence intensityis the sum of the contributions from both beams. As the laser frequency v is tunedtowards the true transition frequency vo , 1), decreases, and the two holes move towardseach other; meanwhile, because of the Gaussian distribution of the molecular velocities,the total absorption of both the pump and probe beams increases, which results in anincrease in the total fluorescence intensity. At the line center, where v = v o , the two holesoverlap and the two counterpropagating waves interact with the same molecules (v, = 0).Since both beams are competing for the same molecules, the total power absorbed, andtherefore the fluorescence intensity, decreases at the exact line center. Therefore, a smalldip, called a Lamb dip [34] is observed at the center of the Doppler profile of a spectralline when either the absorption or the fluorescence are monitored (see Fig 3.2(b)).Chapter 3. Laser Induced Fluorescence Spectroscopy^ 75I nk (vz)^EKh vEi^sowvzni (vz )0 vzFigure 3.1: "Hole burning" in the lower level velocity population distribution of anabsorbing transition and generation of a corresponding population peak in the upperlevel [36]./-11.11^I1^11^111111(b)Chapter 3. Laser Induced Fluorescence Spectroscopy^ 760vFigure 3.2: (a) Two Bennett holes burned symmetrically into the velocity populationdistribution (n(vz )) by two counterpropagating laser beams of frequency v v o . (b)Lamb dip formed at the center (v = v o ) of the profile of intensity versus laser tuningfrequency [361.Chapter 3. Laser Induced Fluorescence Spectroscopy^ 773.iv Doppler-free saturation fluorescence spectroscopySuitable modulation permits the Doppler profile to be suppressed, and the Lambdips to be observed directly. Because the width of a Lamb dip signal can be severalorders of magnitude smaller than the Doppler width, a spectrum containing only theLamb dip signals gives a huge increase in resolution, allowing the measurement of closelyspaced features such as hyperfine splittings, which are usually totally hidden in Doppler-limited spectroscopy. Lamb dips can be detected in both absorption and fluorescenceexperiments. In cases where the saturation is small, the change in absorption is difficultto detect and the small Lamb dips may be buried in the noise of the Doppler background.In these situations, fluorescence spectroscopy provides a much more sensitive method forthe detection of the Lamb dip signals. In what follows, we will discuss two Doppler-freefluorescence techniques based on the Lamb dip phenomenon.3.iv.A Intermodulated fluorescenceSorem and Schawlow [35] developed a sensitive modulation method for picking out theLamb dips in fluorescence experiments. Consider the experimental arrangement shownin Fig. 3.3, where two counterpropagating laser beams derived from the same laser aremechanically chopped at different frequencies, fi and f2 , and the fluorescence signal isdetected by phase sensitive detection at reference frequency (fi f2).The basic concepts behind this Doppler-free technique are as follows. Assume thatthe intensities of the two laser beams areIi = 2—1/0 (1 + cos 2i fi t),1/2 = —2/0 (1 + cos 271 - f2i),(3.15)(3.16)Chapter 3. Laser Induced Fluorescence Spectroscopy^ 78Figure 3.3: Schematic drawing of the intermodulated fluorescence experiment used inthis laboratory.Chapter 3. Laser Induced Fluorescence Spectroscopy^ 79where /0 is the intensity of each beam before modulation. The intensity of the laser-induced fluorescence isIFL = C ns(h + 12),^ (3.17)where C is a proportionality constant and n, is the saturated population density of theabsorbing state, given by [36]n s = no(1 — So)= no [1 — a(h + /2)]•^ (3.18)In this equation, n o is the initial population density of the absorbing state and S0 is thesaturation parameter which is proportional to the total laser intensity (I i + /2 ). InsertingEq. (3.18) into Eq. (3.17), we obtain/FL = C [no(Ii + h) — a no (/1 + h) 2 i•^(3.19)Eq. (3.19) shows that the fluorescence intensity contains linear terms, representing signalsmodulated at L and f2 , and quadratic terms representing signals with modulations(1]. + 12) and (II — 12 ). The linear terms give the Doppler profile of the normal laser-induced fluorescence, while the quadratic terms describe the saturation effects resultingfrom simultaneous interaction of the two laser beams with the same molecules, in otherwords the Lamb dip signals. As a result, phase sensitive detection of the fluorescence atthe reference frequency of (fl + 12) produces only the saturation signals, with most ofthe Doppler background suppressed.The sub-Doppler technique of intermodulated fluorescence is nowadays widely usedin spectroscopy to study hyperfine structure. The attainable resolution is limited only bypressure broadening and the natural linewidth. At less than one torr pressure, the spec-tral linewidth obtained with IMF is usually much less than 100 MHz, which is sufficientChapter 3. Laser Induced Fluorescence Spectroscopy^ 80to resolve the hyperfine structure in most small molecules. When two hyperfine transi-tions sharing a common level lie within the same Doppler profile, a saturation signal mayappear exactly midway between the two hyperfine lines. This so-called "crossover reso-nance" signal is artificial and occurs when molecules are Doppler-shifted into resonancefor the first transition by the pump beam and for the second transition by the counter-propagating probe beam. A schematic diagram showing the generation of crossover reso-nances is given in Fig. 3.4. The presence of crossover signals is sometimes advantageoussince they can be used to assign transitions with a common level and thereby separatethe upper and lower state splittings.3.iv.B Intracavity fluorescenceThere are some disadvantages when applying intermodulated fluorescence to high-resolution spectroscopic studies. First, in order to achieve saturation, the laser beamsmust be focused to small diameters so that sufficient power density can be obtained.As a result, the saturation volume for the interaction between the laser beams and themolecules may be too small to allow observable saturation signals for weak electronictransitions. Next the two counterpropagating laser beams must be slightly misalignedfrom each other, or an optical isolator inserted, to avoid feedback to the laser. Thismisalignment of the two beams, or the reduction in the available power from the presenceof the optical isolator, can greatly reduce the saturation effect. These shortcomings ofIMF are reduced when the sample molecules are inside the cavity of a standing-wave laser,in the method of intracavity fluorescence spectroscopy. The experimental arrangementfor this technique is shown in Fig. 3.5 schematically. The total laser power availableinside the laser cavity can be up to two orders of magnitude higher than that obtainedoutside the cavity from the same laser. Therefore, even though the diameter of the laserbeam inside the resonator is about ten times bigger than that of a focused beam inA l F1F2 F2b)AF=^XAJ -16F= AJ =0X AF=L1J+1I^Pfigure 3.4: a) The formation of crossover resonances (F1 + A2 and F2 + A1) as the result of allowed AFtransitions (A 1 and A 2 ) occurring within the same Doppler-broadened velocity profile as forbidden AF^A,/transitions (F1 and F2 ). The diagram shows the laser scanning toward the non-Doppler-shifted AFtransition (occurring at A l + A 2 ) and beyond toward higher frequency to the AF AJ + 1 transition (F1 + F2 ).If the F's and A's are exchanged, the first central Lamb dip is the AF = OJ — 1 transition. b) Stick diagram ofthe spectrum of the four forbidden transitions that can accompany a AF AJ = 0 Q transition (X denotes acrossover). With an R line, the AF 0 and AF = —1 transitions and the associated crossovers occur to the redof the AF AJ + 1 transition, while with a P line the forbidden transitions lie to the blue of the OF = OJ — 1transition.00tuningdevicefluorescencedetectorM I^Laser^absorption cell^Etalon M 2Figure 3.5: Schematic drawing of an intracavity fluorescence experiment.Chapter 3. Laser Induced Fluorescence Spectroscopy^ 83the IMF arrangement, the power density of an intracavity laser beam is still sufficientto achieve saturation for most molecules. Because of the large volume in which thelaser beam interacts with the molecules, intracavity saturation spectroscopy is a muchmore sensitive technique than intermodulated fluorescence; thus it is an ideal method forstudying the hyperfine structure of weak electronic transitions.Since amplitude modulation of a tunable laser disrupts the frequency scanning cir-cuitry, phase sensitive detection of the saturation signals is achieved by modulating thefrequency of the laser. Frequency modulation is realized by superimposing a sine wavevoltage at low frequency on the ramp voltage that drives the frequency scanning mech-anism. Lock-in detection at the frequency of the sine wave modulation produces signalsthat are derivatives of Lorentzian line profiles. For the purpose of illustration, Fig. 3.6shows the excitation spectrum of the rR 3 (16) line of a band in the electronic spectrumof gaseous H 2 CS, taken with a fluorescence-based intracavity laser spectrometer. Fig.3.6(a) was obtained by recording the total fluorescence intensity against the laser fre-quency; two Lamb dips are clearly seen in the Doppler profile of the rotational line. Thecorresponding sub-Doppler spectrum of Fig. 3.6(b) is taken as the second derivative ofFig. 3.6(a); this is done by phase sensitive detection of the fluorescence signal at twice themodulation frequency. In this particular spectrum, the linewidth is limited by pressurebroadening to about 15 MHz.3.v Resolved fluorescence spectroscopyThe fluorescence-based saturation spectroscopy discussed so far is essentially absorp-tion spectroscopy, since the spectra recorded are plots of the total fluorescence intensity(which is proportional to the absorbed power) against the laser frequency. Anotherr R 3 (1 6) (a) Doppler SpectrumI (b) Sub—Doppler Spectrum —.....—.—...w.wwwwie"wwwwwww"40.01**010441kI^ I0.0^0.5 1.0^1.5^2.0FREQUENCY (GHz)Figure 3.6: The Doppler-limited and sub-Doppler spectra of the rR 3 (16) line of H2CS,recorded by a fluorescence-based intracavity laser spectrometer.Chapter 3. Laser Induced Fluorescence Spectroscopy^ 85useful technique is wavelength-resolved fluorescence spectroscopy, in which the total flu-orescence signal is resolved into its component wavelengths. The spectra obtained thisway represent all the allowed transitions from the upper level excited by the laser to thevarious lower levels. These data are of great value in assigning the lines of a complicatedspectrum.The experimental arrangement for wavelength-resolved fluorescence is relatively sim-ple. Fluorescence, induced by a pump laser tuned to a single line of an electronic transi-tion, is directed to a monochromator at right angles to the direction of the laser beam.There are two possible arrangements. In one, a photomultiplier tube at the exit slit ofthe monochromator records the intensity as a function of wavelength when the grating isrotated. In the other the exit slit is replaced by an array detector, which records all thewavelengths simultaneously with the electronic equivalent of a photographic plate. Laser-induced fluorescence spectra are quite weak, so it is necessary to use a microchannel-plateimage intensifier in front of the array detector. In this case, the fluorescence spectrum isa plot of signals recorded on the individual elements against the position in the spectralwindow. Calibration of the spectral window of the monochromator is done from knownatomic emission lines in the spectrum of a hollow cathode lamp.Applications of wavelength-resolved fluorescence spectroscopy range from the assign-ments of molecular spectra, measurements of molecular constants, transition probabilitiesand Franck-Condon factors to studies of the dynamics of chemical reactions. Those ap-plications which are particularly relevant to the present work are discussed below.i). Line assignmentsSpectra of heavy molecules with high spin multiplicity are usually complicated byoverlapping of different spin-vibrational subbands, and perturbations. The complexity ofthe spectra is sometimes so great that no recognizable branch structures can be observedin the spectra. As a result, rotational analysis becomes almost impossible without aChapter 3. Laser Induced Fluorescence Spectroscopy^ 86method of determining the quantum numbers of the individual rotational lines and theirrelations to the other spectral lines. In this situation, resolved fluorescence provides ameans for assigning the individual rotational lines.When a molecule that has been selectively excited to an upper state rotational leveldecays spontaneously, the resulting fluorescence spectrum consists of a small number ofrotational lines for each vibrational band. One of the lines always lies at the same fre-quency as the laser. If only two lines appear and the other line is at a shorter wavelength,the excited line belongs to a P branch; if the other line appears at longer wavelength, theexcited line belongs to an R branch. A Q line may appear, in between the R and P lines,depending on the selection rules implied by the type of transition. If the excited level hasthe rotational quantum number J, the observed R and P lines will be the R(J — 1) andP(J + 1) rotational lines, and their separation, labelled 0 2F"(J), is given for a linearmolecule as [20]02F"(J) ' vR(J — 1) — vp(j + 1 ) P.--_, (4/3: — 6D:',) (J + 2) , (3.20)where B: and D;', are the effective rotational and centrifugal distortion constants ofthe lower state respectively. Measurements of some successive separations allow theassignments of the J quantum numbers to the rotational lines and simultaneously give arough B value for the lower state.ii). Measurements of vibrational frequency and observation of new electronic statesWhen a laser excites a single upper state rotational level, fluorescence may occurto the original vibronic state, or to other vibrational levels of the same electronic state,or to vibrational levels of other electronic states depending on the selection rules forelectronic transitions. The emission to other vibrational levels of the same electronicstate allows assignments of the vibrational quantum numbers for the lower electronicstate and measurements of the vibrational intervals. The vibrational intervals are givenChapter 3. Laser Induced Fluorescence Spectroscopy^ 87in terms of the vibrational constants by [20]AG(v + 1 = G(v + 1) — G(v)= We — 2we x e (v + 1),^ (3.21)where G(v) is the term value of vibrational level v, w e is the vibrational frequency andwe x e is the anharmonicity constant.Transitions to other electronic states may also occur, and can be used to give infor-mation on new electronic transitions. Accurate measurements of the fluorescence spectramay then give the positions of the new electronic states and their approximate rota-tional constants. In addition, measurement of the relative intensities of the R and Plines allows accurate determinations of the 52 values of the two electronic states involved.Resolved fluorescence spectra often show a very good signal to noise ratio. Therefore,weak electronic transitions or so-called "forbidden" transitions which are not obvious inthe excitation spectra may be observable in resolved fluorescence spectra. This can bevery valuable in determinations of the spin-orbit splittings of case (a) electronic states be-cause, for transitions between two case (a) electronic states, the "spin-forbidden" AE 0transitions may be too weak to be seen in the absorption spectra, but may become ob-servable with the more sensitive resolved fluorescence, so that direct measurement of thespin-orbit splittings becomes possible.Chapter 4Laser Spectroscopy of NbNPart IUnusual Electron Spin and Hyperfine Effects in the Electronic Statesof the (4d6) 1 (5sa- ) 1 and (4d8) 1 (4d70 1 Configurations4.i IntroductionThe electronic spectroscopy of Nb-containing molecules in the gas phase is of greatimportance in quantum chemistry. Niobium 41Nb has the largest magnetic momentof any known non-radioactive nucleus (6.1435 nuclear magnetons) and, with a nuclearspin 1, it produces some extraordinarily large hyperfine effects in the optical spectra ofgaseous free radicals such as NbN and NbO. The large hyperfine splittings allow accuratemeasurements of the hyperfine parameters which carry valuable information about theelectronic structure and bonding in the molecule.Dunn and Rao [38] first reported large nuclear hyperfine splittings in the spectrumof gaseous NbN in 1968; with the moderate resolution of a grating spectrograph, theyobserved extremely broad low-J rotational lines in the outer two subbands of a 3 4) -30 transition in the red region of the emission spectrum. The broad linewidths wereattributed to large Nb hyperfine splittings in the 3 ,6, lower state, which was assignedto the electron configuration (4d8) 1 (5sa) 1 . In 1975, Femenias et al [39] calculated the88Chapter 4. Laser Spectroscopy of NbN^ 89hyperfine parameters a and (b + c) for the 3 (1) and 36, states from partially resolvedhyperfine structure in the 3 0 - 3 6. system (now labelled B — X) which they photographedat higher resolution; their results indicated that the B 3 (I) excited state makes a non-negligible contribution to the observed hyperfine structure. In 1988, the B30 — X30system was reinvestigated by the Spectroscopy group at the University of B.C.; laserexcitation spectra recorded at sub-Doppler resolution showed almost completely resolvedhyperfine structure, from which extremely precise rotational and hyperfine parameterswere derived for the upper and lower states [40]. It was found that the middle spincomponents of the X 30 and B3 (I) states are perturbed, respectively, by the OA and POstates from the same (4d8) 1 (5scr) 1 and (4d8) 1 (4dir) 1 electron configurations, by second-order spin-orbit/Fermi contact interaction. This second-order effect is particularly severein the ground state and causes extensive distortion to the hyperfine structure whichcould not be handled by the standard case (a o ) formalism for a triplet state. As aresult, five effective hyperfine parameters, instead of the normal three, were required tofit the distorted hyperfine energy levels of the X30 state, while four were needed for theB343. state. Meanwhile, a recent rotational and hyperfine analysis of the C 3 II — X36,system carried out by the same group has revealed further dramatic second-order spin-orbit/Fermi contact interactions [41, 42]; in this case, the 3 111 spin-component was foundto have been pushed down from its expected position by 650 cm -1 [41], apparentlyby the e 1 II state from the same electron configuration (4c/(5) 1 (4c/r) 1 . Because of thisextraordinarily large second-order spin-orbit effect and the lack (at that time) of datafor the elII perturbing state, the hyperfine structures of the three spin components ofthe C3II state were treated separately to derive effective hyperfine parameters for theindividual substates. Although the spin-orbit/Fermi contact interaction between the ellland C 3 II states could be studied indirectly, given the distorted molecular parameters ofthe C state, by second-order perturbation methods as in Ref. [42], a full investigation ofChapter 4. Laser Spectroscopy of NbN^ 90this interaction has required data from the 1 H state.In Part I of this chapter, we report an analysis of the second-order spin-orbit/Fermicontact interactions between the 1 11 and 3 H states of NbN using the sub-Doppler data ofthe C311 — X30 (0, 0) band from Ref. [42] and the newly accquired high-resolution dataof the OH — X 302 (0, 0) band at 18457 cm -1 . The 1 H — 30 2 intermultiplicity transitionhas been observed in laser excitation spectra and assigned using lower-state combinationdifferences together with wavelength-resolved fluorescence studies, as will be reported inPart II. The present analysis shows that the energy levels of the 1 11/ 311 complex can bewell represented by a full Hamiltonian matrix constructed according to case (ag) coupling;the least squares results give deperturbed rotational and hyperfine constants for the 1 11and 3H states as well as the interaction parameters, which can be satisfactorily describedin the single configuration approximation. Also in this Part, we report a rotational andhyperfine analysis of the newly observed (4c16) 1 (4c/7r) 1 PO — (4c/(5) 1 (53o- ) 1 a 1/ transition;the effective hyperfine parameters of the upper and lower states are found to have beenmodified by the spin-orbit/Fermi contact cross-terms between the singlet and tripletstates from the same electron configurations, confirming the perturbation mechanismproposed in Ref. [40].4.ii Experimental detailsGaseous niobium nitride (Nb 14N) was prepared in a flow system by reacting NbC1 5vapour with a trace of active nitrogen in a 2450 MHz microwave discharge operatingat 100 W power. The NbC15 vapour was produced by sublimation of NbC1 5 powder,contained in the side-arm of a quartz tube, at about 80 °C; it was then entrained ina mixture of N2 and Ar carrier gas and passed through the microwave cavity. Themicrowave discharge produced a 10-cm long lavender-colored flame, which was pumpedChapter 4. Laser Spectroscopy of NbN^ 91across an observation chamber fitted with Brewster-angle windows and light baffles. Thetotal gas pressure in the chamber was maintained at between 300 and 1000 millitorrs. Itwas found that, within this pressure range, high pressure favours the production of NbNmolecules in the ground state, while low pressure enhances the production of moleculesin the low-lying excited states.Laser excitation spectra of NbN were recorded using a Coherent Inc. model 599-21 standing wave tunable dye laser pumped by a 3-watt argon ion laser (Coherent Inc.Innova 90-6). The new ell — X 3A2 (0, 0) band near 18457 cm -1 and fic1) — a 1 A (0,0)band near 17115 cm -1 were recorded at both Doppler-limited and sub-Doppler resolutionswith the dyes rhodamine 560 and 590 respectively. The technique of intermodulatedfluorescence (IMF) (see Section 3.iv.A) was used to eliminate the Doppler-profiles andgave spectral lines of about 100 MHz linewidth. Recording the sub-Doppler spectra ofthe weak 1 H — 30 2 (0, 0) transition turned out to be very difficult since it lies in theextreme blue end of the range of the laser dye rhodamine 560, where the output power isvery low indeed. Eventually it was found that a small amount of concentrated ammonia,added to the dye solution, shifted the gain curve of the dye about 150 cm -1 to the blue,permitting a laser output power of about 15 mW for this band; this was only adequatefor IMF studies of the strongest rotational lines of this weak intermultiplicity band.The NbN spectra were recorded as 35 GHz continuous scans of laser frequency underthe control of a MicroVAX computer; the controlling programme also acquired the iodineexcitation spectrum and the fringe patterns (interpolation markers) from a 299 MHz freespectral range Fabry-Perot interferometer to provide calibration. Successive scans over a10 cm -1 region were concatenated and the frequencies of the 12 lines in the region, takenfrom Gerstenkorn and Luc's iodine atlas [43], were fitted to a linear expression in themarker positions to derive the frequency of the first marker and the exact free spectralrange of the etalon; this was then used to convert the marker positions of the NbN lines toChapter 4. Laser Spectroscopy of NbN^ 92absolute frequencies. Although the frequency of an individual I 2 line has an uncertaintyof +0.0025 cm -1 because of the unresolved quadrupole structure, our calibration methodproduced consistent NbN line frequencies with relative precisions of +0.0003 cm -1 forthe sub-Doppler lines.4.iii The spin-orbit and hyperfine structures of the (46) 1 (4d7r) 1 1 11/ 3 11—(4c16) 1 (5so- ) 1 X30 system4.iii.A Description of the spectraSub-Doppler spectra of the three subbands of the C 3II — X30 (0, 0) transition wereacquired before the start of the present project; detailed descriptions of these spectrahave been given in my M.Sc. thesis. Thus only a brief outline is provided here. Fig. 4.1shows the Q head regions of the three spin-components of the (0, 0) vibrational band atsub-Doppler resolution. In all three subbands, the hyperfine widths of the rotationallines are largest at low J and decrease with increasing J except in the C3II0 —X30 1 (0, 0)subband where the J" = 14 lines show the smallest linewidths. These hyperfine patternsare consistent with both the upper and lower states being in case (a n ) coupling, wherethe hyperfine splitting is proportional to [aA (b c)E]Sl in first order. The very largehyperfine widths observed in the C 3H2 — X 303 subband reflect the importance of theSI coefficient in governing the hyperfine splittings. The smaller hyperfine widths of the— X302 subband are attributed to the smaller f2 values, and particularly to thefact that E = 0 in both substates so that the Fermi contact and dipolar interactionsdo not contribute. Because of its zero ft quantum number, the C 3II0 substate has nohyperfine splittings in first order; therefore the hyperfine widths of the C 3 I10 — X301subband essentially represent the hyperfine splittings in the lower state. It is known fromhyperfine analysis of the B — X system [40] that, under the influence of spin-uncouplingQ(14)8110 — 3A1a---LiAlioutkuo9Vis Q(13)in - 302Lid smiadt Lod L.,061Figure 4.1: The Q heads of the three C 3I1 - X30 (0, 0) subbands of NbN.ccCADChapter 4. Laser Spectroscopy of NbN^ 94effects, the hyperfine components of the X 30 levels eventually reverse their energy ordersat high enough J; this happens very early in X 30 1 and causes the hyperfine splittingsof the rotational levels to pass through zero at J = 14. The 'spiking' observed at Q(14)in the C3II0 — X30 1 subband is direct evidence for this hyperfine structure reversal inX3A i .The e 1 II — X302 band has similar branch structure to the C 3II 1 — X302 subband,with comparatively narrow hyperfine widths. Three branches (R, Q, 13) are observed;Fig. 4.2 shows part of the Q branch of the (0, 0) vibrational band at Doppler-limitedresolution. The R and Q branch lines are comparatively clear, but the P lines are verybadly blended with other spectral lines in the region. Eventually the P lines were assignedby matching the observed line frequencies with those calculated from the positions of theR and Q lines. Small A-doublings are observed in the high J R lines with J' > 35. Therotational assignments and the electronic assignment of the lower state of the new bandwere based on the lower state combination differences, which were found to agree withthose obtained from the C 3II 1 — X30 2 (0, 0) subband to within experimental error. Theassignment of the upper state is not obvious from the laser excitation spectra since theR lines appear to be slightly stronger than the P lines, but the observation of a 1 11 — 1 E+emission following selective excitation to the upper state (see Part II) secures the upperstate assignment.It is seen in Fig. 4.2 that the hyperfine widths of the rotational lines of the OH —X30 2 band are also largest at low J and decrease as 1 /J; consequently the hyperfinestructures of the very low J lines should be most easily resolvable and contain morehyperfine information than the higher J lines. Unfortunately, under our experimentalconditions, the low-J structure of this intermultiplicity band is just not strong enough tobe measurable at sub-Doppler resolution. As an alternative, we recorded the sub-Dopplerspectrum of the comparatively strong Q(11) line. In order to obtain a spectrum with aFigure 4.2: The Q head of the OH — X302 (0,0) band of NbN at Doppler-limited resolution.Chapter 4. Laser Spectroscopy of NbN^ 96reasonable signal-to-noise ratio, two spectra recorded under identical conditions of laserfrequency scan were co-added to give a single spectrum The result is shown in Fig. 4.3.The ten hyperfine lines observed in the spectrum are assigned as the ten AF = AJhyperfine transitions between the ten hyperfine components of the J = 11 levels of theupper and lower states. Because no OF AJ lines are present, the hyperfine splittingin the J = 11 level of the 1 11 state cannot be measured directly. But since the hyperfineenergies of the J = 11 level of the 302 state have been measured in the hyperfine analysisof the B3 (1)—X30 system [40], the energies of the hyperfine components of the 1 11, J = 11level can be calculated from the line positions of the ten OF = LJ Q(11) components.The measured line frequencies of the hyperfine transitions of the Q(11) line are given inTable 4.1 along with the calculated energies of the OH, J = 11 hyperfine components;the assigned rotational lines of the OH — X 30 2 (0, 0) band are given in Table 4.2.Although the present hyperfine data on the OH state are quite limited, their observationnevertheless calls for a full rotational and hyperfine analysis of the (4dS) 1 (4c/r) 1 e 1 rl/C311complex.4.iii.B Hamiltonian matrix for the 1 11/311 complex.The standard expressions for the effective rotational and hyperfine Hamiltonian fora case (a) electronic state have been derived in Chapter 2. Since different forms ofHamiltonian operators can be used under different circumstances, it is necessary to specifythe exact form of the Hamiltonian which we have used in our full matrix treatment ofthe 1 11/3 11 complex. To account for the interaction between the 1 11 and 3 H states, theelectron spin-orbit term, and the Fermi contact and dipolar hyperfine operators must bewritten in their microscopic forms. Thus the total molecular Hamiltonian is writtenH = Hrot + Ils so + fI -88 + -1-1-s sr + fiel,CD + HLD + flmag.hf s + HQ^(4.1)18456.1921 cm -1F" = 6.5 7.5 8.5 95 10.5 11.5 12.5^13.5^14.5^15.518456.1012 cm-1\I\Figure 4.3: Hyperfine structure of the Q(11) line of the ell -I — X36.2 (0, 0) band of NbN. The transitions observedare the OF = AJ type.caChapter 4. Laser Spectroscopy of NbN^ 98Table 4.1: Observed hyperfine transitions of the elll - X 3 0 2 (0, 0) Q(11) line and calcu-lated energies of the 1 11, J = 11 hyperfine levels (in cm -1 ).F' - F"Q(11)Frequencye111,FJ = 11Energy15.5 - 15.5 18456.1013 15.5 18923.987914.5 - 14.5 18456.1159 14.5 18923.993613.5 - 13.5 18456.1282 13.5 18923.997512.5 - 12.5 18456.1401 12.5 18924.001411.5 - 11.5 18456.1514 11.5 18924.005110.5 - 10.5 18456.1617 10.5 18924.00849.5 - 9.5 18456.1711 9.5 18924.01138.5 - 8.5 18456.1791 8.5 18924.01347.5 - 7.5 18456.1854 7.5 18924.01436.5 - 6.5 18456.1921 6.5 18924.0162Chapter 4. Laser Spectroscopy of NbN^ 99Table 4.2: Rotational line positions of the 011 - X 3A 2 (0, 0) band of NbN (in cm -1 ).J R Q P8 18465.433*9 18466.271*10 18467.195* 18446.361*11 18468.050* 18456.140* 18445.246*12 18468.886 18456.007 18444.12313 18469.726 18455.851 18442.990*14 18470.533 18455.681 18441.827*15 18471.355 18455.501 18440.63416 18472.146 18455.304 18439.474*17 18472.937 18455.105 18438.273*18 18473.716 18454.886 18437.062*19 18474.469 18454.654 18435.853*20 18475.203 18454.410 18434.60721 18475.943 18454.156 18433.38022 18476.666 18453.890 18432.11523 18477.373 18453.611 18430.824*24 18478.068 18453.316 18429.56225 18478.761* 18453.011 18428.240*26 18479.423 18452.69127 18480.078 18452.361 18625.59728 18480.724 18452.023 18424.322*29 18451.667 18422.968*30 18481.978* 18451.29831 18482.585* 18450.91932 18483.169 18450.52533 18483.734* 18450.113*34f 18484.29134e 18484.32435f 18484.83535e 18484.87336f 18485.375*36e 18485.419*.blended. e and f refer to the parities of the upper levels.Chapter 4. Laser Spectroscopy of NbN^ 100where the terms, written using operators appropriate for a case (ap) basis, are= B (:I2 -^,§2 - -^_ - LS+ )- D (:I 2 - +^S!. -^-^ (4.2). E aiii • i i^(4.3)i.52 ,1t (3:§? — ,§2)--=^ (4.4)= 7^- '§2 ) + 7(j+'§-- + 3—`;'+)^ (4.5)= —1 AD [Eli • i i , :1 2 —,/". + .§ 2 —^— i+L — i__ +]2 ^i ++ -di AD [3,'? - ,§ 2 , ./ 2 — ,/, + .;.2 — :s5, — i+ L — i.-§+ ]^(4.6)1= -2 (o+p+q)(S_2 +,.'!)1^ 1 -- 2 (p 2q)(4.S+ + S_)^2q^-1- (4.7)a^E b,i I.8i — E gitBgri fin,r -3 \./1.0T 1 (i) T 1 (:s i , 61 2)(4.8)2^3i? - i2HQ2^e"4I(2I -1) .The terms in Eq. (4.1) can be identified by their subscripts. 1-1„ t is the rotationalHamiltonian; H30 , f/33 and _kr are respectively the electron spin-orbit, spin-spin and spin-rotation interaction terms, while fiei ,cp represents the centrifugal distortion correctionsto fi„ and fiss ; _km is the A-doubling operator; and finally Hmag.hfs and HQ are thenuclear magnetic and electric quadrupole hyperfine Hamiltonian operators respectively.The symbol [x, y] + xy yx is the anticommutator, which is required to preserve theHermitian form of the matrices [11]. The parameter ai in Eq. (4.3) is the spin-orbitinteraction constant of electron i, not to be confused with the coefficient of the I•L termin the nuclear magnetic hyperfine Hamiltonian. In Eq. (4.8), the second term is the Fermicontact interaction, while the third term includes the dipolar interaction represented by-"rot-fisofissHsrHLDilmag.hfs(4.9)Chapter 4. Laser Spectroscopy of NbN^ 101the parameter c,c= 3gitB9nit. < A ir -3TO(C) i A >,^(4.10)and a hyperfine A-doubling term represented by the coefficient d:d = — V69 iuB9ri tin, < A ± 1 Ir -372 2 (C)1A + 1 > .^(4.11)The matrix elements of the operators in Eq. (4.1), acting within a single case (a)state, are given in Section 2.iv.B. Matrix elements of the electron spin-orbit and nuclearFermi contact and dipolar interaction operators between the 1 II and the 311 state fromthe ST- configuration have been derived in Appendix B. The structure of the Hamiltonianmatrix that we have used for the (4d8) 1 (4dir) 1 1 11/ 311 complex of NbN is given as Table4.3 with the non-zero matrix elements indicated by various labels; the exact meaningsof these labels are given in Table 4.4. There are eight basis functions for each J in thegeneral case, representing the combinations A = ±1, 1 11 and 'Ho. The 8 x 8 matrix canbe factorized into two 4 x 4 matrices by transforming to the Hund's case (a) parity basisIJI>= 01(IA;SE;JS2> ± —A;S—E;J—S2>).^(4.12)In Eq. (4.12), the sum and difference (s and d) functions correspond respectively to thee and f parity levels in the 1 H state, but f and e parities in the 3 11 state. It is importantto note that the parities of the basis functions alternate with J, as can be seen in Table4.3.The Hamiltonian matrix for the X 30 state has been derived in Ref. [40] and will notbe discussed here.4.iii.0 Fitting the spectral data to the energy expressionsSince accurate molecular parameters of the X 30 state have been derived from hyper-fine analysis of the strong B3 (I) — X30 system [40], no attempt was made to refine theChapter 4. Laser Spectroscopy of NbN^ 102Table 4.3: The structure of the 1 11/3 11 matrix in the Hund's case (a) parity basis.< 1 11()1< Ji^< 3112(d)i< 3111(al)I< 3110(i)11 111(!) >IJ1 31120 >>1 31110 > i 31100 >A(1) T -Po — b a b—b A 13 ±•-20a 13 A ± .Fi,_i B T kiob +.F2o B + kio A ± 0< 111 (i,)1< J — 11^< 3 112(”-)1< 3111(11)1< 3110(”)1—C (1) + g (1) —d12 —c —d10d21 —C2 —1)21 +c20—c D12 —C1 T gll —D10 + C10d01 Tg02 7,01 + LO1 —Co< 1H(6.15 f)1< J — 21^< 31-12(iDI< 3111(i Di< 311001e(1) + 7,(1)E2 ±H20el ± 1-111+7-(02 4The superscript (1) refers to the 1 11 state; the subscripts are used, whenever necessary,to indicate the ft values of the two combining basis functions. The upper and lower signsrefer to the relevant (e or f) parity functions; the symbols .s and d indicate the sum anddifference functions respectively (see Eq. (4.12)).Chapter 4. Laser Spectroscopy of NbN^ 103Table 4.4: Matrix element representations of the labels used in Table 4.3.A = < A; SE; Al IF I --ftrot^-Liss+Hel,CD flmag.hfs^I A; SE; J11 IF >B = < A; SE +1; J11 ± 1 IF I _firo t^ ----fimag.hfs I A; SE; J52 I F >—C = < A; SE; J — 152 IF I Limag.hfs HQ I A; SE; JCI IF >TV'- < A; SE ± 1; J — 152 ± 1 IF I fimag .hfs I A; SE; J1l IF >- < A; SE; J — 2S/ I F I _HQ I A; SE; JSZ IF ><A+2;SE;JS2+2IFIft LD+ flQ1A;SE;J9IF>Tg•< A±2;SE;J —112+2IFAIA;SE;JCIIF >7-1^<A+ 2;SE;J— 2/1+2/Flf/Q 1A;SE;Jf2/F><A±2;SE+1;f0±1IFAD IA;SE;J52IF>±,C^< A ± 2; SE 1; J — 15 +1 F I Limag .hfs I A; SE;^F >±0^<A±2;SE±2;J52/Fliii,DIA;SE;JSI/F>a = < A; S — 1E; Alf F I so -I-. ---limag.hfs I A; SE; HI I F >< A; S — 1E ± 1; Jf2 ± 1 IF fimag .hfs I A; SE; Kt IF >c = < A; S — 1E; J — 152/F I fi--mag.hfs I A; SE; <MI F >d = < A; S — 1E ± 1; J — 15 +1 F^91 A; SE;^F >The matrix elements within the 11-1 and 3H basis functions, labelled by the upper caseletters, are calculated using the standard equations given in Section 2.iv.B, while thematrix elements between the 1 11 and 3 H states, labelled by the lower case letters, havebeen derived in Appendix B. The signs of the script letters are to be compared with thesigns of the explicit formulae; for example, the ± signs of D are to be compared with the+ signs of Eq. (2.196).Chapter 4. Laser Spectroscopy of NbN^ 104lower state constants in the present work. Our procedure has therefore been to fit thecomplete sub-Doppler data set of the C — X (0, 0) transition obtained from Ref. [42], andthe sub-Doppler and Doppler-limited data of the new OH — X 30 (0, 0) transition, to thedifferences between the eigenvalues of Table 4.3 and the energies of the X30, v = 0 statecalculated from the Hamiltonian matrix and molecular parameters given in Ref. [40].The final set of parameters for the 1 11/311, v = 0 complex, determined by an iterativeleast-squares fit, is given in Table 4.5. Most of the 1400 data entries in the least-squares fitare the hyperfine transitions from the three 3H - 30,6, (0,0) subbands, accurately calibratedto within 10 MHz; the data from the 1H — 302 (0, 0) transition include the ten Q(11)hyperfine lines and the less accurate Doppler-limited rotational lines (which were givena weight of 0.05). The standard deviation of the fit is 0.0005 cm -1 (15 MHz), which iscomparable to the uncertainty of the spectral data.4.iii.D DiscussionIndeterminacies in the rotational and electron spin parametersIt is known that second-order spin-orbit effects give rise to matrix elements that havethe same form as the spin-spin operator. In Eq. (4.4), consequently, A eff = Ass +As°, where A SS is the direct spin-spin parameter and As° is the second-order spin-orbitcontribution. The Ass and As° parameters are indistinguishable and cannot be determinedseparately. Since the second-order contrbution has been directly measured using thespin-orbit cross term (the A(P) term) in the present full matrix treatment of the III/ 3 11complex, the spin-spin interaction term becomes unnecessary in our Hamiltonian matrix.Similarly the inclusion of Ar, the centrifugal distortion correction to the spin-orbitcoupling between 1 II and 'II, eliminates the need for the AD parameter.Brown et al[11] have shown that an indeterminacy exists among B, AD, AD and 7Parameter 3ll 1 llT 17674.3694 ±1 17776.9493 ±10B 0.4962579 ±4 0.494886 ±9106 D 0.4905 ±3 0.517 ±6A 201.4711 ±1104 AD 0.213 ±3'7 0.0 fixeda 0.0113 ±10 0.0064 ±21b 0.0 fixedc 0.0058 ±10d —0.0030 ±10e2Qqo 0.0025 ±6od-p-Fq 3.3053 ±1p + 2q 0.0625 ±5q —0.00135 ±13 0.0010 ±12Interaction ParametersA( P) 697.8873 ±3103 A (1 ) —0.535 ±12b(P) —0.0949 ±8C(P) 0.0126 ±19Chapter 4. Laser Spectroscopy of NbN^ 105Table 4.5: Deperturbed rotational and hyperfine constants for the e 1 II and CM statestogether with the interaction parameters. Values in cm -1 .aErrors quoted are three standard deviations, in units of the last significant figuregiven. a = 0.00050 cm'.Chapter 4. Laser Spectroscopy of NbN^ 106for a case (a) 3 11 state, essentially because there are only three effective B values forthe three spin-orbit components, but four parameters to be determined from them. Thesame argument applies to the 1 11/311 complex, where five rotational parameters, includingB( 1 11) and AD ) (which replaces AD), need to be determined from the four effective Bvalues of the 1 11 and 311 states. To avoid this indeterminacy, we have set 7 to zero andfloated the other four parameters in the least squares fit. The reason for setting y to zerois that the direct spin-rotation interaction is normally very small and the second-ordercontribution to -y, which is usually much larger than the first-order contribution, hasbeen accounted for by the spin-orbit parameter A(P).The spin-orbit effects in the (4d8) 1 (4dir) 1 1 11/ 311 complexAs shown in Appendix B, the matrix elements of the microscopic electron spin-orbitoperator can be calculated from the Slater determinant wave functions for the 111 and3I1 states in the single configuration approximation as< in I II. I 3111 > = (as + lair),< 3 112 1 140 1 3112 > = - < 3 110 I fiso 1 3110 > - (as - - air)^(4.13)In Eq. (4.13), a s and a, are respectively the spin-orbit parameters of the 8 and relectrons; (a 6 - a,) and (a s + 2a,) can be identified with the experimentally determinedspin-orbit parameters A and A(P) respectively. From the values of A and A() given inTable 4.5, we obtainas = 449.7 + 0.2 cm -112a, = 248.2 + 0.2 cm-1^(4.14)The present as value is almost identical to the value of 446 cm -1 given in Ref. [40] whichwas derived from the spin-orbit separation of the X 30 state [44] from the (4dS)1(5so-)1Chapter 4. Laser Spectroscopy of NbN^ 107configuration. However, the value of a a, is quite different from that derived from the(4c16) 1 (4c1r) 1 B3 4) state ('357 cm -1 ). This shows that the single configuration descrip-tion of the electronic states of the NbN molecule using simple Slater determinant wavefunctions works very well in some cases, but obviously is not perfect.The hyperfine parametersJust as the spin-orbit parameter A can be related to the parameters of the individualunpaired electrons in the single configuration approximation, so can the hyperfine pa-rameter a, the coefficient of the I•L term in the magnetic hyperfine Hamiltonian. In thissection the symbol a is used with a different meaning from the previous section. As shownin Appendix B, the hyperfine a parameters for the (4d8) 1 (5scr) 1 X3/, (4c16) 1 (4thr) 1 B3 (1)and (4c/S) 1 (4thr) 1 C3 1-1 states can be approximately expressed asaA(3A) = 2a5 ,aA(3(I)) = 2a 6 + a ir ,aA(3H) = 2a6 — a ir .From Eq. (4.15), we have the following relation:2 aA( 30) — aA(3 (1)) = aA( 3 1-1)Substituting the aA values of the 3 ,6, and 3 4) states from Ref. [40], we find2 aA(30) — aA( 3(1)) = 0.00963 cm-1(4.15)(4.16)(4.17)which is fairly close to our experimental aA value for the C 3 II state: 0.0113 cm -1 . Inthe single configuration approximation, the 1 H and 3H states from the same electronconfiguration should have the same aA parameter; the different aA values obtained forChapter 4. Laser Spectroscopy of NbN^ 108the two electronic states as shown in Table 4.5 are probably another indication of thedeficiency of the single configuration approximation.It is clear from our data that the Fermi contact parameter b of the C311 state is effec-tively zero since floating b increases the standard deviation marginally. The cross-termFermi contact parameter MP) between the singlet and triplet II states has a surprisinglylarge value: -0.0949 cm -1 . It is not certain if this is real or whether it is merely an ap-parent value for the Hamiltonian model used, but the fact that the sign of the b constantis negative indicates that the Fermi contact effect arises from spin polarization, with nometal so unpaired electron present; this is consistent with the (46) 1 (4c/r) 1 electron con-figuration for the 1 H and 3H states. According to Appendix B, the hyperfine parametersc and cUl are related to the dipolar parameters cb and c„ of the 46 and 4dir electronsby1c =-2 (c8 + e')c(P) = 1-2 (c5 - c7).Therefore from the values of c and co D) given in Table 4.5, we obtain2 cs^ 1= 0.0092,^2-c, = -0.0034 cm-1 .(4.18)(4.19)A-doubling parameters of the C 311 stateThe general definitions of the A-doubling parameters o, p and q are given in Section2.iv.C. In the approximation that the A-doubling of the C 311 state is caused by a single3 E state, expressions (2.218) - (2.220) simplify to1 < 3H ./ L+ I 3 E > 2o -^ (4.20)4^En- EEp = 2 < 3IIIAL+I 3 E >< 3 1-11BL+1 3E > (4.21)En — EEChapter 4. Laser Spectroscopy of NbN^ 109q = 4< 3111 B L A_ 1 3 E > 2 (4.22)Ell - EE From Eqs. (4.20) and (4.21), we haveo A.p^8B •(4.23)From Table 4.5, we find that o/p equals to 49.72, which is almost identical to theratio A/(8B) (=50.75). This remarkable agreement between the two ratios suggeststhat the "unique perturber" expressions for the A-doubling parameters given in Eqs.(4.20)—(4.22) hold to a surprising degree of accuracy in the present case. Howeverit must be pointed out that the C3II state can also interact with the yet-unobserved(46)2 1 E+ state; thus the excellent agreement between o/p and A/8B may be just afortunate accident.4.iv Hyperfine analysis of the PO — OA system4.iv.A Spectral characteristicsAn emission band at 17115 cm -1 , first observed by Dunn [45], was originally assignedas arising from a 1 11 — 1 E+ electronic transition. However a recent wavelength-resolvedfluorescence analysis of the rotational lines (see Part II) shows that the upper state is a 1 4)state and the perpendicularly polarized nature of this transition (AA = —1) indicates thatit is a 1 4) — 1 A transition. The fl (I) state originates from the same electron configuration,(46) 1 (4thr) 1 , as the B3 1 state, while the OA state comes from the same (46) 1 (58a) 1configuration as the X3 0 state. Studies of the hyperfine structure of this PO — OAtransition will be informative since it will certainly provide additional information on thesecond-order spin-orbit/Fermi contact interactions between the a 1 / and X30, f1 4) andB3 1 states reported in the analysis of the B3 4) — X30 system [40].Chapter 4. Laser Spectroscopy of NbN^ 110The 141) — 10 (0, 0) band lies between the 3 0 2 — 30 1 (1, 0) subband at 17131 cm -1and the (2, 1) subband at 17075 cm -1 ; it is very simple in appearance, with single R,Q and P branches and no A-doubling. The hyperfine widths of the rotational lines aremoderate at low J and decrease as 1/J. Figs. 4.4 and 4.5 show respectively the hyperfinestructures of the Q branch head and the R(3) line at sub-Doppler resolution. Most ofthe resolved hyperfine lines are the OF = AJ hyperfine components. They are usuallythe strongest lines within a rotational transition and show the Lande-type pattern: thehyperfine components open out on the high F side and the highest F component hasthe highest intensity. In the first two R and Q lines, the weak OF .6,J hyperfinesatellites are also observed, as can be seen in Figs. 4.4 and 4.5; these hyperfine satellitesare grouped with the hyperfine main lines with the same value of F', this indicates thatthe hyperfine splittings of the lower state are much smaller than those of the upper state.The relative intensity of the OF = AJ main lines and the hyperfine satellites changesdramatically with F at very low J; this was very important to note when making thehyperfine assignments. For instance, in the Q(3) line of Fig. 4.4, the hyperfine mainlines are fairly strong in the F' = 72, 61 and 51 groups, while the hyperfine satellitesare almost unrecognizable, but in the F' = 41 and 31 groups, the hyperfine main linesbecome very weak and in the F' = 21- group, the main line disappears and the two linesobserved are the F' — F" = 21 — q, 22 — 31 satellite lines. The disappearance of the21 — 21 component is caused by an intensity cancellation of the type reported in variousQ lines of the NbO spectrum [46], this occurs when the vectors J and F would be atright angles in a classical model.4.iv.B Determination of molecular constants of the a 1 / and f1 statesFor a singlet electronic state, where the spin angular momentum S is zero, the effectiveHamiltonian consists only of the rotational part and the I • L hyperfine term. Thus HI II$7.5^4.5 1.5.6.5 II 114.5^2.5 0.5IIIQ(4)I1.15.5Q(3 )F' = 7.5 4.5^3.5 2.5 1.56.517115.0800 cm-1IIAFigure 4.4: The Q head of the flAto — a 1 L (0,0) band of NbN at sub-Doppler resolution.P' - F"^8.5-7.57.5-6.54.5-4.56.5-5.5^5.5-4.5 4.5-3.55.5-5.5^3.5-3.52.5-2.5t116.5-6.57.5-7.5eitSiFigure 4.5: The hyperfine structure of the R(3) line of the P1) — a 1 L (0, 0) band of NbN.Chapter 4. Laser Spectroscopy of NbN^ 113may be written asii- = B R2 — DR4 + a i • i, (4.24)where R = J — L — S is the rotational angular momentum, I is the nuclear spin angularmomentum and L the electron orbital angular momentum. The Hamiltonian matrix fora singlet state in case (a,3 ) coupling is rather simple; it includes the diagonal rotationaland hyperfine matrix elements given by Eqs. (2.172) and (2.193), and the off-diagonalhyperfine matrix elements of the type LW = +1, given by Eq. (2.194). Thereforefor each F quantum number, there are either (2/ + 1) or (2F + 1) basis functions,representing the possible I J > functions for each F, and the Hamiltonian matrix is eithera (2/ + 1) x (2/ + 1) or (2F + 1) x (2F + 1) matrix, whichever is smaller. A simple non-linear least-squares fitting program was accordingly written to fit the measured hyperfineline frequencies of the f1 0 — OA (0, 0) transition to the differences of eigenvalues of theupper and lower state matrices. The derived rotational and hyperfine constants for the1 0 and 'A states are given in Table 4.6, while the hyperfine lines used in the fit are listedin Table 4.7.The centrifugal distortion constants D for both the 1 0 and 1 ,6, states were not variedin the fit because the highest J value of the hyperfine data was only 6; they were heldto values derived from a separate rotational fit of the P0 — a 1 / (0, 0) band using therotational lines given in Table 4.8.The B constant for the OA state is slightly larger than that of the .VA ground state;this is similar to what is found in TiO and VO, where the lower spin-multiplicity statesarising from the same electron configurations as the ground states have slightly larger Bconstants [47, 48].Chapter 4. Laser Spectroscopy of NbN^ 114Table 4.6: Molecular constants of the PO and a 1 A states of NbN (Values in cm -1 ).Parameter 1,6, 14,T(v = 0) 0.0' 17118.10656^+3B 0.50827^±4 0.49509^+5106 D 0.420'^±147 0.490^+149aA 0.0054^+3 0.0369^±2Standard deviation=0.00042Error limits (in parentheses) are three standard deviations in units of the last significantfigure quoted. 'Absolute energy position of OA, v = 0 is 5197.2 cm -1 , determinedfrom analysis of the C3H 1 - OA (0, 0) transition [44]. b Absolute rotationless energyof PO, v = 0 is 22312.9 cm -1 . 'Values obtained from a linear least-squares fit of therotational lines of the 1 (1) - lA (0, 0) band.Table 4.7: Hyperfine transitions of the f1 (1) - a 1 /. (0, 0) band of NbN.J" F' - F"RFrequency (cm -1 ) J" F' - F"QFrequency (cm -1 )2 7.5 - 6.5 17118.6825 3 7.5 -7.5 17115.63672 6.5 - 5.5 17118.6250 3 6.5 -5.5 17115.58082 6.5 - 6.5 17118.6138 3 6.5 -6.5 17115.57462 5.5 - 4.5 17118.5757 3 6.5 -7.5 17115.56772 5.5 - 5.5 17118.5650 3 5.5 -4.5 17115.52532 4.5 - 3.5 17118.5327 3 5.5 -5.5 17115.51972 4.5 - 4.5 17118.5251 3 5.5 -6.5 17115.51432 3.5 - 3.5 17118.4912 3 4.5 -3.5 17115.47952 3.5 - 4.5 17118.4837 3 4.5 - 5.5 17115.46982 2.5 - 2.5 17118.4671 3 3.5 - 2.5 17115.44062 2.5 - 3.5 17118.4585 3 3.5 - 4.5 17115.43353 8.5 - 7.5 17119.5729 3 2.5 - 1.5 17115.4113Continues on next pageChapter 4. Laser Spectroscopy of NbNTable 4.7, ContinuedJ" F' - F"RFrequency (cm -1 ) J" F' - F"QFrequency (cm -1 )3 7.5 - 6.5 17119.5328 3 2.5 - 3.5 17115.40583 7.5 - 7.5 17119.5264 4 8.5 - 7.5 17115.51433 6.5 - 5.5 17119.4980 4 8.5 - 8.5 17115.51003 6.5 - 6.5 17119.4925 4 7.5 - 7.5 17115.46703 6.5 - 7.5 17119.4855 4 7.5 - 8.5 17115.46313 5.5 - 4.5 17119.4665 4 6.5 - 6.5 17115.43113 5.5 - 5.5 17119.4617 4 6.5 - 7.5 17115.42573 4.5 - 3.5 17119.4405 4 5.5 - 4.5 17115.40123 4.5 - 4.5 17119.4370 4 5.5 - 5.5 17115.39803 3.5 - 2.5 17119.4177 4 5.5 - 6.5 17115.39413 3.5 - 3.5 17119.4156 4 4.5 - 3.5 17115.37253 3.5 - 4.5 17119.4117 4 4.5 - 4.5 17115.37013 2.5 - 2.5 17119.3990 4 3.5 - 2.5 17115.34993 2.5 - 3.5 17119.3964 4 3.5 - 4.5 17115.34513 1.5 - 1.5 17119.3876 4 1.5 - 0.5 17115.31923 1.5 - 2.5 17119.3851 4 1.5 - 2.5 17115.31613 0.5 - 1.5 17119.3794 5 9.5 - 8.5 17115.36615 9.5 - 9.5 17115.36305 8.5 - 8.5 17115.33115 7.5 - 7.5 17115.30345 6.5 - 6.5 17115.27845 5.5 - 5.5 17115.25735 4.5 - 3.5 17115.24075 4.5 - 4.5 17115.23905 3.5 - 3.5 17115.22405 2.5 - 3.5 17115.21106 10.5 - 10.5 17115.19416 9.5 - 9.5 17115.16926 8.5 - 8.5 17115.14656 7.5 - 7.5 17115.12696 6.5 - 6.5 17115.10926 5.5 - 5.5 17115.09366 4.5 - 4.5 17115.08006 3.5 - 3.5 17115.07056 2.5 - 2.5 17115.0612115Chapter 4. Laser Spectroscopy of NbN^ 116Table 4.8: Rotational lines of the f 1 - a 1 L (0,0) band of NbN (in cm-1 ).J R Q P5 17121.2266 17122.080*8 17123.666 17114.7509 17124.413* 17114.509*10 17125.117 17114.246 17104.35511 17125.822 17113.953 17103.06212 17126.484* 17113.662 17101.74813 17127.142* 17113.306* 17100.437*14 17127.759 17112.937 17099.07015 17128.345 17112.515 17097.683*16 17128.927* 17112.085 17096.26317 17129.431* 17111.636 17094.82218 17129.949* 17111.164* 17093.36519 17130.433 17110.671 17091.86320 17130.896 17110.136 17090.355*21 17131.324 17109.578 17088.82222 17131.738 17109.011 17087.21623 17132.115 17108.398 17085.65024 17132.453 17107.763 17084.00925 17132.759 17107.076 17082.36726 17133.052 17106.384 17080.69527 17133.320 17105.662 17079.003*28 17133.556 17104.937 17077.26129 17133.783 17104.16030 17133.970* 17103.36131 17134.132 17102.52932 17134.259 17101.66433 17134.357 17100.77134 17134.433 17099.87335 17134.480 17098.93536 17097.97237 17096.97638 17095.96839 17094.927*40 17093.84741 17092.736*42 17091.583*.blendedChapter 4. Laser Spectroscopy of NbN^ 1174.iv.0 The hyperfine structures of the fl (I) and elA statesAs pointed out in Ref. [40], the spin-orbit/Fermi contact interation between thesinglet and triplet states from the same electron configuration, if treated as a second-order effect, can cause substantial corrections to the measured rotational and hyperfineparameters of the singlet and triplet states, depending on the magnitude of the crossterms. Consequently the presently measured hyperfine parameters aA for the alA andPO states are only apparent parameters, since they have undoubtedly been modifiedby the second-order spin-orbit/Fermi-contact effects, considering the extensive hyperfinedistortions observed in their triplet counterparts. The < 1 A1f-/FA 1 > and < 1 401i/1 303 >cross terms can be derived in the same way as the < 1 101 3 11 2 > cross terms given inAppendix B. Using second-order perturbation theory, it can be shown that the cross termsproduce an equal and opposite correction to the aA parameters of the singlet state andthe middle component of the triplet state. Analysis of the hyperfine structure of the X 30state in Ref. [40] showed that the apparent aA value of the 30 2 spin-orbit componentwas raised from the "actual" value 0.02485 cm -1 to 0.04586 cm -1 , an increase of 0.02101cm -4 . Thus if the deperturbed aA parameter of the alA state is assumed to be thesame as that of the X 30 state, (which is a correct assumption in the single configurationapproximation, as has been shown in Appendix B), the apparent aA parameter for a 1 6,can be predicted to be (0.02485-0.02101)=-0.00384 cm -1 . The predicted value is quiteclose to the experimental value of 0.0054 cm -4 ; the slight difference between the twovalues probably reflects the difference between the true aA parameters of the lA and30 states. The second-order spin-orbit/Fermi-contact interaction between the PO andB3 (1) states from the (4c/(5) 1 (4c/r) 1 configuration is much smaller since the 1 (1) state liesmuch further away from the 3 0 3 state; this is reflected by the much smaller increase ofthe apparent aA value (0.0411 cm -1 ) of the B3 1'3 substate from the true value (0.0401Chapter 4. Laser Spectroscopy of NbN^ 118cm -1 ) [40], calculated from the aA values of the unperturbed B 3 (1) 2 and B3(1)4 substates.4.v ConclusionTo summarize the first Part of Chapter 4, the present study has investigated the elec-tron spin-orbit and nuclear hyperfine interactions between the singlet and triplet states ofNbN arising from the (4c16) 1 (4thr) 1 and (46) 1 (5.5(7) 1 configurations in considerable detail.The interaction between the (46) 1 (4d7) 1 elll and C3I1 states has been quantitativelyanalyzed to give direct measurements of the spin and hyperfine interaction parametersbetween the two states; the parameters obtained can be understood satisfactorily inthe single configuration approximation by comparison with the molecular parameters ofother known triplet states of NbN. The measurement of the hyperfine parameters of thea 1 / and PO states has provided a direct confirmation that extensive second-order spin-orbit/Fermi contact interactions between the singlet and triplet states from the sameelectron configurations produce substantial corrections to the coefficients of the I • Lmagnetic hyperfine interaction. The corrections to the apparent aA parameters of the(4c10 1 (5scr) 1 a 1 L and (4d8) 1 (4dir) 1 fl(I) states are found to be about the same as thoseof the central components of the (4ch5) 1 (5so-) 1 X30 and (46) 1 (4(170 1 B3 (1) states butopposite in direction, which is consistent with the second-order perturbation treatmentof the spin-orbit/Fermi contact cross terms. Because of the large spin-orbit parametersof the 4th5 and 4dir electrons, as shown in Eq. (4.14), spin-orbit/Fermi contact interac-tions between other NbN singlet and triplet states from the same electron configurationsinvolving the 4d8 and 4dir electrons are also expected to be significant; in the worst cases,the interactions will be so severe that the resulting singlet/triplet complex is essentiallyin case (c) coupling, as is the case for the e 1 l1/C311 complex.Since this work completes the determination of the molecular constants for the fourChapter 4. Laser Spectroscopy of NbN^ 119Table 4.9: Spin-orbit constants AA and hyperfine constants aA for the electronic statesfrom the configuration (4d8) 1 (4dir) 1 of NbNa.States To AA aA22315.3 0.0369343, 16518.5565 807.39 0.04007311 17674.3694 201.47 0.0113111 17776.949 0.0064Values in cm -1 . aConstants for the 3 4:0 state are obtained from Ref.[40].electronic states derived from the configuration (4d8) 1 (4c/r) 1 , it is appropriate to tabulatetheir electron spin-orbit and hyperfine parameters for comparison purposes. This hasbeen done in Table 4.9. As has been discussed in Section 4.iii.C, the electron spin-orbitparameters fit quite well into a simple single configuration description of the electronicstates, but the hyperfine parameters do not fit as well. Obviously more sophisticatedelectronic wave functions are required for more accurate representation of the electronicsystem of NbN. It is hoped that the present results will inspire further ab initio studieson molecules like NbN, specifically the electron spin-orbit coupling constants and thehyperfine parameters, which are very sensitive tests of ab initio wave functions.Chapter 4. Laser Spectroscopy of NbN^ 120Part IIResolved Fluorescence Study of Two Low-lying Singlet Electronic States ofNbN: the (530.)2 bi..-7-1-L and (4d8) 2 cl F Excited States4.vi IntroductionThe most challenging feature of the molecular spectroscopy of transition metal contain-ing compounds (TMC) is the large number of close-lying electronic states of different spinmultiplicities arising from the many unpaired metal d electrons. The relative positions ofthese electronic states are almost impossible to calculate by ab initio methods because ofthe complicated electron correlation effects. Experimental data of this type are thereforevery valuable in theoretical understanding of the electronic structure and chemical bond-ing in these TMC molecules. The best understood transition metal diatomic molecule,spectroscopically, is probably titanium oxide, TiO; 11 of its 16 expected low-lying elec-tronic states, including both singlet and triplet states, have been observed and analyzed[47]. Our understanding of other TMC molecules is not as extensive. In most molecules,usually only one spin-multiplicity system has been observed; when two spin systems areknown to coexist, their relative positions mostly cannot be determined because of theabsence of intermultiplicity transitions.The electronic spectra of NbN have only been observed comparatively recently. Al-though the transitions involving the three triplet states, X 30, B3 (1) and C311, have beenwell characterized, the low-lying singlet states, especially their positions relative to thetriplet states, are mostly unknown. Thus it is desirable to explore the new low-lyingsinglet electronic states and the related electronic transitions of NbN so that we have abetter understanding of the spectroscopy of this molecule.Chapter 4. Laser Spectroscopy of NbN^ 121Like titanium oxide, NbN has two electrons in open shells. The ground electronicstate, X30, is now known to come from the electron configuration (4d6) 1 (5so) 1 [40].Promotion of the 5sa electron to the metal 4dir orbital gives rise to a number of perpen-dicularly polarized electronic systems in the visible region; these include the prominentB30 — X30 and C311 — X 30 systems whose rotational and hyperfine structures havebeen extensively studied [40][Part I]. Two other electronic systems which result fromthe same electron promotion involve the singlet electronic states of NbN; they are the(46) 1 (4chr) 1 e l II — (4c16) 1 (5sa) 1 X30 2 and (46) 1 ochry f1 a l A transi-tions analyzed in Part I. The observation of the e 1 II — X302 intermultiplicity transitionis not surprising since the e 1II state is known to be heavily mixed with the C 3 1I 1 spin-component, as described in Part I, so that it contains considerable triplet character.Similar intermultiplicity transitions from C 311 1 to singlet states are also expected to beobservable; in fact, a recent study of the resolved fluorescence emitted from 03 11 1 [44]has identified the C3 1I 1 — a 1 A transition in the infrared. The same study also revealeda new C3Il — A3 E - transition near 12150 cm -1 and the spin-satellites of the C — Xtransition which provided a direct measurement of the spin-orbit intervals of the X 30ground state. As a result, we have now established the absolute energies of the alA, e 1 11and f1 (I) states at 5197.2, 18857.4 and 22315.3 cm -1 respectively.In this second Part of Chapter 4, we report the analysis of two new NbN singlet elec-tronic transitions observed in wavelength-resolved fluorescence following laser excitationto the e 1 11 and PA, states; these are the e 1 II — b1 E+ and f 1 (1) — cif transitions in theinfrared region. The present study gives the energies of the two new singlet states andtheir approximate rotational constants, and also proves the singlet characters of the otherthree known singlet states, a, e and f, as mentioned above.Chapter 4. Laser Spectroscopy of NbN^ 1224.vii Experimental detailsThe NbN source was the same as that described in Part I. The laser employed wasthe same Coherent 599-21 cw dye laser operating with an output power between 15 to100 milliwatts depending on the laser dyes used and the spectral region. The laser-induced fluorescence was excited when the laser beam was passed through the tail of theNbN discharge flame; it was then focused onto the aperture of a Spex-1702 0.75 meterspectrometer by a system of two lenses. Wavelength-dispersed fluorescence was detectedwith a microchannel-plate intensified array detector (PAR model 1461) mounted in placeof the exit slit of the spectrometer. In the visible region, the spectral width of the detectoris about 200 A with a resolution of 0.2 cm -1 . The fluorescence spectra were taken withexposure times ranging from a few seconds for strong NbN lines to half a minute forweak spectral features. The whole process from data acquisition to spectral calibrationis performed under the control of a PDP 11/23 minicomputer.The calibration of a resolved fluorescence spectrum is provided by the emission linesfrom a Fe-Ne hollow cathode lamp recorded at the same spectral window. The vacuumwavelengths of the Fe and Ne atomic lines, taken from Crosswhite's atlas [49], were fittedto a three-term polynomial to obtain the calibration constants. The accuracy of thecalibrated spectral lines is limited by the characteristics of the array detector to about+0.2 cm-1 .4.viii Data analysis4.viii.A Transitions from the (4d6) 1 (4dir) 1 OH stateFig. 4.6 shows the resolved fluorescence spectrum near 7300 A obtained following laserexcitation of the Q(22) line of the eln — X 30 2 band at 18453.89 cm -1 . Four fluorescenceChapter 4. Laser Spectroscopy of NbN^ 123NbN: 1 11 — 1 A (0,0)Q(22)NbN: 1 11 — 3 E(1+ (0,0)Figure 4.6: The OH — alA and iVE GT+ transitions near 7300 A from laser excitation ofthe Q(22) line of the OH — X 3A 2 transition.Chapter 4. Laser Spectroscopy of NbN^ 124lines are observed; they are assigned as the R(21), Q(22), P(23) rotational lines of theOH — a 1 /. (0,0) transition and the Q(22) line of the &II — A 3 EcTf. (0, 0) transition. Sincethe rotational constants of the e 1 II, a 1 ,6, and A3 E,T+ states have been obtained fromother spectral analyses (see Part I and [44]), the rotational line positions of the two newtransitions were calculated from the known constants and then compared to the linefrequencies measured from the fluorescence spectrum. The calculated and the measuredline positions agree to within experimental error and thus confirm the line assignments.The intensity difference between the two transitions clearly indicates that the emissionto the OA state is more favorable than the emission to the A3 E- state, reflecting thedominant singlet character of the upper state. The absence of the R and P lines fromthe OH — A3 EcT+ emission is also consistent with the transition being a weakly allowedintermultiplicity transition.About 650 cm -1 to the red of the OH — a 1 / transition lies another emission from theexcited OH state. The fluorescence patterns illustrated in Fig. 4.7 immediately identifythis as a 1 II — 1 E+ transition. The reasoning is as follows. When the A doubling isnot resolved in the elli — X 302 band, the laser excites both parity components of theupper state, and a regular three line (R, Q, P) fluorescence pattern is obtained, as shownin Fig. 4.7(a); when the A doubling is resolved and the laser excites only one of theA-components of a rotational line, two different fluorescence patterns are obtained: inFig. 4.7(b), a single fluorescence line, Q(36), is observed following laser excitation of thelower A-component of the R(35) line, while in Fig. 4.7(c) two rotational lines, R(35)and P(37), are observed in the resolved fluorescence following excitation of the upperA-component of the same rotational line. These fluorescence patterns are characteristicof the lower state being a non-degenerate state, namely a 1 E or 3 E0 state. Since nolow-lying 1 E- state is expected in this region and the expected low-lying A 3E0 state hasbeen located a few hundred cm -1 below, the new non-degenerate state must be a 1E+ P(23)R(21)12969.9 cm-113014.9 cm -1(b) Excitation of R(35),lower A-componentChapter 4. Laser Spectroscopy of NbN^ 125(a) Excitation of 'H — 3A2 Q(22),A-doubling not resolved Q(22)(c) Excitation of R(35),^R(35)upper A-componentP(37)Figure 4.7: State-selected^— PE+ emission near 7700 A from laser excitation of the— X30 2 transition.Chapter 4. Laser Spectroscopy of NbN^ 126state, presumably the (5su)2 b1 E+2, state judging by its low excitation energy.In order to derive the term value and rotational constants for the new 1 E+ state,we have systematically excited the comparatively strong Q rotational lines of the elII —X30 2 (0, 0) band with laser radiation and measured the emission lines of the OH — 0E+transition. No attempt was made to derive rotational constants for the e 1 II state usingthe present data since precise values have been derived in Part I using much more accuratespectral data. We have therefore used the available e 1 II state parameters to calculatethe upper state energies and then fitted the measured line frequencies of the OH — b 1 E+transition to the following energy expressions:v(Rj) = E'(J + 1) — [n, + B"J(J + 1) — D"J 2 (J +1) 2 ]v(Qj) = E'(J) — [n, + B"J(J +1) — D"J2 (J +1) 2 ]v(PP ) = E'(J — 1) — [T,',' + B"J(J + 1) — D"J 2 (J + 1) 2 ] (4.25)where E' is the calculated energy of the e 1 II level, while T,C, B" and D" are respectivelythe term value, rotational constant and centrifugal distortion constant of the b 1 E+ state.The fitting results, along with the line frequencies used in the fit, are given in Table 4.10.4.viii.B The fl(1) — c1 F transitionConsidering all the possible electronic states that can arise from the low-lying electronconfigurations of NbN and their estimated energy order, there are only two spin-allowedelectronic systems that can be reached in emission from the (4c/(5) 1 (4dr) 1 f1 4) state: theknown (4d6) 1 (4c/r) 1 f1 — (4d6) 1 (5so) 1 a 1 0 system observed in the visible region and anew system, assigned as (4c18) 1 (4c1r) 1 f 1 4) — (4d8) 2 c1 F. Following laser excitation of thepo — a 1 L (0, 0) band near 17115 cm -1 , we observed a new electronic transition in theinfrared region in resolved fluorescence; it was easily identified as the expected PO — c11'Chapter 4. Laser Spectroscopy of NbN^ 127Table 4.10: Rotational constants and line positions for the OH - bl-E+ transition of NbN(in cm -1 ).state^Ta^B^106D111 18857.46^0.4949a^0.517`1E+^5862.8 (1)^0.4979 (5)^-0.11 (37)Standard deviation=0.11Error limits in parentheses are three standard deviations, in units of the last significantfigure quoted. aValues relative to X 3 A 1 . b Value derived from the origin of the 011 -X30 2 (0, 0) band, held constant. c Values obtained from Table 4.5, held constant.J R Q P9 12994.3410 13005.39 12984.1911 13006.08 12994.2112 13007.06 12994.10 12982.1413 13008.05 12994.00 12981.1914 13008.78 12993.93 12980.0415 13009.81 12993.78 12979.0016 13010.56 12993.70 12977.7117 13011.38 12993.66 12976.7618 13012.51 12993.64 12975.7719 13013.16 12993.16 12974.6820 13014.17 12993.26 12973.2621 13014.86 12993.14 12972.4222 13015.71 12992.98 12971.0623 13016.48 12992.73 12969.9024 13017.38 12992.59 12968.8225 13018.20 12992.39 12967.7126 13018.95 12992.24 12966.2927 13019.68 12991.98 12965.1328 13020.11 12991.96 12963.9929 13021.17 12991.47 12962.8630 13021.83 12991.1531 13022.58 12990.94 12960.0432 13023.36 12990.82 12959.0033 13024.19 12957.8734 12956.4835 12955.27Chapter 4. Laser Spectroscopy of NbN^ 128electronic transition from the observed fluorescence intensity patterns. Fig. 4.8 shows thefluorescence spectra recorded near 8000 A after laser excitation of the Q(7), Q(21) andQ(34) lines of the PO — a 1 0 (0, 0) band. The dramatic difference in intensity betweenthe low-J R and P lines, as illustrated in Fig. 4.8(a), is characteristic of the high 12values of the electronic states involved in the transition. It can be shown quantitatively,using comparatively noise-free emission lines, that the measured intensity ratios betweenthe R(J — 1) and P(J + 1) lines agree with those calculated for a 1 (I) — 1F transition. Forexample, from Fig. 4.8(b), the intensity ratio /(P(22))//(R(20)) is measured to be 1.71,while calculation using the line strength formula given in Eq. (2.234) gives the ratioas 1.65. The excellent agreement between the measured and calculated values proveswithout any doubt that this is a 1 (I) — IF transition.Rotational constants for the c 1 F state were derived from a large number of measuredrotational lines of the fl(1) — clF transition by the method of least squares. Since accuraterotational constants for the PO state are available from analysis of the PO — ct lA (0, 0)transition as reported in Part I, we again used the available PO state parameters tocalculate the upper state energies and fitted the frequencies of the PO — c1 F transitionto Eq. (4.25). The derived rotational constants for the c 1 F state and the fluorescenceline frequencies used in the fit are given in Table 4.11.4.ix Results and discussionFig. 4.9 shows the energy positions of the new PE+ and c 1 F states relative to theother known electronic states of NbN. The c 1 F electronic state is assigned to the (4d5) 2electron configuration since only this configuration can give rise to a low-lying 1 F state.F electronic states are very rare in spectroscopy; to our knowledge, this is possibly thefirst time that a 1 1' state has been positively identified. The rarity of F electronic statesQ( 7)L ne excited: (a)P(8)(b) (c)P(35)P(22)Q(21) Q(34)E1-1001-1R(33)Er-+Yvvit NovviFigure 4.8: Fluorescence patterns of the P (I) — c1 1' transition of NbN: emissions from (a) J' = 7, (b) J' = 21and (c) J' = 34 levels, showing the large differences in intensities between the R and P lines.Chapter 4. Laser Spectroscopy of NbN^ 130Table 4.11: Rotational constants and line positions for the PO - OF transition of NbN(in cm -1 ).state^Ta^B^106D10^22312.9b^0.4951b^0.490bi t^9918.7 (3)^0.4999 (10)^0.59 (83)Standard deviation=0.31Error limits in parentheses are three standard deviations, in units of the last significantfigure. a Values relative to X30 1 . b Values obtained from Table 4.6, held constant.J R Q P6 12400.51 12393.807 12401.85 12393.80 12387.058 12402.76 12393.80 12386.049 12403.47 12393.73 12385.0110 12393.68 12383.9711 12382.8318 12411.0919 12411.79 12392.4420 12412.69 12392.32 12372.7321 12413.44 12392.11 12371.4822 12414.14 12391.87 12370.4023 12415.00 12391.64 12369.1824 12415.72 12391.44 12367.9325 12416.35 12391.17 12366.6826 12417.03 12390.88 12365.5327 12417.80 12390.72 12364.2828 12418.49 12390.42 12363.0129 12419.18 12390.14 12361.7130 12419.83 12389.85 12360.4331 12420.48 12389.55 12359.1732 12421.21 12389.26 12357.8633 12421.80 12388.96 12356.5834 12388.59 12355.2835 12353.98Charge transfer.: states!,arrII57relll 1 •bar(; 3rt Fl± ^4J.36.7 B3 (1)252cir 40.2111E+ 082 A3E-a al A 03baX3A 2120343.818857.418306.417902.281 8.89117751.3 17458.016943.516144.6489918.75862.85604.55197.25112.1891.0400.502 0±••Infra-red systemsVisible region bandsNIL•Chapter 4. Laser Spectroscopy of NbN^ 131ftis/rfl,f,^3 ^ 22312.9 cm-1v.< Figure 4.9: Electronic states and transitions of NbN known to this date; discussions ofthe new Y311 and charge transfer states are given in Part III.Chapter 4. Laser Spectroscopy of NbN^ 132results from the unusually large A value which normally only occurs in transition metal-containing compounds where large numbers of unpaired d electrons are present.The comparatively low energy position of the WE+ state (only 700 cm -1 above theal-A state) indicates that it arises from the (530 2 electron configuration since other1 E+ states from other electron configurations are expected to lie higher in energy byanalogy to the singlet states of TiO [47, 50]. What this implies is that the observed@ay wry on — ( 530.)2 bi, 1-2, transition involves a forbidden two electron processwhich would usually only occur in the presence of configuration interaction. It is believedthat the (5sa)2 bi.,-+Li state is mixed with the as-yet-unobserved d 1 E+ state from the(4d5) 2 configuration by electrostatic perturbation [51] so that the (4d(5) 1 (4d7r) 1 — (5.9.7) 2electronic transition becomes allowed. Moreover, the (5sa)2 bi,-F2., state is believed tointeract also with the (46) 2 A3E - electronic state, resulting in the A3 E0 componentbeing shifted out of position. As can be seen in Fig. 4.9, the two spin (or more preciselySI) components of the A3E - state, A3ET and A3E01_, lie about 491 cm -1 apart; theenergy separation is too large for a case (b) 3 E - state, and would normally be describedas case (c) coupling. Case (c) coupling often arises when the spin-orbit matrix elementsbetween electronic states from the same configuration are large relative to their energyseparation, as is the case between the OH and C311 states of NbN discussed in PartI. Initially the case (c) coupling in A 3 E - state was thought to be caused by the spin-orbit interaction between the A 3 E0 spin component and the unseen d 1 E+ state fromthe same (4d(5) 2 configuration. But after considering the high energy position of theunseen (4d6) 2 d1 E+ state (expected to lie higher than the c'F state) and the estimatedmagnitude of the perturbation matrix element, < d 1 E+1.//30 1A3E,T >= 2a 6 P.,' 890 cm -1 ,it was realized that the (4d02 d 1 E+/A3 E0- interaction alone could account for no morethan 187 cm -1 of the observed shift, and therefore most of the observed 491 cm -1 shiftmust be caused by the nearby (5 ,90.)2 1)1E+ state. It is believed that the b 1 E+ stateChapter 4. Laser Spectroscopy of NbN^ 133acquires considerable (4d6) 2 d1 E+ character through electrostatic interaction, as alreadymentioned, so that it actually perturbs A 3 E,T more strongly than the d 1 E+ state, becauseit lies much closer. Because of the mutual interactions among the b 1 E+, A3E0 and dl E+states, the exact matrix elements between them are impossible to measure; the net effectof these complicated interactions is that A 3 E0 is pushed down 491 cm" by the othertwo 1 E+ states.As a result of the strong mixing between the (5su) 2 and (4d6) 2 electron configura-tions, electronic states arising from these two electron configurations are expected tohave similar bond lengths and therefore similar rotational constants. Therefore it is notsurprising that the rotational constants of the (5so-) 2 b1 E+ and (4d8) 2 OF states are verysimilar, as can be seen from Tables 4.10 and 4.11.The observation of the (5sa) 2 b1 E+ state finally ends the speculation that the groundstate of NbN may arise from the (5so-) 2 electron configuration [54]. It is known that inzirconium oxide, which is isoelectronic to NbN, the 5scr orbital lies below the 4c1(5 orbitalso that the ground state of ZrO is the 1 E+ state from the (5sa) 2 configuration [52], notthe 3A state from the (46) 1 (530) 1 configuration [53]. On going from Zr to Nb, thestability of the 4d orbital increases substantially as compared with the 5s orbital; this isreflected by the 4d45.5 1 ground state for the Nb atom as compared with the 4d 2 5s 2 groundstate of Zr. Furthermore, in the NbN molecule, the (55(5) 2 configuration produces onlya single 1 E+ state, while (4d6) 1 (5so) 1 produces two states, 3 ,6, and 1 A, which are spreadover a 5000 cm' range. This, together with the increased stability of the Nb d electron,is the reason why the ground state of NbN is (4dS) 1 (5so-) 1 30, not (5so.)2 1E+.Chapter 4. Laser Spectroscopy of NbN^ 134Part IIIRotational Analysis of the B — X, C — X Vibrational Sequence Bandsand Some New Subbands of NbN: Evidence for a New (5so- ) 1 (4chr) 1 3 11Electronic State and Some Charge Transfer States4.x IntroductionConsider the molecular orbital scheme of NbN given in Fig. 4.10, where the valenceelectrons are arranged in the ground state configuration. Low-lying excited electronicstates arise when the 5so- or 4d8 electron is promoted to other Nb-centred molecularorbitals. Table 4.12 lists the expected low-lying electronic states arising from electronconfigurations containing at least one 5scr or 4d8 electron. Nine of the sixteen expectedelectronic states have been observed so far, as shown in Fig. 4.9; electronic states fromthe (4d6) 1 (4do) 1 , (5so) 1 (4da) 1 and (5sa) 1 (4dir) 1 electron configurations have not beenaccounted for experimentally. The reasons for this deficiency are that electronic statesarising from the (5scr) 1 (4do) 1 configuration are not likely to be observed in transitionsfrom the ground state because the (5sa) 1 (4da) 1 — (4c/(5) 1 (5so) 1 transition is a forbiddenAA = 2 transition, while states from the (4d8) 1 (4da) 1 configuration are expected to liequite high in energy since an electron is promoted to the energetic antibonding 4dcr orbitalso that transitions from the ground state to these excited states probably require theenergy of ultraviolet radiation. But electronic states from the (5scr) 1 (4dr) 1 configurationare expected to lie close to states from the (4c/S) 1 (4thr) 1 configuration since the 5scr and4d6 orbitals are known to lie very close in energy; transitions from the ground state tothe (5so- ) 1 (4thr) 1 excited states should therefore be in the visible region. A search for the(5.50) 1 (4dr) 1 311 — X30 transition in the visible spectrum of NbN seems desirable.Chapter 4. Laser Spectroscopy of NbN^ 135Nb orbitals N orbitalsNbN orbitals4du4d5s4dir5sa t4d82p21"r titi2po- t iFigure 4.10: The relative energies of the molecular orbitals of NbN, formed from linearcombinations of the atomic orbitals of Nb and N.Table 4.12: Expected low-lying electronic states of NbN.Electron configuration Electronic state5so-1 4d6 1^3Ar^105sa2 1E+4d82^3 E_^1 E+^ir4d8 1 4dri 3or^31r^i ll^1 05so-1 4dr'^3fir^ill4d8 1 4dol 3Ar^lA5sa 1 4da l^3E+^1E+Subscript r refers to regular (opposed to inverted) multiplet structures.Chapter 4. Laser Spectroscopy of NbN^ 136The electronic transitions discussed above are called metal s — d or d — d transitionssince they involve only molecular orbitals derived predominantly from the metal atomicorbitals; charge transfer transitions, which involve promotions of an electron from aligand-centred orbital to a metal-centred molecular orbital can also occur, though theyusually require higher energies because the ligand orbitals (2pir and 2po) lie some waybelow the metal orbitals. In some late 3d transition metal oxides such as MnO andFeO [55, 56, 57], charge transfer transitions have been observed in the visible and nearinfra-red. The whereabouts of the charge transfer transitions depend on the exact relativepositions of the metal d orbitals and the 0 or N 2p orbitals. Recent theoretical calculation[58] indicates that charge transfer states of NbN may lie as low as 17600 cm -1 relativeto the ground state, as a result of the particularly stable Nb 4d orbitals (implied by its4d1 53 1 ground state electron configuration).In this Part of Chapter 4, we report a high-resolution study of some new subbands inthe 18200-18500 cm -1 region of the Doppler-limited spectrum of NbN observed by laserexcitation spectroscopy. Two of the subbands have been identified as belonging to theexpected (5s o- ) 1 (4d7r) 1 3 11 — (46) 1 (5so-) 1 X3 0 transition, while the other subbands werefound to result from charge transfer transitions. To secure the assignments of the lowerstates of these new subbands, which were found to be excited vibrational levels of theX30 state, a rotational analysis of the vibrational sequence bands of the strong B — Xand C — X systems was performed.4.xi ExperimentalThe experimental arrangement followed closely that described in Parts I and II. Laser-induced fluorescence corresponding to the new NbN bands and the vibrational hot bandsof the B — X and C — X systems was found to be strongest when the total gas pressure wasChapter 4. Laser Spectroscopy of NbN^ 137maintained at about 300 millitorrs. Doppler-limited spectra of NbN were recorded overmost of the 15500 — 18500 cm -1 region with a single-frequency cw dye laser operatingwith an output power ranging from 15 to 100 milliwatts, using DCM, R590 and R560 laserdyes (Exciton Inc.). The resolution of the spectra is about 750 MHz, which is sufficientto resolve most of the rotational lines except for some hyperfine broadened low,/ lines.Resolved fluorescence spectra of selected rotational lines of the new subbands were alsorecorded to assist in the rotational and vibrational assignments of the new transitions.Calibrations of the laser excitation spectra were again provided by iodine absorptionspectra recorded simultaneously; the centre positions of the individual NbN rotationallines are expected to be accurate to within 100 MHz. The resolved fluorescence spectrawere calibrated against Fe emission lines and the calibration accuracy is about 0.2 cm -1 .4.xii Appearance of the low-resolution spectraThe NbN spectra in the 15500 — 18000 cm -1 region are completely dominated bythe vibrational sequence structures of the B — X and C — X electronic systems. TheAv = 0 sequences of both systems are very strong and clearly recognizable in the low-resolution spectra; the Av = +1 sequence bands are much weaker and form most of theremaining band structure observed in the 15500-18000 cm -1 region. There are no strongbands above 18000 cm -1 ; only a few weak bands are observed, as shown in Fig. 4.11which illustrates the low resolution spectrum in the 18200 — 18500 cm -1 region. Threesubbands in this region belong to the known electronic transitions; they are the (1,0) and(2,1) bands of the C 3II2 — X303 transition at 18400 and 18318 cm -1 , and the the (0,0)band of the OH — X 3L2 transition at 18457 cm -1 , as discussed in Part I and II. Therest of the band structure observed in Fig. 4.11 belongs to the new subbands discussedin this Part.OOkr,00r-I Q0311 2 — X303 (2, 1)C3I12 — X3i13 ( 1, 0)e lII — X302 (0, 0)EO00Chapter 4. Laser Spectroscopy of NbN^ 138Figure 4.11: Broadband laser-induced fluorescence spectrum of NbN in the 18200-18500cm-1 region, taken with the intracavity assembly removed, and using the dye R110.Chapter 4. Laser Spectroscopy of NbN^ 1394.xiii Analysis of the vibrational sequence bands of the B — X and C — Xsystems4.xiii.A Sequence structures of the B — X and C — X systemsThe origins of the (0,0) vibrational subbands of the three B 3 (1) — X30 transitions havebeen known for some time [38]; a recent study [41] has also located the (1,0) and (1,1)subbands of the 3 4) 2 -3AI. transition which allowed the vibrational frequencies of the 3 4)and 3 ,6, states to be determined. Given these vibrational frequencies and the (0,0) bandpositions, one can easily predict the origins of the (1,0) and (0,1) subbands of the othertwo B — X components. Once the (0,0), (1,0) and (0,1) bands were located, the Av =0, +1 vibrational sequence bands of the B3 4 — X30 transition could be easily identifiedsince subbands in the same sequence always lie about 50 cm -1 apart. Most of the Av = 0sequence bands observed in our Doppler-limited spectra show clear rotational structurewith strong R, Q and P branches, as illustrated by a portion of the 3 (1) 4 — X303 (3, 3)subband in Fig. 4.12. The rather weaker Av = 1 sequence bands are overlapped by thestrong C — X Av = 0 sequence bands, making branch analysis a bit more difficult; theAv = —1 sequence bands of the 3 (1) 2 - 3 A 1 and 34)3 -3A2 transitions lie too low in energy,and only the 3 4)4 — 303 Av = —1 subbands are observed within our wavelength coverage.In all we have identified and rotationally assigned 36 Ay = 0, +1 vibrational sequencesubbands of the B3 (1) — X30 transition; the assigned lines are given in Appendix C. TheAv > 2 vibrational bands are expected to be very weak and have not yet been observedin our laser excitation spectra.The C — X system is about an order of magnitude weaker than the B — X system,and also has very short vibrational sequences and progressions. In fact no lAv I > 1sequence bands of the C 3110 — X30 1 and C311 1 — X30 2 transitions were observed, whileonly two Av = 1 subbands were observed in the C3 11 2 — X303 transition. The Ay = 0Figure 4.12: A portion of the B3(I)4 — X36.3 (3,3) subband at Doppler-limited resolution. The huge hyperfinewidths of the low-J rotational lines are characteristic of the B 3 (I)4 — X303 transition. Weaker high J P lines ofthe (2,2) subband occur also in this region; their assignments are not marked.Chapter 4. Laser Spectroscopy of NbN^ 141sequence bands are relatively stronger; 4 subbands have been identified and rotationallyassigned for the 3112 - 303 transition, while 3 subbands were identified for each of theother two transitions. The assigned lines of the C — X vibrational subbands are alsogiven in Appendix C.4.xiii.B Term values and rotational constants for the vibrational levels ofthe X3 0, B3 4) and C311 statesSince the B — X and C— X systems both involve the X 3 A ground state, it is desirableto perform a grand rotational fit of all the subbands from the two systems to deriveaccurate molecular constants for the three electronic states involved. However since nospin satellite subbands have been observed at high resolution in either the B 34)— X30 orthe C311 — X 3 0 transitions, there are essentially three separate spin-orbit systems: the34)2,3110 -3,61, 34)3,3H1 - 302 and 3 4)4 , 3H2 — 303 systems; each one needs to be treatedseparately. We have therefore fitted the rotational line positions of all the subbands ina single spin-orbit system to the differences of the upper and lower rotational energiesgiven byEj = Tt, + B,J(J + 1) — Dv [J(J + 1)] 2 (4.26)where Ty is the term value of vibrational level v of a given substate, By is the effectiverotational constant and D, the centrifugal distortion correction constant. Followingrecent low resolution measurements of the spin-orbit intervals of the X 30, v = 0 state[44], we give the term values of the three spin-orbit components of the 3 0, v = 0 stateas 0 ( 30 1 ), 400.5 (30 2 ) and 891.0 cm -1 (303 ). The least-squares results for the threerotational fits are given in Tables 4.13, 4.14 and 4.15; the standard deviations are about0.004 cm -1 (120 MHz), which is comparable with the experimental error.Also given in Tables 4.13, 4.14 and 4.15 are the derived vibrational frequencies w e , theChapter 4. Laser Spectroscopy of NbN^ 142Table 4.13: Term values and effective rotational constants of the vibrational levels of theX32 1 , B30 2 and C3110 substates of NbN (in cm -1 ).V 71,12 Bv,f1 1 06 D ,,,0 Derived Parameters0 0.0' 0.50012 (2) 0.458 (16)1 1033.734^(8) 0.49758 (2) 0.475 (13) we = 1042.392 2059.196 (10) 0.49496 (3) 0.459 (15) we x, = 4.223 3076.312 (11) 0.49237 (3) 0.469 (15)X30 1 4 4085.046 (13) 0.48976 (3) 0.483 (20) Be = 0.501425 5085.368 (14) 0.48709 (3) 0.469 (20) a, = 0.002556 6077.184 (19) 0.48445 (5) 0.487 (33) -y, = -0.91 x 10'7 7060.461 (20) 0.48179 (5) 0.506 (35)8 8035.147 (22) 0.47908 (6) 0.507 (36)0 16144.648^(1) 0.49532 (5) 0.497 (19)1 17130.995^(6) 0.49260 (3) 0.511 (15) we = 995.902 18108.850^(9) 0.48985 (2) 0.506 (14) we x, = 4.453 19078.121 (10) 0.48707 (3) 0.501 (14)B302 4 20038.704 (11) 0.48429 (3) 0.511 (16) Be = 0.496665 20990.502 (14) 0.48144 (3) 0.504 (20) a, = 0.002686 21933.290 (17) 0.47860 (4) 0.535 (28) -y, = -0.157 x 10'7 22866.846 (20) 0.47567 (5) 0.522 (36)8 23790.447 (20) 0.47260 (5) 0.598 (34)0 17902.281^(2) 0.49647 (3) 0.507 (16)C3110, 1 18878.458^(8) 0.49284 (3) 0.619 (16) We = 993.182 19837.630 (11) 0.48737 (3) 0.881 (21)^, we x, = 8.500 17908.891 (3) 0.49655 (3) 0.507 (17)C3 110f^1 18885.992 (8) 0.49305 (3) 0.558 (18) co, = 994.472 19845.720 (20) 0.48721 (19) 0.622 (46) we x, = 8.69Standard Deviation=0.0038 cm -1 . Error limits in parentheses are three standarddeviations, in units of the last significant figure quoted. 'Fixed at zero.Chapter 4. Laser Spectroscopy of NbN^ 143Table 4.14: Term values and effective rotational constants of the vibrational levels of theX302 , B303 and 0 311 1 substates of NbN (in cm -1 ).X3A 2v Tv711 Bv,f2 106D,,0 Derived Parameters0 400.5a 0.50172 (2) 0.466 (12) we = 1042.441 1434.556^(5) 0.49913 (2) 0.474 (11) we x e = 4.172 2460.326^(6) 0.49652 (2) 0.475 (12) Be = 0.503013 3477.753^(8) 0.49390 (3) 0.472 (21) a e = 0.002584 4486.794 (11) 0.49128 (5) 0.500 (50) ye = —0.65 x 10-5B3 43v 71,,,0 B,,,0 106D02 Derived Parameters0 16943.466^(3) 0.49581 (2) 0.494 (12) we = 992.981 17927.646^(4) 0.49307 (2) 0.505 (12) we x e = 4.362 18903.235^(6) 0.49028 (2) 0.503 (12) Be = 0.497173 19870.106^(8) 0.48748 (3) 0.504 (20) a, = 0.002724 20828.122 (11) 0.48466 (5) 0.524 (43) -ye = —0.132 x 10 -4C3 /11v Tv,c Bv,c 106/3.„,0 Derived Parameters0 17457.964^(3) 0.49529 (3) 0.494 (14)1 18437.985^(6) 0.49223 (2) 0.510 (12) we = 993.952 19404.082^(7) 0.48864 (3) 0.568 (17) we x e = 6.96Standard Deviation=0.0040 cm -1 . Error limits in parentheses are three standarddeviations, in units of the last significant figure quoted. aValue obtained from separatemeasurements of the X 3A spin-orbit intervals [44], and held constant.Chapter 4. Laser Spectroscopy of NbN^ 144Table 4.15: Term values and effective rotational constants of the vibrational levels of theX303 , B304 and C31I 2 substates of NbN (in cm -1 ).X3A3V 71,0 Bv, 0 1 06 Dv ,0 Derived Parameters0 891.0a 0.50253 (2) 0.473^(7)1 1924.976^(5) 0.49995 (2) 0.480^(6) we = 1042.592 2950.658^(6) 0.49731 (2) 0.477^(6) wex, = 4.223 3967.973 (10) 0.49468 (3) 0.480^(7)4 4976.882 (12) 0.49202 (2) 0.480^(9) Be = 0.503815 5977.335 (14) 0.48933 (3) 0.478 (10) a, = 0.002576 6969.272 (15) 0.48664 (3) 0.481 (11) -ye = -0.109 x 10 -47 7952.635 (16) 0.48391 (3) 0.479 (13)8 8927.73^(32) 0.4830^(8) 1.12^(46)B3 04v Tv,C2 Bv,f/ 106D,,0 Derived Parameters0 17751.383 (18) 0.49634 (4) 0.511 (19)1 18733.539^(4) 0.49355 (2) 0.508^(6) we = 992.342 19706.997^(7) 0.49073 (2) 0.510^(6) we xe = 4.653 20671.610 (10) 0.48788 (2) 0.513^(7)4 21627.219 (13) 0.48500 (2) 0.514^(9) Be = 0.497685 22573.582 (14) 0.48206 (2) 0.517 (10) a, = 0.002726 23510.258 (14) 0.47904 (2) 0.528 (10) ye = -0.225 x 10 -47 24435.835 (15) 0.47572 (3) 0.664 (13)8 25348.65^(59) 0.4718 (12) 1.5^(6)C3I-12V Tv,f) Bv,s-2 106Dv,0 Derived Parameters0 18306.443^(8) 0.49618^(2) 0.506 (7)1 19291.017^(7) 0.49227^(2) 0.440 (9)2 20243.578^(6) 0.48586^(2) 1.104 (8)3 21243.056 (33) 0.48362 (12) 1.47^(8)Standard Deviation=0.0043 cm -1 . Error limits in parentheses are three standarddeviations, in units of the last significant figure quoted. aValue obtained from separatemeasurements of the X 30 spin orbit intervals [44], and held con3tant.Chapter 4. Laser Spectroscopy of NbN^ 145anharmonicity constants wex„ the equilibrium rotational constants Be and the coefficientsa, and -y, for the X 30 and B34 substates. we and we x, are derived from the vibrationalintervals AG,4-1/2 of a given spin-orbit substate according to the energy formula for ananharmonic oscillator [20]:1^1G, = we (v + —2) — wex,(v + —2)2'(4.27)while Be , a, and -y, are obtained by fitting the effective rotational constants B e, of a givensubstate to the expression [20]:1^1By = Be — a e (v + —2) + -ye (v + —2) 2 . (4.28)According to Eq. (4.27), the dissociation energy De of a diatomic molecule can beexpressed in terms of the vibrational frequency, we , and the anharmonicity constant,we x e , as2co,^De = ^ (4.29)4 we x,Using the average co, and we x, values of the X30 and B 343. states from Tables 4.13— 4.15,the dissociation energies of the NbN molecule in these two electronic states are calculatedto be^De(X3 A) = 8.02 e.V.,^De (B30) = 6.85 e.V.^(4.30)The calculated De values for NbN probably overestimate the actual values since Eq.(4.27), from which expression (4.29) is derived, usually overestimates the energies of highvibrational levels [20]. If the true D, is taken as 20% less than the calculated value assuggested in Ref. [20], the ground state dissociation energy of NbN will be about 6.4e.V, which is very close to the theoretical value given by Langhoff and Bauschlicher [58].It is very clear that there are vibrational perturbations in the excited vibrational levelsof the C311 state since the term values of the C 31I2 levels do not obey formula (4.27) atChapter 4. Laser Spectroscopy of NbN^ 146Table 4.16: Comparisons of the observed and calculated term values of the 0 311 2 vibra-tional levels (in cm -1 ).v Toas^Tool Tobs — Teal0 18306.4431 19291.017^19291.4 —0.42 20243.578^20267.6 —24.03 21243.056^21235.0 8.14 22193.65 23143.4Tea l, t, = Tobs,v=0 + We V—WeX e[( V + -1) 2 — 0.25]co, = 993.8, as 3 110wexe = 4.4, as 311all. Perturbations to the C3110 and C311 1 vibrational levels are not so obvious, althoughtheir large anharmonicity constants, 8.60 and 6.96 cm -1 , suggest some irregularity in thetwo substates. Table 4.16 compares the observed and "deperturbed" term values of the0311 2 vibrational levels. As can be seen, the v = 2 and 3 levels are clearly perturbed,with v = 2 being pushed down by 24 cm -1 and v = 3 pushed up by 8 cm -1 . The oppositedirections of the perturbations to the v = 2 and 3 levels indicate that the perturbingstate has a smaller vibrational frequency than the C 311 2 substate. Dunn first proposed[59] that the C3 11 2 , v = 2 level was perturbed by what he called the Y electronic state,lying about 100 cm -1 above, following his identification of a subband at 19450 cm -1 asthe (1,0) band of a transition involving the X 30 2 , v = 0 level; however he could notdetermined the nature of the Y perturbing state from his emission spectra. We now haveevidence that the perturbing state is the SZ = 2 spin-orbit component of a new 311 stateChapter 4. Laser Spectroscopy of NbN^ 147from the electron configuration (5sa) 1 (4dir) 1 . The analysis of this new electronic stateand its subbands follows.4.xiv The new (53a) 1 (4c170 1 3112 — (4d5) 1 (530) 1 X303 subbands at 18419 and18276 cm -14.xiv.A Rotational analysisTwo subbands with origins at 18419 and 18276 cm" have been assigned to a new(5300 1 (4c/7) 1 Y3 11 2 — (46) 1 (530) 1 X3 .6,3 transition. The 18419 cm -1 subband is com-paratively strong, with single R, Q and P branches; part of it is shown in Fig. 4.13,along with various other overlapping subbands; the 18276 cm" subband is much weakerand is overlapped by the strong (2,1) band of the C 311 2 — X303 transition and two othersubbands. The Y — X subbands show R heads at 18431 and 18284 cm" respectively.Rotational assignments are straightforward to make, except for the blended lines nearthe R heads. Some high J rotational lines are clearly perturbed, since J' > 30 linesin the 18419 cm -1 subband are shifted to lower frequency and double lines appear atJ' = 36, while in the 18276 cm -1 subband double lines appear at J' = 29 and 30. Theperturbation in the 18419 cm -1 subband is puzzling since the branch structure suddenlycuts off at J' = 37 even though two quantum numbers back, at J' = 35, the R(34), Q(35)and P(36) rotational lines are still comparatively strong.The low J rotational lines of the two Y — X subbands are all broadened by hyperfineeffects; the broadening of the first few rotational lines is particularly severe so that nounambiguous first lines of branches can be recognized; this prevents direct assignmentsof the C2 values of the combining states. The lower states were eventually assigned bycomparing the lower state combination differences with those obtained in the rotationalanalysis of the B — X system. It was found that the lower state of the 18419 cm'NbN•Int11111 I II 1 01 no — X3 A1P^1 1^111 ^Ii^IIR 1 11 2 — X303(v% 2), •^1r,4',2)^I^IIIIIIN1111111 .11^I^ill! ^ll I II^1 1 1^11111^III II^11^IIII :114101111 1111 11^Jill III 1^i c3112-x3A30,0) ft35^I^33 11^125^1 1 1 1 " 1^1^11^1 15 1^1^1 1^! I 111 1^Y3112 - X3A3(1, I 1) QI^II^10 I^0 1^1 20 ^P 5^1 1^11814 tN‘fivt%IIJQ5 1I 1^r^1 1^1 1^I I125IP 13 15112 — .106,3(v% 3)^204 4^4/k,^tiek^°aft'111111 kJ! n ul l' ti ll^Li^11^1^III II^II II I "I 1pQ C1 31:1 ? Ti1: 3(11 'P ) ^1 1^P PI P I^IN^I III^II 'I^I HI 1 .13 115 11 11 1 20^1 (III^25^1^11 II^r^In^I^I^1 35,^I I^I^II II^30^I^I^II^I III^35^11 1 11^I^1 1^1^11?'I11 25Figure 4.13: Laser excitation spectrum of NbN in the region 18384 —18433 cm -1 . The two strong subbands areY3 11 2 — X303 (1,1), with origin at 18419 cm -1 , and C3112 — X303 (1,0) at 18400 cm -1 . There are also threeoverlapping charge transfer subbands in the region, which will be discussed in Section 4.xv.QP1 I^313 1^I^I^I271610 — x3A3 2)II30^113 — X3L3(v', 3)^P^20111IIr11 351II351II■4=-00Chapter 4. Laser Spectroscopy of NbN^ 149subband is the v = 1 level of the X 303 state, while that of the 18276 cm -1 sub-band is theX3L3 , v = 4 level. The lower state assignments are consistent with the large linewidthsobserved in the low J lines of the two subbands since X 303 is known to have largehyperfine splittings. The vibrational assignments of the lower levels were confirmed bywavelength-resolved fluorescence studies which show emission from the selectively excitedY levels to various lower vibrational levels of the X 303 state to the higher frequency sideof the excited subbands. Emission to a single vibrational level of the X state alwaysdisplays stronger P lines than R lines in resolved fluorescence; this indicates that theY — X transition is a AC/ = —1 transition according to the HOnl-London intensity factors[60]. The fact that both Y — X subbands show comparatively strong Q branches in thelaser excitation spectra indicates a perpendicularly polarized transition. If Hund's case(a) selection rules are assumed to apply to the new transition, the above evidence pointsto a 3H2 — 3A3 transition with the two subbands being the Y — X (vi, 1) and Y — X (v4,4)vibrational bands respectively. The vibrational numberings of the upper states cannot bedirectly assigned in the present work, but an earlier 15 N isotope study of the 19451 cm -1band [59], which has the same upper state as the 18419 cm -1 band, suggested v' = 1,which gives the vibrational assignment of the 18419 cm -1 subband as (1,1). From theupper state rotational constants of the 18276 and 18419 cm -1 subbands given in Table4.17 and, assuming the rotational constants of the vibrational levels decrease at the samerate as the nearby 0311 2 state, the upper state vibrational quantum number of the 18276cm -1 subband was found to be 5. Thus the 18276 cm -1 subband is assigned as the (5,4)vibrational band.Rotational constants of the upper and lower states were derived from the line posi-tions of the 18419 and 18276 cm -1 subbands by the method of linear least squares; theconstants obtained are given in Table 4.17, while the line frequencies are listed in TableChapter 4. Laser Spectroscopy of NbN^ 150Table 4.17: Rotational constants derived from the 18419 and 18276 cm -1 subbands ofNbN (in cm -1 ).Subband vo Be f f 106 D'e f f^Beef f 106 D i f f3 1123112— 303(1, 1)a- 303(5, 4)c18418.84918276.027(6)(12)0.48113 (5)a = 0.00980.46453 (24)1.45 (3)^0.49985—0.48 (13P.49201(4)(25)0.44860.47 (13)cr = 0.0086Error limits are three standard deviations, in units of the last significant figure quoted.aLines with J' > 30 are perturbed and thus excluded from the fit. 'Value fixed. V' =28, 29 and 30 lines are not included because of perturbations.4.18. Only the unperturbed rotational lines are included in the least squares fits. As ex-pected, the lower state B constants are essentially identical to those of the X 3L3 , v = 1and v = 4 levels given in Table 4.15. By adding the energies of the v = 1 and v = 4 levelsof the X303 state to the measured subband oringins, we obtain the vibrational energiesof the new Y3 11 2 substate to be 20343.8 cm -1 for v = 1 and 23253 cm -1 for v = 5.4.xiv.B The electrostatic perturbation between the Y311 2 and C311 2 statesA difficulty with the assignment of Y as a triplet state is that its other two spin-orbit components have not been observed in either emission or laser excitation spectra.Also one can question the upper state vibrational assignments of the 18419 cm -1 and18276 cm -1 subbands since the v = 0 level has not been seen, nor have any othervibrational levels besides the two observed Y levels. The explanation for these missingsubbands can be found in the mechanism of the interaction between the Y state and theChapter 4. Laser Spectroscopy of NbN^ 151Table 4.18: Line positions of the Y3 11 2 - X303 (1, 1) and (5,4) bands of NbN (in cm -1 ).J ,,Y 3112R- X343(1,1)Q P7 18411.093*8 18426.16 * 18417.517* 18409.8129 18417.179 18408.52010 18427.364* 18416.791* 18407.19 *11 18427.914* 18416.384* 18405.79012 18415.888* 18404.37013 18428.867 18415.400* 18402.920*14 18414.867* 18401.41415 18428.680* 18414.293 18399.880*16 18430.000* 18413.678* 18398.305*17 18430.300* 18413.021 18396.69118 18430.575 18412.333 18395.04819 18411.585 18393.355*20 18430.950* 18410.808 18391.59521 18431.107 18410.003* 18389.840*22 18409.121 18388.01523 18408.216* 18386.15024 18431.218* 18407.269* 18384.25025 18406.271 18382.30026 18405.226 18380.30627 18430.950* 18404.132 18378.24628 18430.757 18402.990* 18376.17929 18430.466 18401.791 18374.03730 18430.123 * 18400.550 18371.83931 18429.689** 18399.177 * 18369.58532 18429.179** 18397.763** 18367.261 *33 18428.494** 18396.232 * 18364.826 *34 18427.658 * 18394.576** 18362.304 *35 18426.539** 18392.740 * 18359.652 *35 18426.492** 18392.740 * 18359.652 *36 18390.644 * 18356.841 *36 18390.599 * 18356.841 *37 18353.753 **.blended. *=perturbed.Chapter 4. Laser Spectroscopy of NbN^ 152Table 4.18, ContinuedJ"173 112 - X343(5, 4)Q P11 18272.410 18262.19812 18271.77 * 18260.62 *13 18271.056 18258.98014 18270.298* 18257.296*15 18269.480* 18255.55616 18268.623 18253.74617 18267.699 18251.910*18 18266.726 18249.99819 18265.710* 18248.060*20 18264.642 18246.07021 18283.994* 18263.541 18244.00722 18283.771* 18262.367* 18241.89823 18283.470* 18261.140 18239.76124 18259.882 18237.54425 18282.757* 18258.558 18235.296*26 18282.314 18257.197* 18233.00127 18281.818** 18255.794 18230.65628 18281.332** 18254.269 * 18228.26728 18281.17 ** 18254.269 * 18228.26729 18280.802 18252.830 * 18225.820 *29 18252.673 * 18225.820 *30 18251.341 * 18223.353 *30 18251.289 * 18223.197 *31 18279.480 18249.708 18220.888 *31 18279.480 18249.708 18220.835 *32 18278.791 18248.064* 18218.27333 18278.064 18246.386 18215.66034 18277.248* 18244.667 18213.00135 18276.433 18242.902* 18210.30236 18241.097 18207.548*.blended. *.perturbed.Chapter 4. Laser Spectroscopy of NbN^ 153(4d6) 1 (4dir) 1 C3II state. The (5so- ) 1 (4dr) 1 Y3 11—(4d8) 1 (5su) 1 X30 electronic transitionis believed to be very weak since it is basically a metal d — d transition, and it appearsthat the transition only becomes observable when the vibrational levels of the Y statehappen to be mixed with the vibrational levels of the 0 311 2 substate so that transitionmoment is transferred from the intense C — X system to the Y — X system. This Y — Cmixing can be clearly seen in the resolved fluorescence spectrum shown in Fig. 4.14. Forthis spectrum the laser was pumping the Q(17) line of the Y 3 I12 — X303 (1, 1) subbandat 18413.0 cm -1 ; the resolved fluorescence near 5770 A shows three sharp R, Q, andP lines of the Y — X (1, 2) subband emitted from the J' = 17 level together with theconsiderably stronger branch stucture of the C 3112 — X303 (2, 2) transition, which isinduced by collisional effects and state mixing between the Y3 11 2 , v = 1 and C 3 11 2 , v = 2levels.The interaction between electronic states of the same symmetry from different electronconfigurations is called an electrostatic perturbation, and is caused by the matrix elementof the electrostatic Hamiltonian [51]:< Y31I, v' Hei C311, v > = H, < v' Iv > ,^(4.31)where the electrostatic parameter He is given byHe = < Y3 I1 I Het I C3Il > .^ (4.32)The effect of the matrix element (4.31) (denoted H12 hereafter) is to shift the energies ofthe interacting states in opposite directions by an amount 6E, given by [51]:sE = ( AE0 ) 2^AE°+ Hi222 2 (4.33)where AE° is the zero-order energy separation of the two states. From the observedenergy positions of the Y 3 11 2 , v = 1 and C3 11 2 , v = 2 levels and the calculated energyLaser pumps Q(17) of Y 3112 — X303 (1, 1)Q BranchY3112 — X363 (1, 2)Q(17)R(16)Chapter 4. Laser Spectroscopy of NbN^ 154Figure 4.14: Laser-induced resonance fluorescence of the Y 3 11 2 — X303 (1,2) band, ac-companied by the collisionally induced C 31-12 — X303 (2,2) emission (near 5770 A).Chapter 4. Laser Spectroscopy of NbN^ 155shift of the C3II 2 level given in Table 4.16, the zero-order separation of the two levels iscalculated to be about 50 cm -1 . Substituting the values of AE° and SE into Eq. (4.33),we obtain the perturbation matrix element H12 = He < v = 11V = 2 > as 42 cm". Ifthe eigenfunctions of the two mixed states are taken asC1 I 1 > + V1 — C? 1 2 >TO — C?1 1 > + CO > ,(4.34)(4.35)the square of the coefficient C 1 , which represents the contribution of the unperturbedstate 11 > to the mixed state I+ >, can be written asC? = -2- [1 + v(AE0/2)2 + ,H12AEO/2 ^]1From the values of AE° and H12, it is found that the observed Y3 II2 , v = 1 state contains22% of the character of the C311 2 , v = 2 state. Thus it becomes clear that the Y 3 II2 , v = 1level is heavily mixed with the v = 2 level of the C3II 2 substate. Application of the samearguments to the mixed Y3 II 2 , v = 2 and C3 1-1 2 , v = 3 levels, where the C level is shiftedup by 8.1 cm", gives only 5% C 3II 2 character to the Y3 II2 , v = 2 level. The interactionbetween the v = 0 level of the Y3II 2 state and C3II2 , v = 1 level is even smaller asjudged by the very small shift of the C 311 2 , v = 1 level shown in Table 4.16. As forthe higher vibrational levels of the Y3II 2 state, the v = 3 and 4 levels do not lie closeto any C 3 II2 level, but the v = 5 level is expected to lie about 100 cm' above theC3II 2 , v = 5 level, according to the estimated vibrational frequency of the Y state (-- , 750cm"), and is therefore believed to interact strongly with the C 3 II 2 , v = 5 level. From theabove discussion, it becomes clear that although the Y — X transition is intrinsically veryweak, transitions involving the Y 3 11 2 , v = 1 and 5 vibrational levels become comparativelystrong through intensity borrowings from the strong C—X transition because these two Ylevels are heavily mixed with the C vibrational levels. Other vibrational levels of the Y3 II2 (4.36)Chapter 4. Laser Spectroscopy of NbN^ 156state do not interact strongly with the C state, therefore the Y — X transitions remainweak and difficult to observe. Similarly the vibrational levels of the Y 3 I10 and Y311 1 spinsubstate probably do not lie close to the vibrational levels of the other C 311 substatesand therefore have small interactions with the C state. The smaller Y — C mixings,compounded by the weakness of the C3 110 — X 30 1 and C311 1 — X302 transitions, areprobably the main reasons why we have not observed the Y 3 110 —X30 1 and Y311 1 — X322transitions.In our experiments on collisional transfer we find that collisional transfer follows areasonably strict selection rule AS2 = 0, and that it is particularly efficient if the transferoccurs between states with closely similar properties. These observations are consistentwith the very great intensity of the collisionally-induced C 3 1I2 — X3L3 (2, 2) subband inFig. 4.14, which is actually stronger than the resonance transition Y 3 I12 — X303 (1, 2)itself.4.xiv.0 The rotational and hyperfine structure of the (5sa) 1 (4dir) 1 Y3 1-1 ex-cited state.Rotational constantAs shown in Table 4.1 7, the new Y 311 electronic state has B '-, 0.48 cm -1 , whichis smaller than B(v = 0) for any known electronic states of NbN other than a chargetransfer state. The possibility that the Y state represents excited vibrational levels ofthe known B 3 (I) or C3 I1 state can be eliminated since none of the vibrational term valuesof the B and C states given in Tables 4.13 — 4.15 match those of the Y 3 II2 , v = 1 andv=5 levels. Therefore, from its B values, it seems certain that the Y state must be oneof the two unseen triplet electronic states, (5s0) 1 (4d7r) 1 3H or (4d6) 1 (4da) 1 30, that cangive rise to electronic transitions from the X 3 0 ground state by promotion of the 46 orChapter 4. Laser Spectroscopy of NbN^ 1575so electron to the 4d7r or 4da orbital. Since the 18419 and 18276 cm -1 subbands areperpendicularly polarized, judging from the comparatively strong Q branches, they areonly consistent with the (5sa) 1 (4dir) 1 3H — (4d6) 1 (5sol 1 X30 transition.Hyperfine structure of the Y 3H2 substateIf the Y state is indeed the 3H state coming from the (5so - ) 1 (4d7r) 1 electron config-uration, it should show comparatively large hyperfine splittings since the 5sa unpairedelectron is in a molecular orbital derived from the metal 5s orbital, which is known tohave a large positive Fermi contact interaction. In what follows, we will show that a pos-itive Fermi contact parameter can be derived for the new Y 3II state from the rotationallinewidths of the Y3H 2 — X303 (1, 1) subband at 18419 cm -1 . The Y3H2 — X303 (5, 4)subband at 18276 cm -1 shows similar linewidths to the 18419 cm -1 subband, but becauseits rotational lines are much weaker, it will not be included in the discussion.Shown in the second column of Table 4.19 are the rotational linewidths (FWHM)of some clearly unblended low-J P lines of the 18419 cm -1 subband. Because of theunresolved hyperfine structure within a rotational line, the observed linewidth is thesum of the hyperfine width and the contributions from Doppler and pressure broadeningeffects. In our Doppler-limited spectra, the Doppler broadening is about 750 MHz (0.025cm -1 ), while pressure broadening is negligible. Subtracting the Doppler width from therotational linewidths, we obtain the hyperfine widths of the individual rotational lines aslisted in the third column of Table 4.19. Column four lists the hyperfine splittings of therotational levels J = 6 — 14 of the X 303 , v = 1 state obtained from hyperfine analysisof the C3 H2 — X3A3 (1, 1) subband in Ref. [42]. The hyperfine width of a rotational lineis the difference of the hyperfine splittings of the upper and lower rotational levels. Infirst approximation, the hyperfine energy of a case (a) electronic state is given by theChapter 4. Laser Spectroscopy of NbN^158Table 4.19: Calculation of the hyperfine widths of the Y 3 I12 levels.JLinewidthP(J)HF linewidtha =AEhfs(X, J)— AEhf s (Y, J-1)AEhfs(J) AEhfs(Y, J)X3 6,3 , v = 1b 1'314 AEhf s (X, J)6 0.4578 0.2448 0.5357 0.174 0.149 0.3938 0.2098 0.5338 0.160 0.135 0.3448 0.1829 0.5309 0.148 0.123 0.305910 0.2742 0.1419 0.51811 0.131 0.106 0.2479 0.1336 0.53912 0.117 0.092 0.225613 0.2064 0.1068 0.51714 0.108 0.083 0.1898All values in cm -1 . a(HF linewidth)=(linewidth) —(Doppler width, 0.025 cm -1 ). bValuesobtained from Ref. [42]. cAEhf s (Y, J) = AEhf s (X, J^—(HF linewidth of P(J + 1)).diagonal element of the hyperfine Hamiltonian,h [F(F + 1) — /(/ 1) — J(J + 1)]Ehf s = 2J(J + 1)where the hyperfine parameter h is given byh = aA (b c)E.Thus the hyperfine splitting of the rotational level J can be written asS2 h [Fmax (Fmax + — &in (Pm in + 1)] AEhf s^ 2J(J + 1)(4.37)(4.38)(4.39)Obviously AEhf, will be negative if S1 h is negative. If the CI h values of the upper andlower levels happen to have opposite signs, the hyperfine width of a rotational line willactually be the sum of the hyperfine splittings of the combining levels; in this case, theP lines will always have larger hyperfine widths than the Q and R lines with the samevalues of J". Since the P lines in the Y3 11 2 — X30 3 subband are narrower than theChapter 4. Laser Spectroscopy of NbN^ 159corresponding Q lines with the same J" values (R lines cannot be used for comparisonsbecause of heavy blendings), it is concluded that the Y 3 11 2 substate must have a positiveh, and smaller hyperfine splitting than the X 303 state. Therefore the hyperfine widthsof the P rotational lines of the Y — X subband equal [AEhf s (X, J) — AEhfs (Y,J — 1)]; inother words, the hyperfine splittings of the Y state, AEh f s (Y, J — 1), are the differencesof the hyperfine splittings of the lower state, AEhf,(X, J) (column 4 of Table 4.19), andthe measured hyperfine widths of the P lines (column 3). The calculated values forAEhfs (Y) are listed in the second last column of the Table. From Eq. (4.39), we havefor the same J values:AEhf,(Y3112, J) _ 2 h(Y3 11 2 ) (4.40)AEh f s (X3 A 3 , J) — 3 h(X303 ) .Therefore from the ratios of the upper and lower state hyperfine splittings at the sameJ, given in the last column of Table 4.19, and taking the h constant of the X303 , v = 1substate as 0.1115 cm' from Ref. [42], the h constant of the Y3H2 substate is calculatedto be about 0.089 cm -1 , using Eq, (4.40). Substituting the h constant into Eq. (4.38)and taking the parameter aA (from the aI • L term in the nuclear magnetic Hamiltonian),asaA (58o-4dr; Y311) = a, = 0.0144 cm -1from Eq. (4.15), we have(b + c) = h — aA = 0.075 cm'.^ (4.41)The nuclear dipolar parameter c is normally quite small (< 0.01 cm -1 ), so that it issafe to say that the large value of (b + c) obtained in Eq. (4.41) is mainly due toa large positive Fermi contact parameter b, which is only consistent with the electronconfiguration (4d6) 1 (58a) 1 for the new Y311 state, where an unpaired metal 5s electronis present. Since both the (5sa) 1 (4dir) 1 Y3H and (46) 1 (580) 1 X3 A states involve theChapter 4. Laser Spectroscopy of NbN^ 160same 5sa unpaired electron and therefore have similar Fermi contact interactions, it isno surprise that the values of the (b + c) hyperfine parameters of the two electronic statesare quite close (0.075 vs 0.08642 cm -1 ); the difference is probably attributable to thedifference of the nuclear dipolar parameters c between the 7r and 6 electrons, as shownin Eq. (4.19).4.xv The charge transfer transitions4.xv.A Branch analysisThe new charge transfer subbands of NbN are located in two relatively isolated re-gions; 18200-18350 and 18350-18500 cm -1 . We shall consider the two subbands in the18200-18350 cm -1 region first.The NbN spectrum in the region 18200-18350 cm -1 is fairly crowded because itcontains the strong C 311 2 — X303 (2,1) subband near 18318 cm -1 , the Y311 2 — X303 (5, 4)subband near 18276 cm -1 and two comparatively weak charge transfer subbands withorigins at 18285 and 18298 cm -1 . A portion of the Doppler-limited spectrum, with therotational assignments marked is shown in Fig. 4.15. Because of the overlapping fromthe other two subbands, the branch structures of the two charge transfer subbands arenot easy to recognize at first sight, especially in the extremely crowded region at thelow frequency end of the spectrum. Eventually each subband was found to consist ofa comparatively strong Q branch, where the lines could be observed to J" 35, andrelatively weak R and P branches with lines observable to J" 25. The presence ofstrong Q branches indicates that both subbands result from perpendicularly-polarizedtransitions with Alt = +1. The rotational quantum numbers were assigned to thespectral lines using lower state combination differences, initially assuming B" 0.5cm -1 . The branch structures of the 18285 and 18298 cm -1 subbands were found to beNbN41•4 910 OfQ1 1 1 I 11^1 1^III^I 111 11^11 11^III^1111^III^I^I^II^1)11111 111^11 p1111 1 1 c3n2—x3A3(2,1)1^I 40 8^I I^11^111111^IIv,(11 20^I^I^I^11^1111111IR 20^25^• 30.5OiiI0^I^4 oi\NYIA1 i t^1 1^1^I 1 111 1 II 1 111C3112—x3A3(2 1)1 II IIPao^11 no l'ixjizi l:I1 1 1 1 11^II 11 1^35 11 IIQ^I^I^I^I^1 1^II^1 2.5^(III^III 1^11^III20Q 10Y3 11 2 — x3 ,13 (5,4) P 15 25^i^1^1^1 30^ 1 15 1810igure 4.15: Laser excitation spectrum of NbN in the region 18249-18303 cm -1 . The three overlapping subbandsre Y3 I12 — X303 (5,4), with an R head near 18284 cm -1 , and the two A-components of the 110 — X30 1 (v', 4)barge transfer subbands, with R heads near 18303 and 18290 cm -1 respectively. Some comparatively strongigh J lines spread over the whole spectrum belong to the C3 11 2 — X303 (2, 1) subband.I^I^ks^I^I110 X3 A1(1/, 4)11^iR^15 201111^ , 1 11^I^II^I 1 I^1 -` 15^I^1II^II^11 11^1 25 11111 11 III 1111^1111^1 I^III^I^III^I^111111119^1^1^1^1 1^1 5 11 ^I^JI^1^11111^1 III 11^1^riip^11 p I25 I 1139 ^1 10Igi^ 114I I X,3411,(V11) ^Q 6^1 1^10 1^1 ^13R^15 2° 111 I I I 1 313ri^X303(5,4)^134PQI^1 15^I^111^11_11^1111^1111^III^NI^29.^11^1^II^1 11 1^II^I^111^1.24I^111 -0^15 1^1^1^1^1 1^1 1 20 II^Ho.— X 3A1(1/0)^I 1^1^25 11^II^I^I^II^1^28^I 1PIII^I^ql II^11III^1^III^I^111 1^III^1^11001 441)14Amiii1 41)4:4A4Ii^1^1^1111 1^114^1111^11111^11 ^11111^1111^11^1111^1111 1^11 I^HI1 1^1 3° 1 11 l^III^11^1^III^I^NI^I^11 3?Chapter 4. Laser Spectroscopy of NbN^ 162very similar, with both forming R heads at J" = 11. By comparing the lower statecombination differences of the two subbands with those of the X 30 substates obtainedfrom rotational analysis of the B — X and C — X systems discussed earlier, the two lowerstates were identified as the same v = 4 level of the 30 1 substate. Again wavelength-resolved fluorescence was used to help with the vibrational assignment of the lower states.The fluorescence patterns also indicate that both subbands are ASZ = —1 transitions,since the P lines are stronger than the R lines. As a result, the 18298 and 18284 cm -1bands were assigned as the two A-components of an CV = 0 — X30 1 (v', 4) transition. Thelinewidths of the 18285 and 18298 cm -1 subbands are quite modest at very low J anddecrease rapidly with increasing J, passing through a minimum at J = 15, in a mannercharacteristic of the hyperfine structure reversal in the X 30 1 state near J = 15 [40].Assuming the upper states follow Hund's case (a) coupling, like the X 3 0 lower state, theupper states of the two new subbands must be either 3 110 or 5 110 states according to thecase (a) selection rules, AE = 0 and AA = Aft. The spin multiplicities of the new IIstates will not be known until all the spin components have been accounted for.The Doppler-limited spectrum of NbN in the 18350-18500 cm -1 region is very com-plicated, with as many as five different subbands overlapping each other at a givenfrequency. A portion of the Doppler-limited spectrum in this region has been shown inFig. 4.13. Three relatively strong subbands, with origins at 18400, 18419 and 18457cm -1 , can be recognized rather easily in the spectra; they are respectively the knownC311 2 — X303 (1, 0), Y3 112 — X3A3 (1, 1) and e 1 11 — X3 L2 (0, 0) subbands discussed ear-lier. Six more subbands were also identified in this region after extensive analysis; theirorigins are measured to be 18432, 18441, 18442, 18443, 18488 and 18499 cm -1 , and theybelong to new NbN charge transfer transitions. Again the new electronic transitionsare identified as perpendicular transitions since all the subbands show comparativelystrong Q branches. The two relatively isolated subbands at 18488 and 18499 cm -1 haveChapter 4. Laser Spectroscopy of NbN^ 163very similar branch structures, and are reminiscent of the two 110 — X'A i subbands at18285 and 18298 cm -1 . The branch structures of the other four subbands are highlyirregular with rotational perturbations observed in all the upper states. In the 18442cm-1 subband, only rotational lines between J" = 25 and 42 are observed; all linesshow very small linewidths and the line spacing in each branch is of the order of 1.5-4.0cm-1 . The 18432 cm -1 subband behaves more regularly, though only rotational lineswith J' > 15 are observed, with comparatively broad linewidth. The 18441 and 18443cm -1 subbands have almost identical branch structures: both subbands have very shortbranches with only J" = 8 — 18 lines present, both show R heads at J" = 13 and mini-mum linewidths at J" = 15. These branch structures are characteristic of a 11 0 — X301transition. Again using evidence from lower state combination differences, resolved flu-orescence and hyperfine widths, we have assigned the 18488 and 18499 cm -1 subbandsas the two A-components of a 110 — r A i (v', 4) transition, the 18442 cm -1 subband as11 2 — X303 (v', 2), the 18432 cm -1 subband as 11 2 — X303 (vi, 3), and the 18441, 18443cm-1 subbands as a Ho — X301 (v', 2) transition. Because of the complexity of the spec-tra in the region where the new subbands are observed and the presence of rotationalperturbations, rotational assignments of the new subbands are not straightforward. Usu-ally several resolved fluorescence spectra needed to be taken for each branch in orderto establish the rotational numbering and the branch type. Eventually almost all therotational lines in the 18200 — 18500 cm -1 region were assigned either to the new chargetransfer subbands or to other known NbN subbands. The assigned rotational lines of thecharge transfer subbands appear in Appendix D.Chapter 4. Laser Spectroscopy of NbN^ 1644.xv.B Determination of rotational constantsRotational constants were derived from the new subbands by least-squares fitting ofthe line frequencies of each subband to the energy expressionV = v0 + [Bef f f(f +1) — D'ef ff2 (f +1) 2 J—[B:f fJ"(J" + 1) — D'," ffP2 (J" + 1) 2 ]^(4.42)In Eq. (4.42) vo is the band origin for zero rotation, while Beff and De f f are respectivelythe effective rotational and centrifugal distortion constants. The rotational constants forthe various subbands are given in Table 4.20. As can be seen in the table, the upperstates of the new subbands all have small rotational constants B compared to the lowerstate, characteristic of charge transfer transitions.4.xv.0 DiscussionAs can be seen in Table 4.20, the new NbN transitions correspond to decreases in therotational constants B of about 10%, which translates as increases of about 5% in bondlength. Since other known electronic transitions of NbN between the metal-centeredmolecular orbitals produce at most a 2% increase in bond length, the 5% bond lengthincrease must result from promotion of an electron from a nitrogen-centred bondingmolecular orbital, such as the 2pr or 2su orbital as shown in Fig. 4.10, to a Nb-centrednon-bonding or anti-bonding orbital. Such electronic transitions are referred to as chargetransfer transitions, since electron charge is transferred from the ligand (N) to the metal(Nb); the corresponding electronic states are called charge transfer states. Fig. 4.16 showsthe observed charge transfer transitions from the X 30 ground state and the energies ofthe charge transfer states derived from the band origins given in Table 4.20. At firstsight, the five new electronic states look like the successive vibrational levels of the HoChapter 4. Laser Spectroscopy of NbN^ 165Table 4.20: Rotational constants of the charge transfer subbands of NbN (in cm -1 ).Subband vo Be f f 106D'eff Reif f^106Def fII0 - 3A 1 (v', 4) 18284.778 (13) 0.44904 (55) 2.02 (72) 0.49024 (56) 1.04 (73)Ho - 301(v/, 4) 18298.205 (8) 0.44749 (33) 1.55 (33) 0.48965 (34) 0.38 (34)II0 - 3A 1 (v',3) 18488.425 (28) 0.45715 (74) 1.83 (95) 0.49198 (77)-0.15 (102)110 - 30 1 (v', 3) 18499.262 (52) 0.4568 (18) 0.85 (203) 0.4913 (17) -0.45 (185)110 - 3A 1 (vi, 2)a 18441.023 (71) 0.4569 (13) 11.5 (30) 0.4948 (15)^0.61 (352)110 - 3 ,6, 1 (v', 2)a 18443.112 (44) 0.4628 (9) 10.4 (22) 0.4946 (10) -0.36 (230)11 2 - 3/.3 (v', 3) 6 18432.268 (21) 0.45991 (37)-0.45 (30) 0.49461 (34) 0.32 (26)H2 - 303(2) 1 , 2)c 18441.94 (30) 0.4582 (13)^0.96 (58) 0.4976 (13)^0.58 (57)o- ranges from 0.009 to 0.021 cm -1 . Error limits in parentheses are three standard devi-ations, in units of the last significant figure quoted. aOnly J" < 20 lines are found, linesare very sharp near J" = 15, and the upper state is perturbed. b Only J' > 15 lines areobserved, upper state is perturbed. Only J' > 25 lines are observed, lines are sharp;upper state is perturbed.22383.3221575.6^ 2369.821564.720502.320500.21844218494182914085.13076.32059.21033.22240021392.8tes4977.218442184323968.32950.91925.3891.3v = 4v = 3v = 2v = 1v = 0110±,v + 1I10±, '1,harge Transfer Stav = 4v = 3^ — 2^ — 1Chapter 4. Laser Spectroscopy of NbN^ 1660.0 ^ — 0 X3A1^ X3A3Figure 4.16: Energy levels of the charge transfer states and transitions of NbN. Valuesin cm -1 .Chapter 4. Laser Spectroscopy of NbN^ 167and 11 2 spin components of a quintet or triplet state; however close examination of theirenergy separations and rotational constants shows that they cannot all belong to thesame electronic state. The two II 0 states at 21570 and 22376 cm -1 may belong to thesame electronic state since their energy separation of 806 cm -1 is about correct for thevibrational interval of a charge transfer state, while their transitions to the X30 1 lowerstate show similar branch structures with about the same A-doublings. The other 11 0level, at 20501 cm -1 , definitely appears to belong to a different electronic state sinceit lies more than 1000 cm -1 below the 21570 cm -1 level and has a much smaller A-doubling (2 cm -1 ). The two 11 2 levels shown in Fig. 4.16 seem to be unrelated sincetheir energy separation (about 1007 cm -1 ) is too large to be the vibrational spacing ofa charge transfer state. Whether the observed 11 2 levels are related to the 110 levels isimpossible to tell for certain from the available information. If the II states are assumedto arise from the N(2/371- )3 Nb(4d8) 2 (5so) 1 configuration, which was suggested by a recenttheoretical calculation [58] as giving rise to the lowest lying charge transfer states, it canbe argued that the 110 level at 21570 cm -1 and the 11 2 level at 21393 cm -1 may belongto the same vibronic state. The reasoning is that both levels have similar rotationalconstants (see Table 4.20), and the inverted energy order of the II state (11 0 lying above11 2 ) is consistent with the r 382a 1 configuration. Judging by their rather different Bconstants, the 22376 (11 0 ) and 22400 (11 2 ) cm -1 levels are probably unrelated. Thereforeit appears that the observed electronic substates might belong to three different chargetransfer states of H symmetry; these new states might be the single 511 state and two 311states corning from the N (2/371-) 3 Nb (4c18) 2 (5so) 1 electron configuration. If so, all four Helectronic states should show comparatively large nuclear hyperfine splittings because thepresence of the 5so unpaired electron should give rise to large Fermi contact interactions.Although no hyperfine data concerning the charge transfer subbands have been obtained,one can roughly estimate the hyperfine widths of the charge transfer states from theChapter 4. Laser Spectroscopy of NbN^ 168linewidths of the observed subbands and the known hyperfine widths of the lower states.Of course these arguments only apply to II 2 — X303 subbands because the Ho substatesof the 110 — X304 subbands must have essentially zero hyperfine widths since they haveii = 0. Although only high J (> 25) lines are observed in the 11 2 — X30 subbandat 18442 cm -1 , the extremely small linewidths suggest that the hyperfine splittings inthe 112 state are essentially identical to those of the X303 state; this indicates thatthe hyperfine splittings in the charge transfer state are large and presumably caused byFermi contact interaction just as in the X3 0 state. The rotational lines of the 11 2 —X303subband at 18432 cm -1 show almost identical widths to the corresponding lines in theY3 11 2 — X303 (1,1) subband, which means that the 11 2 charge transfer state must havesimilar hyperfine splittings to the Y state. It was shown earlier that the linewidths ofthe Y — X subband lead to the conclusion that an unpaired 5w electron is present inthe Y3 I1 state. Accordingly the same conclusion applies to the upper state of the 18432cm-1 subband. Thus the evidence from the hyperfine structures of the 18442 and 18432cm -1 charge transfer subbands seems to suggest the presence of large Fermi contactinteractions in the observed charge transfer 11 states, which is only consistent with theN (2pir) 3 , Nb(4c/(5) 2 (5.scr) 1 electron configuration where a Nb 5so- unpaired electron ispresent.Because the present data on the charge transfer transitions are quite fragmentary, ouranalysis of the charge transfer states of NbN is still incomplete. More charge transferbands are expected to occur above 18500 cm -1 . Studies of these even higher-lying chargetransfer subbands will be essential for confirmation and better understanding of thepresent results.Chapter 5Intracavity Laser Spectroscopy of VOPart IHyperfine Parameters and Electron Configuration of the B 4II State5.i IntroductionVanadium oxide, VO, is important in the astrophysics of cool metal-rich stars, whereit provides the basis for the spectral classification of the late M-type stars [61]. It isalso one of the few transition metal compounds where a comparatively large number ofknown electronic states can be understood within the single configuration approximation[62]. In contrast, some of the late 3d oxides such as CoO, Ni0 and CuO show such strongconfiguration interaction effects that our understanding of the excited states is still veryimperfect [47] .The evidence used to assign the electron configurations in Ref. [62] was a combinationof hyperfine parameters, spin-orbit constants and bond lengths. In particular, a key pieceof evidence was the observation that the main branch lines of the A 4II — X4 E- becomevery sharp at high N [63]. Since the ground state is known to have fairly wide hyperfinesplittings in all four of its electron spin components [64, 65], this means that the hyperfinewidths are essentially the same in the two electronic states. The A 4 Il state has quite asmall spin-orbit coupling constant, A = 35.19 cm -1 [63], so that it is almost completely 169Chapter 5. Intracavity Laser Spectroscopy of VO^ 170uncoupled to case (b) at high N values. Evidently, therefore, the same unpaired 4so-electron is present in the configurations of the two electronic states. It is logical to assigntheir configurations as (4so-) 1 (3c/8) 1 (3thr) 1 A411 and (4so) 1 (36)2 X4E-.Nevertheless, although a consistent picture can be built up from this basis in whichthe other low-lying 'II state, B4 II, is assigned to the configuration (3dS) 2 (3chr) 1 , the mostdirect evidence, in the form of hyperfine parameters, has not been available. The reasonsfor this deficiency are that the A4 11 — X4 E- system at 10500 A lies too far to the redfor sub-Doppler laser work to be practical, while the B 4 11 — X4 E- system at 7900 A hasunusually complicated structure because of the large number of branches with differenthyperfine widths, compounded by the presence of rotational perturbations in the B 4 IIstate. The aim of this work is to secure the foundation of the assignments of the excitedstates of VO, using the very sensitive technique of intracavity laser spectroscopy [66] tomeasure the principal hyperfine parameters of the B 411 state. The B — X (0, 0) bandlies outside the available wavelength coverage of our intracavity system, but adequatespectral data could be obtained from the (1,0) band near 7400 A.5.ii Experimental detailsThe intracavity spectrometer is based, like that of Ref. [66], on a Coherent, Inc.,Model 599-21 single-mode standing-wave tunable dye laser. The body of the laser isfirmly bolted to an optical table, while the output coupler mirror is placed on a heavyoptical mount 50 cm beyond its normal position. The beam splitter and transfer opticsfor the reference cavity which controls the frequency scanning are mounted outside thecavity, just in front of the output coupler mirror, such that the two beams are directedthrough the reference channel and the Fabry-Perot etalon by two large plane mirrors thatreplace the small mirrors that come with the laser. The "tweeter" mirror of the laserChapter 5. Intracavity Laser Spectroscopy of VO^ 171cavity has to be focused about 1 m beyond the output coupler and beam splitter.A fluorescence cell fitted with Brewster-angle windows was placed in the extendedcavity of the laser, with a microwave discharge source producing the VO sample moleculesat one side. The laser-induced fluorescence was recorded by a photomultiplier tube onthe top of the cell. The VO molecules were made by a low power discharge in a flowingmixture of VOC13 and argon. Frequency calibration was obtained by splitting the outputof the dye laser, with one part going to a pressure and temperature stablized Fabry-Perotetalon with a 750-MHz f.s.r [67], and the other part to a uranium-neon hollow cathodelamp, where the uranium spectrum was recorded optogalvanically.Even with its cavity approximately doubled in length, the dye laser still gives single-frequency continuous scans of 35 GHz, though much more care has to be taken duringits alignment. The VO spectra were obtained by scanning the laser frequency with acomputer-generated ramp voltage, on which was superimposed a sine wave modulationat 70 Hz. The sine wave causes frequency modulation of the laser, whose excursion waschosen to be twice the expected linewidth. The component of the fluorescence at twicethe sine wave frequency was demodulated by a lock-in amplifier to eliminate the Dopplerprofiles of the lines, and gave second-derivative line-shape sub-Doppler signals. Spectrain the 7400 A wavelength region were obtained using the laser dye pyridine 2 (Exciton,Inc.).5.iii ResultsThe B4II state of VO is the one major gap remaining in our understanding of itslow-lying electronic states. Absorption spectra taken in a high temperature King furnace[65] have been partially analyzed by Veseth [68], but a recent detailed analysis of thedischarge emission spectrum and laser excitation spectrum (see Part II and Ref. [79])Chapter 5. Intracavity Laser Spectroscopy of VO^ 1 72showed that there are several rotational pertubations in the B411 state. The perturbingstate is a 2 E+ state coming from the same electron configuration, o- 82 , as the X4 E-ground state. A plot of the B 411 state rotational energy levels, with scaling to magnifythe details, against (J+ 2) 2 = J(J +1)+1 is illustrated in Fig. 5.1; this plot shows thatthe low-J structure of the B411, v = 0 level is essentially unperturbed, as is the low-Jstructure of the 12 = 2and .. components of the v = 1 level. The intense B — X (0, 0)band lies too low in frequency for our dye laser spectrometer, but the somewhat weaker(1,0) band is accessible. We have therefore recorded some of the low-J lines of theB411312,512 _ X4E- (1, 0) subbands at sub-Doppler resolution.The appearance of the hyperfine structure depends strongly on J and on which elec-tron spin component of the ground state is involved. As explained in Ref. [64], the case(1313) coupling of the ground state gives hyperfine level widths in the ratio 3:1:-1:-3 forthe four electron spin components F1 , F2, F3 and F4 (that is, J = N + 1 to J = N — 1).There are also internal hyperfine perturbations between the F2 and F3 electron spin com-ponents, whose level structures would cross at N = 15 in the absence of hyperfine effects[64, 69]. The B411 state, on the other hand, is in case (a o ) coupling, where the hyper-fine widths initially decrease as 1/J in each f component. A typical line involving theJ = N + 1 components of the ground state therefore has an eight-line hyperfine patternsuch as that of the RP31 (1 5) line shown in Fig. 5.2, where the splitting is comparativelywide, and is caused almost completely by the 51 V hyperfine structure in the ground state.The low-J patterns are very different. We illustrate part of the TQ41 (1) line in Fig.5.3, and part of the overlapping 7' Q41(2) and uR41(0) lines in Fig. 5.4. These figuresshow the small hyperfine splitting in the B4 1-1 51 2 component and the presence of centredips, or cross-over resonances, which are an artifact of this technique just as they are inintermodulated fluorescence [35].V-2v=15/2 f ....... ...f not found. .......... --.... ------ •.•••.1 ....................^e perte not foundf-1/2 5/2 ee 3/25/2 V = 03/2 1/2 e-1/2...^.. fifer, ......................... .as". ........................................................ - • • .•pert1000 2000^3000(J +1 )22.^11,f pert^—1/2e not found.........^...........^--------------et1460014300137001340012800OLil12400Figure 5.1: Energy levels of the Rill state of VO, as presently known, plotted against (J + 1) 2 . The courses ofthe 2E+ perturbing levels (which are discussed in Part II) are shown as dashed lines. Letters e and f refer tothe parities of the levels.F"^13^14^15^16^17^18^19^20IIIVFigure 5.2: Hyperfine structure of the RP31 (15) line of the B411 — X4 E- (1,0) band of VO; The lines havesecond-derivative profiles.6-6F' - F"4-4^3-4^5-4cd 1 cdI^1•II5-54-5^6-55-6Icd cdII$tcdi1 .1361 .494 cm-11-2 3-2cdII3-32-3 4-31ift,I113614.197 cm-11 1vfigure 5.3: Part of the hyperfine structure of the TQ 41 (1) rotational line of the B4 11— X4 E- (1, 0) band of VO;"cd" indicates a center dip or cross-over resonance. The F' = F" = 2 component is missing because of intensityancellation effects (see text).5-5cd6-5•13617.482 cm -1R410)13617.596 cm -1cd7-74-4F' - F"^3-4 cd^5-4 It lQ41 ( 2 )^6-6Figure 5.4: The overlapping uR41 (0) and TQ41(2) lines of the B411 — X4 E- (1,0) band of VO. The UR41 branchs extremely weak and has only been seen in intracavity experiments.Chapter 5. Intracavity Laser Spectroscopy of VO^ 177The very great sensitivity of the intracavity technique is emphasized by the obser-vation of the lines shown in Fig. 5.3 and 5.4. There is no sign of any line from theserotational branches in the emission spectra that were used to establish Fig. 5.1; theonly branch running in this region in the emission spectra is the TR42 branch. The ro-tational assignments of the new lines were, however, easily made using the ground-statecombination differences [64], as were the hyperfine assignments.The pattern of center dips in Fig. 5.3 should be noted. The hyperfine structure of theTQ41 (1 ) line consists of groups of lines with the same value of F", since the ground-statesplittings are much larger than those of the upper state. The center dips between themiddle component and the two outer components are prominent in the F" = 4 group,less so in the F" = 3 group, and absent in the F" = 2 group. It happens that the F" = 2group represents an intensity cancellation of the type reported in various Q lines of theNbO spectrum [46]; these cancellations are characteristic of high nuclear spin angularmomenta and cause certain hyperfine components to disappear when the vectors J andF would be at right angles in a classical model. The central feature of the F" = 2group is a center dip between the F' — F" = 1 — 2 and 3 — 2 components; it lies halfwaybetween the these two components, which is not quite at the expected position of the2 — 2 component.The hyperfine widths are very small in the B 4 II31 2 component, as can be seen in Fig.5.5, which illustrates the SR32 (1) line.5.iv Hyperfine parameters for the B 41I, v = 1 levelNo attempt has been made in this work to refine the electron spin and hyperfineparameters of the ground state: good values are available from the analysis of the C 4 E- —X4 E - system at sub-Doppler resolution [64], and their accuracy was sufficient to predictFigure 5.5: The sR32 (1) line of the B 411312 — X4 E- (1, 0) band of VO, showing the verysmall upper state hyperfine widths.Chapter 5. Intracavity Laser Spectroscopy of VO^ 179the microwave transitions recently observed by Suenram et al [70] to within 11 MHz.We have therefore used the calculated ground-state hyperfine energies [64] to obtain theupper-state energy levels, and fitted them to an expression consisting of the rotationalenergy for each substate together with the diagonal elements of the Frosch and Foleyhyperfine Hamiltonian [17],< J9IFIkhf s I JS2IF >= To + BJ(J + 1) — DJ2 (J +1) -, + C2 hW(FIJ)'2J(J + 1) (5.1)whereW(FIJ) = F(F + 1) — /(1- + 1) — J(J+ 1).^(5.2)The parameter h for each substate is related to the Frosch and Foley parameters by therelationh = aA + (b + c)E .^ (5.3)As is well known, the Fermi contact and dipolar parameters, b and c, respectively, cannotbe separated in case (a) coupling, such as applies to the low-J lines of the B 411 state, andthere are effectively only two determinable parameters: a, the nuclear spin-electron orbitinteraction and the combination (b+c). Spin-uncoupling cannot be neglected even at thelowest J values of high spin-multiplicity states [71]; it was therefore necessary to includean apparent centrifugal distortion parameter, ph, which is essentially a second-order crossterm between the spin-uncoupling operator, —2BJ • S, and the Fermi contact operator,bI • S. The diagonal matrix elements of the nuclear electric quadrupole interaction [72]were also included, but the parameter e 2 Qqo is not well determined.The final values are given in Table 5.1, while the hyperfine energy levels of the B 4 115 1 2and B41131 2 substates which made up the data set are given in Table 5.2.With the use of Eq. (5.3) the two h parameters can be converted to the Frosch andChapter 5. Intracavity Laser Spectroscopy of VO^ 180Table 5.1: Rotational and hyperfine constants for the B 4II, v = 1 level of VO derivedfrom the 4 115/2 and 4113/2 substates (Values in cm -1 )Parameter B411512 B411312To 13610.3014^(4) 13539.0750 (13)B 0.519605^(32) 0.51461 (14)105 D 0.14^(5) 0.64 (30)h —0.00978^(13) 0.00317 (69)Ph —0.0019^(1) —0.0010 (4)c2(2,70 0.0017^(16) 0.0016 (45)a- 0.00019 0.00040Derived parameters:a = 0.00964 cm'b+c = —0.01295 cm'Errors, in parentheses, are two standard deviations in units of the last significant figurequoted.Chapter 5. Intracavity Laser Spectroscopy of VO^ 181Table 5.2: Hyperfine energies of the v = 1 vibrational level of B 411 5 12 and B4 H31 2 sub-states of VO (in cm -1 ).B4 115 1 2J^F^EnergyB4113/2J^F^Energy2.5654313614.825413614.840813614.853613614.8642 2.5654313543.582613534.579613543.575713543.57282 13614.8719 2 13543.57061 13614.8773 1 13543.56887 13618.46773.5 6 13618.47745 13618.48584.5876543213623.148013623.154613623.160213623.165113623.169213623.172313623.17434.587654313551.811413551.809613551.808013543.806213551.805013551.80379 13628.8661 9 13557.46908 13628.8707 8 13557.46697 13628.8745 7 13557.46485.5 6513623.877813628.8808 5.56513547.462813557.46124 13628.8832 4 13557.46013 13628.8850 3 13557.45922 13628.8864 2 13557.458610 13635.62149 13635.62498 13635.62786.5613635.630313635.63245 13635.63414 13635.63553 13635.6362Chapter 5. Intracavity Laser Spectroscopy of VO^ 182Foley forma = 0.00964 cm-1 ;^b + c = —0.01295 cm-i .^(5.4)The error limits are set by the uncertainty in h(4 11312 ) and are approximately +0.0007-CM 1 .5.v DiscussionThe comparatively large negative value for b + c in the B4 11 state is fairly strong evi-dence that the electron configuration is (3c/5) 2 (3c/r) 1 . Even though we cannot determineb and c separately, it is possible to estimate the value of c for the two possible configura-tions (3d8) 2 (3dir) 1 and (3dS) 1 (3c/r) 1 (4so- ) 1 , and to show that it must be small comparedto the observed value of b + c, so that b has to be negative; within the Hartree-Fock(single configuration) approximation this is only consistent with the first configuration,where the unpaired 4so electron is not present.Specifically, if we approximate the molecular orbitals as vanadium atomic orbitals,the dipolar parameter c for these two configurations can be taken as a sum over the iunpaired electrons, given by [73, 74]C =^ 3giingnit,^< 3 cos t 0 — 1 >IA471- cc^2^6"^2S < r3 >nrThe angular factors in Eq. (5.5) are2[3) 2 — 1(1+1)] < 3 cos 2 O — 1 >IA = (2/ — 1)(2/ + 3) •Only the radial factor < r -3 >3d is needed, since the angular factor < 3 cost 0 —1 >„= 0prevents the 4sa electron from contributing. We can obtain < r -3 >3d either from theexperimental dipolar parameter for the (36) 2 (4sa) 1 ground state [64], or from Mortonand Preston's ab initio calculation [75]. The two methods happen to give the same value(5.5)(5.6)Chapter 5. Intracavity Laser Spectroscopy of VO^ 183to within the accuracy of the experiment, showing that the 3cm orbital is in first approx-imation unchanged from its atomic character by the chemical bonding. Accordingly wetake< r -3 >3d = 2.10 x 1031 m3 .Straight-forward application of Eqs. (5.5) and (5.6) then givesc( 4 11, 6 27r) = —0.00312 cm'; c( 4 H, 67ro- ) = —0.00104 cm-1 .Since the true Fermi contact parameter is given by [17]2aF = (b + c) — i c,it is seen that aF must be of the order of —0.011 cm -1 whichever value of c is chosen.A large negative value such as this can only be consistent with a spin polarization [76]mechanism, in the absence of an unpaired 4so electron.As discussed in Section 5.iv, it was necessary to include an apparent centrifugaldistortion to the hyperfine parameter h; this takes account of spin-uncoupling. A simplecalculation shows that the magnitude of the parameter Dh is reproduced to within afactor of two by the second-order cross-term between the spin-uncoupling and Fermicontact operators. We consider the value of Dh in the 4 115/2 substate. The off-diagonalmatrix element between 1 4 11 5/2 > and 1 4 113/2 > basis functions, in case (a) coupling, canbe written< 4115/2) <-1 I II I 4113/2, J > = — 153 {(J(J+ 1) —^bW]LT] [B^4 A j+(FIJ)1)^(5.10)This produces a second-order correction to the 4 115/2 energy which consists of three terms.The first term has the form 3B 2 [J(J+ 1) — 15/4]/AE( 4 11 5/ 2 — 4 113/2), and is a correctionto the B value of the substate [20], originally described by Mulliken. The second is thecross-term(5.7)(5.8)(5.9)AE(2) = 6 B b[J(J + 1) — 1:]W(FIJ) (5.11)4AE(4 115/2 — 4 113/2) J(J + 1 ) 1Chapter 5. Intracavity Laser Spectroscopy of VO^ 184which can be identified with the Dh term,AE(2) — C2 Dh J(J + 1) W(FIJ) 2J(J + 1)(5.12)If the coefficient -11 is taken as small compared to J(J+1), which is equivalent to including4its effects in b itself, Dh is seen to be= 3B bDh3 G" A Espin—orbit3(0.5121)(-0.0098)(D(71.23)= —0.000085 cm -1 .^(5.13)The third term involves [bW(FIJ)] 2/AEspin_ or bit , and is a very small correction to thequadrupole parameter, which is not considered further.Similar arguments applied to the 4113/2 state give Dh( 4 I13/2) = —0.000047 cm -1 . Forboth substates the experimental values are about twice the calculated values; it is notcertain whether the discrepancy is a real effect or merely apparent, but the fact that thesign and rough magnitude of the Dh terms are correctly reproduced is encouraging.5.vi ConclusionThe principal hyperfine parameters of the B411, v = 1 state of VO have been deter-mined from intracavity sub-Doppler laser spectroscopy. The technique has been shownto be very sensitive for low concentrations or for weakly absorbing species, and the re-sults have proved that the B411 state is described in single-configuration approximationas (36)2(3thr)1.Chapter 5. Intracavity Laser Spectroscopy of VO^ 185Part IIAnalysis of Rotational Structure and Perturbationsin the B 4II — X4 E- (1,0) Band of VO5.vii IntroductionThe VO B — X system near 7900 A was discovered in stellar spectra [77] a number ofyears before laboratory work by Keenan and Schroeder [78] proved that VO is the carrier.From absorption spectra taken in a high temperature King furnace, it was identified byRichards and Barrow [65] as a 4 1-1— 4 E - transition and later partially analyzed by Veseth[68]. As mentioned in Part I, the structure of the B — X system is complicated by thelarge number of branches with different hyperfine widths and the presence of rotationalperturbations in the B 4II state. Preliminary analysis of the B — X system [62, 81]suggested that the perturbing state is a 2 E+ state arising from the electron configuration(4so) 1 (3d8) 2. In an effort to understand fully the structure of the B4II state and itsinteraction with the (4so-)1 (3d8)2 2E+ state, a thorough investigation of the B 4II — X4 E+system has been carried out by the Spectroscopy group at UBC. Rotational and hyperfineanalysis of the B — X (0, 0) band, using Doppler-limited and sub-Doppler spectral data,has produced precise molecular constants for the B4II, v = 0 state and the perturbing2 E+ state; the results are given in Ref. [79]. In this second Part of Chapter 5, wereport a detailed rotational analysis of the B — X (1, 0) band, which provides additionalinformation for the B4II and 2 E+ states.Chapter 5. Intracavity Laser Spectroscopy of VO^ 1865.viii Spectral dataThe spectral data used in this analysis are largely taken from the Doppler-limited VOemission spectra recorded by Merer and Hocking [80]. To supplement the emission data,laser excitation and wavelength-resolved fluorescence spectra of the weak B411112,-1/2 -X4 E - (1,0) subbands were also recorded; the experimental details follow.The VO molecules were produced in the same way as described in Part I; calibra-tion of the laser excitation spectra was also obtained in the same manner. The sameCoherent-599-21 single-frequency tunable dye laser was used, but in its original configu-ration. The laser power obtained near 7450 A wavelength was 50-70 milliwatts using thelaser dye pyridine 2 (Exciton, Inc.). Most of the output power was sent in a single beamto the fluorescence cell, which was placed outside the laser cavity. The laser-induced VOfluorescence was recorded using phase-sensitive detection to give Doppler-limited laserexcitation spectra. Wavelength-resolved fluorescence spectra were also recorded for someselected spectral lines to obtain unambiguous rotational assignments. In this experimen-tal arrangement, the fluorescence was directed to the slit of a 0.7 m. spectrometer (Spexmodel 1702) using mirrors and lenses, and the dispersed fluorescence was detected witha microchannel-plate intensified array detector (PAR model 1461). Calibration of theresolved fluorescence spectra was provided by the emission spectrum from an iron-neonhollow cathode lamp recorded in the same spectral window as the VO spectra.5.ix Analysis of the branch structure of the B 411 — X4 E - (1, 0) BandMost of the branches of the B 4 II — X 4 E- (1, 0) band in the region 13350 — 13650cm -1 were identified and assigned by Cheung [81] from the emission spectra. How-ever analysis of the branch structure of the comparatively weak 4H1/2, -1/2 - 4 E - ( 1 ) 0 )subbands was incomplete because of complications from the overlapping (2,1) sequenceChapter 5. Intracavity Laser Spectroscopy of VO^ 187band and rotational perturbations. To complete Cheung's branch analysis, wavelength-resolved fluorescence spectra were recorded for most of the unassigned rotational linesin the 13350 — 13450 cm -1 region to obtain absolute line assignmemts. The wavelength-resolved fluorescence spectra give the rotational selection rules for the various rotationallines and also give the rotational and vibrational numberings of the lower levels. Usingthis method and combining our data with Cheung's analysis, we eventually identifiedand assigned 22 sharp branches of the B 4H — X 4 E - (1, 0) band, involving the FT and P3levels. Most of the Fr and F,',' branches, on the other hand, are broadened beyond recog-nition by the lower-state hyperfine structure [64]. The assigned lines of the B — X (1,0)band are listed in Appendix E; only the sharp lines are given, since they are sufficient todetermine the upper state constants.It is worthwhile to note that all the observed FT , PT sharp branches show the char-acteristic internal hyperfine perturbation pattern near N" = 15, discovered by Richardsand Barrow [69]. This internal hyperfine perturbation occurs because the F2 and F3lower levels (J = N ± 1/2) with the same N happen to cross at N = 15, because of theparticular values of the rotational and electron spin parameters; however matrix elementsof the hyperfine Hamiltonian of the type OF = AN = 0, AJ = +1 acting between thetwo spin components cause avoided crossings of the hyperfine levels. As a result, extralines are induced, and since the exact energy pattern of the ground state is known [64],the positions of the extra lines immediately give the assignment of the lower levels of abranch as F2 or F3 and its N" numbering. Another feature worth noting is the relativeintensities of the branches of a given subband. It was found that, in all four subbands,the AN i AJ satellite branches are comparatively strong and, in the B 4 14 3/ 2 , 1 / 2 - X 4 E -subbands, the satellite branches are actually stronger than the AN = /J main branches.This intensity pattern is consistent with a 4 II(case (a))- 4 E - (case (b)) transition, thoughit is difficult to understand why the main branches are weaker than the satellite branchesChapter 5. Intracavity Laser Spectroscopy of VO^ 188in the B 4-312,112 -- X4 E - subbands.The PQ23 branch (involving the B 4- 1 / 2 , v = 1 upper levels with f parity) is inter-esting because it is very obviously perturbed. This comparatively strong branch can befollowed easily from N" = 6 on, but starting at N" = 25 the line separations begin to de-crease with increasing N and the rotational lines become weaker and broader; at N" = 30the lines broaden considerably and after that the branch disappears. It was suggested[62] that the B4 11 1 /2 , f , v = 1 upper level is perturbed by a lower-lying non-degenerate 2 E+state from the (4so- ) 1 (3db) 2 electron configuration, where the presence of an unpaired 4screlectron results in broad hyperfine structure in the 2 E+ state, which, through a spin-orbitperturbation mechanism, broadens the hyperfine widths of the B 4 11 1 1 2 levels in the vicin-ity of the avoided crossing. Perturbation-induced extra rotational lines were eventuallyfound 20 cm -4 to the red of the last N" = 30 line and identified by wavelength-resolvedfluorescence.Fig. 5.6 shows the calculated energy levels of the B4II, v = 1 state and the perturbing2 E+ state, suitably scaled, plotted against (J -F) 2 . The B4II state is shown to be regular(with a positive spin-orbit coupling constant) since no A-doubling is observed in the F4( 4 115/2) component, whereas the other three spin components all show various degreesof A-doubling effects. It can be seen from Fig. 5.6 that the1 -14-1/2,f level is heavilyperturbed by 2 Elf}", while 4111/2 ,, is slightly perturbed by 2 E;1- at high J. Although onlythree extra lines involving the II4-1/2,f levels have been observed, they are sufficient togive a quantitative analysis of the B 4 11/ 2 E+ interaction.5.x The energy levels of the 4 H/ 2 E+ and 4 E - statesRotational energy levels for 4 11 and 4 E - states have been described in detail in theanalysis of the A4 11 — X4 E - system of VO [63]. Therefore it will only be necessary toChapter 5. Intracavity Laser Spectroscopy of VO^ 18913650 -^ f/e- 4-ir5/213600 -T^- - 4Tr fieEf"(\i+I-) 13500 -,T=inail13450 -13400 -I^I^111111111^IIIIIII^II^II0^500 1000 1500 2000(J+1/2)2- 13550 - 3/2Figure 5.6: Calculated energy levels of the B 4II, v = 1 and 2 E+ states of VO, plottedagainst (J + a ) 2 . A quantity 0.5 x (J + 1) 2 has been subtracted to magnify the scale.Letters e and f refer to the parities of the levels.Chapter 5. Intracavity Laser Spectroscopy of VO^ 190extend the Hamiltonian matrix for a 4H state given in Ref. [63] to include a 2 E+ stateand to give the matrices we have used in the analysis of the B - X system.For a single electronic state, we take the spin and rotational Hamiltonian in the formH = B CI - i- ".) 2 -D(. -1,- "')'' + A 4, ,'',2A^. 2^ 1+ —3 (3,?. - S ) + 7 (:/ - i - ,§) • ,A9 + -2 AD [LA, (j - i - .§) 21 +1^A^:92 \ t(J .L :9 + —3AD [(3 2Sz — ),^— — )2] + HLD,+(5.14)where the terms can be identified by their coefficients. B is the rotational constant,and D its centrifugal distortion correction; A and A are the first- and second-order spin-orbit parameters where A includes the dipolar electron spin-spin interaction; -y is thespin-rotation interaction parameter, and AD and AD represent the centrifugal distortioncorrections to the first- and second-order spin-orbit couplings. The symbol [x, y] + =xy + yx is the anticommutator, which is required to preserve the Hermitian form of thematrices. The A-doubling Hamiltonian for the 4 H state is taken in the form appropriatefor case (a) coupling asHLD = 21 (0 + p + q)(,'.1_ + '?_) - 21 (p + 2q) (,)+ ,'1_ + j_ :5t) + 21 q (j._ + j.2_)+ _Do+p+q [(:7 _ i _ :9)2 , (:§2 _L_ :92 1 —)^_ILD4^ " + -1- -) +^4 P+2q+ -LiDg (J - L - s) 2 , (l_f_ + J!) +ka -i -,§) 2 , (,i+ s‘ + +,1_,_)1 4.(5.15)In Eq. (5.15), the ladder operators are assumed to act within the manifold of the 4 H state,and the A-doubling parameters (o + p+ q), (p + 2q) and q are sums over the interactionswith other states as explained in Ref. [27] and Section 2.iv.C. The parameters Do-f-p+q,Dp+2q and Dq represent centrifugal distortion corrections to the three A-doubling terms.From Eqs. (5.14) and (5.15), the Hamiltonian matrices for the 4 II and 4E- states can bederived in a case (a) basis, and the results have been given in Ref. [63]. For our purpose,Chapter 5. Intracavity Laser Spectroscopy of VO^ 191the 4 H matrix needs to be extended to include the interacting 2 E+ state. The part ofthe Hamiltonian which produces matrix elements between the 4H and 2 E+ states is themicroscopic electron spin-orbit interaction,11„ = E a ii i •The matrix elements are calculated to be< 2 E+ , F2 I fir so 1 4111/2, f > < 2 E+ , F1 I Hso 1 4111/2, e > = 2 1 A l (5.16)< 2 E+ , F2 I Hso 1 411-1/27f > — < 2 E+ , F1 I A90 1 4 11_ 1 / 2 , e > = A2 (5.17)where A l and A2 are the measurable parameters. Theorectically A l and A2 should beequal, but we find experimentally that they are marginally different, that is, just outsideexperimental error. It is important to point out that the A l and A2 parameters givenhere are equivalent to Kovacs' constant but divided by a factor of J8/3; moreover,the sign of A2 in Eq. (5.17) is the reverse of that given by Kovacs [82]. The resultingHamiltonian matrix for the 4 11/ 2 E+ complex which we have used is given in Table 5.3,while the matrix for the 4 E- state is taken as that given by Ref. [63].5.xi Least-squares fitting of the dataAs indicated in Ref. [63], the internal hyperfine perturbation in the X 4 E - state causesthe _FT and P-1 ' levels to be shifted from the positions which they would have in the absenceof hyperfine structure. The amount of the shifts produced, as shown in Table 5.4, is quitesignificant even at N values quite far away from the internal perturbation; therefore it isnecessary to correct all the line positions in branches involving PT and F4' levels.After the line corrections given in Table 5.4 were applied to the _FT and F4' branches,the line positions were fitted to the appropriate differences between the eigenvalues of theTable 5.3: Matrix Elements of the Rotational Hamiltonian for a 4 11/2 E+ Complex in Case (a) Coupling.2E1 /e+ B411-112,f le B4111/2,1 /e B41-1312,1 le B411512,1 le2E+T2E+ + B (1 ►[Z ± (J +1)]— D (1) [z±(J +D]2 +17 (1 ►(J + -1 ± 1)442 — • ii5iii4 11 -112T_1/2 + (B — 1 AD +2AD)(z + 1) — D(z2 +5z +1)±3(J +1)Do+p+q—0Z1B — 12-7 — AD —2D(z + 2)1 T ft° +p+q)+(z+2)Do+p+9 +1(22' — 1)Dp+2q j—113(.7 — 1)[2D(J^+I)^T^1(P^+2q) T D o+p+q F 1(z +1)Dp+22 T (z — 2)D9TII(Z — 1)(z — 4)gq+1Dp4. 2q + 1-1),(z — 2 )]4110T112 + (B — lAD —2AD)(z+3)—D(z2+13z+5) ± (J + !')[(p + 29) +3Do+,+9 + Dp4. 2q (z + 3) +D9 (z — 1)]—21T:11B — '7 —2)D-2D(z+2)±t(J+1){q+ p3p4.29 + Dq (z +2))].1/3(z — 1)(z — 4)[-2D±.113,7 (J + 1 )]4113/2— Symmetric1T312 + (B + 1 AD —2AD)(z + 1) — D(z 2 +9z — 15) ± (z — 1)(J+1)Dq—I3(z — 4)[B — .17 +AD — 2D(z — 2)1411 5/21T512 + (B + IAD +2AD)(z — 5 ) — D(z 2 "—7z + 13)z = (J 1-) 2 . Upper and lower signs refer to e and f rotational levels respectively.The basis functions ISJS1 > have been abbreviated to )SS1 >.Chapter 5. Intracavity Laser Spectroscopy of VO^ 193Table 5.4: Corrections applied to the observed F2 and F3 line positions to allow for theinternal hyperfine perturbation shifts (in cm -1 )a.N F2 F3 N F2 F3 N F2 F3 N F2 F34 -0.030 -0.003 14 -0.079 +0.055 24 0.029 -0.026 34 0.020 -0.0175 -0.031 +0.008 15 ±0.080 25 0.027 -0.025 35 0.019 -0.0166 -0.031 0.012 16 +0.075 -0.086 26 0.026 -0.024 36 0.019 -0.0167 -0.033 0.017 17 0.065 -0.075 27 0.025 -0.022 37 0.019 -0.0168 -0.034 0.022 18 0.051 -0.060 28 0.023 -0.021 38 0.018 -0.0159 -0.036 0.025 19 0.047 -0.058 29 0.023 -0.020 39 0.018 -0.01510 -0.053 0.031 20 0.043 -0.043 30 0.022 -0.019 40 0.018 -0.01511 -0.060 0.033 21 0.038 -0.039 31 0.021 -0.018 41 0.017 -0.01412 -0.065 0.034 22 0.035 -0.031 32 0.021 -0.018 42 0.017 -0.01413 -0.070 0.043 23 0.032 -0.028 33 0.020 -0.017 43 0.017 -0.014a Values are obtained from Ref. [63] with N extended to 43.4 11/ 2 E+ and 4 E - matrices. The X 4 E - , v = 0 parameters were not varied in the fit, butwere fixed to values determined from the sub-Doppler spectra of the C4 E - - X 4 E- (0, 0)transition, where the resolution is a factor of 10 higher. Therefore our procedure isequivalent to fitting the term values of the B 4II(v = 1)/ 2 E+ complex to the eigenvaluesof Table 5.3. The least-squares results are given in Table 5.5.The number of spectral lines used was 450, which produced a standard deviation of0.015 cm-1 , limited by the resolution of the spectral data.Chapter 5. Intracavity Laser Spectroscopy of VO^ 194Table 5.5: Spectroscopic parameters for the B 411/ 2 E+ complex of VO derived from ro-tational analysis of the B 4 11 — X 4 E- (1,0) banda (in cm -1 ).B4II, v = 1 2E+, v = 3 6T512 13612.760^(8) To 13429.75 (8)T312 13538.411^(10) B 0.54045 (18)T112 13473.311^(30) 106D 0.650 FIXEDT_112 13420.344^(162) 7 0.020 (5)B 0.509481^(41)106 D 0.672^(14)7 0.022^(11)AD —0.00025^(17)AD 0.000016^(24)o+p-Fq 1.143^(21)p + 2q 0.0364^(10)q 0.000049^(39)Interaction Parameters between 4 11 and 2 E+A( 4 11 1 1 2 ) = 28.34 (36)A(4 11_ 1 12 ) = 28.78 (29)Standard Deviation = 0.015Errors in parentheses are three standard deviations in units of last significant figurequoted. aParameters of X4 E- were not varied in the fit, they were held to values givenin Ref. [64]. bVibrational assignment of the 2 E+ state is discussed in the text.Chapter 5. Intracavity Laser Spectroscopy of VO^ 1955.xii Results and discussion5.xii.A Spin-orbit coupling constantsThe substate origins, To , for the components of a multiplet H state can be expressedin terms of the spin-orbit and spin-spin parameters as2ATo = To + AAE + T [3E2 - S(S + 1)]+ -y PE — S(S + 1)] + 77A {E 3 — (3S 2 + 3S — OE)] ,^(5.18)where ri is the third-order spin-orbit interaction [71, 83]. As pointed out by Brown [11],for a 211 or 'II state, there is a problem of indeterminacy of the five parameters givenin Eq. (5.18); however, for a 411 state, all five parameters can be determined from thesubstate origins since -y can be obtained from the rotational structure. Therefore usingEq. (5.18) and the measured -y constant, the To values given in Table 5.5 are convertedto the following constants for the B411, v = 1 state:To = 13511.262 cm -1 ; A = 64.213 cm-1A = 2.662 cm -1 ; ri = —0.481 cm-1^(5.19)The spin-orbit constant A derived here is almost identical to the value reported by Veseth[68], which was used in Ref. [68] to calculate the spin-orbit coupling of the 4dir electronand to show that the magnitude of the spin-orbit coupling of the B 411 state is consistentwith those measured in the other VO electronic states. The comparatively large second-order A constant, compared to the first-order constant A, is not uncommon in moleculeslike VO; both the A4 11 and X4 E- states also have relatively large A parameters [63, 64].In theory, the spin -orbit interaction parameters A l and A2 should be equal if the B4 11state is perturbed by a single 2 E+ state. The measured values for the two constants, ascan be seen in Table 5.5, show a slight difference, which may be attributed to two possibleChapter 5. Intracavity Laser Spectroscopy of VO^ 196causes. One possibility is that the centrifugal distortion corrections to A l and A2, whichare not considered here, are sufficiently different at the J values of the two spin-orbitinteraction terms that they cause an apparent difference between A l and A2. The otherpossible cause for the difference is that the B4 11 state is also perturbed by a remote 2IIstate, which only interacts with the B411112 spin component, but not the B 4 11_ 1 / 2 spincomponent. Nevertheless, the difference between A l and A2 is very small, and we shalltherefore take the average value, 28.56 cm -1 , as the spin-orbit interaction parameter Al mbetween the /34 11, v = 1 and 2 E+, v state.5.xii.B The 2 E + perturbing stateTable 5.5 shows that the 2 E+ perturbing state has B = 0.5404 cm -1 , which is verysimilar to that of the X 4 E- state, where B = 0.54638 cm -1 ; therefore it is likely thatit comes from the same electron configuration, (4sa) 1 (3d6) 2 , as the ground state. Thevibrational quantum number, v', of this 2 E+ state cannot be determined just from thedata obtained here; a determination becomes possible on combining the present data forthe B4 II, v = 1/ 2 E+, v' system and those of the B4II, v = 0/ 2 E+, v' —1 system measuredfrom the B — X (0,0) band [79]. Specifically this is done as follows.According to the Born-Oppenheimer approximation, the wave function of a vibronicstate may be separated into electronic and vibrational parts; therefore the perturbationmatrix element between the B4II, v and 2 E+, v' state may be written asil,,,, = < B 41I, v I ii„ 1 2 E + , v' >c:.--,< B 411 1-113. 1 2E+ >< V I V I > . (5.20)The vibrational overlap integrals < v Iv' > can be calculated using numerical integrationover potentials described by a modified Morse function given by U(r) = D e [1 — e-0(r) (r — 7-01 2 (5.21)87r 2 c/./B,/3 (r) = 130+ /31(r — r e ) + /32 (r — re ) 2 .I^hre = 1.1 (5.23)(5.24)Chapter 5. Intracavity Laser Spectroscopy of VO^ 197where2weD e = ^4w,x,(5.22) The parameters, 00, 01 and /32, in Eq. (5.24) are given byaoDe rea1 /3o2re^2a0a2^3 0— i3oP1 — Pf/^ — 732 /32 /30 D,71^2^2,30^6where4B,coectea 1 = ^ + 16_/R5 2^2 we x ea2 = 71.a1^3 Be •The data used to calculate the modified Morse functions for the B 411 and 2 E+ states(from Table 5.5 and Ref. [79]) are as follows:411,v = 0 4 11, v = 1 2 E+, v = x 2 E+, v = x 1To (cm -1 ) 12609.750 13511.262 12433.34 13429.98Be f f (CM -1 ) 0.51268 0.50948 0.54326 0.5405.The To and Be f f values for the 4 11 vibrational levels can be reduced to equilibrium values,using the standard formulae [20], asBe = 0.51428 cm';^a, = 0.0319 cm -1 ; we = 910.58 cm -1 ;we x e = 4.6 cm -1 (assumed);^re = 1.6410 A;^De 45, 000 cm-1 ;flo/312aoChapter 5. Intracavity Laser Spectroscopy of VO^ 198and, assuming the value of x as 0, 1, 2, and 3, the 2 E+ constants can be reduced to thefollowing sets of equilibrium values (in cm -1 ):x = 0^1^2^3^Be^0.5448^0.5478^0.5508 0.5538^We^1005.84^1015.04^1024.24 1033.44wexe^4.6^(assumed)a,^0.0026^0.0026^0.0026 0.0026De 45000 (assumed)Then taking v( 4f1) = 0,1 and v'( 2 E+) = 0,1,2,3,4, the overlap integrals are calculatedto be [84]< 010 >= 0.82648; < 011 >= 0.54591; < 012 >= 0.28265; < 013 >= 0.13045;< 111 >= 0.50891; < 112 >= 0.60878; < 113 >= 0.41508; < 114 >= 0.22936.Since < B 4 111B-„1 2 E+ > is independent of the vibrational quantum numbers v and v',we have from, Eq. (5.20)Ao,v,<01v'>=^ (5.25)A1, v1+1^< 11v' + 1 >Experimentally, the ratio A 0 ,,,/A. 1 ,,,+1 equals 20.417/28.56 = 0.71, which is only consis-tent with the ratio < 012 > / < 113 >= 0.68. Therefore, the v = 0 level of the B 4 11state is perturbed by the v = 2 level of the 2 E+ state, while the B411, v = 1 level isperturbed by the 2 E+, v = 3 level.5.xii.0 A-doubling parametersIn the approximation that the A-doubling of the B411 state is caused by a single 4 E-state, the A-doubling parameters o, p and q are given by1 < 41-1 I Al + 1 4 E - > 2o =^ (5.26)2^En—EEChapter 5. Intracavity Laser Spectroscopy of VO^ 199P =q =2 < 4 11 I AL + 1 4E >< 4 11 I BL+ 1 4E > En — EE2 < 4111BL+14E- >2(5.27)(5.28)Ell — EE Since the parameter q of the B4 11, v = 1 state is very small and not very well determined,we will focus our discussion on the other two A-doubling parameters only. From Eqs.(5.26) and (5.27), it follows at once that:From Table 5.5, we obtaino. Ap 4 B (5.29)o = 1.106 cm-1 ;^p = 0.0363 cm -1 ,^(5.30)and thereforeo/p^ = 0.967.^ (5.31)A/(4B)The almost unity ratio calculated in Eq. (5.31) indicates that the A-doubling of the B 4IIstate can be very well described by interaction with a single 4 E - state.5.xiii ConclusionElectron spin and rotational constants of the B 4 II, v = 1 and the perturbing 2 E + , v = 3states have been determined from the spectral data of the B — X (1, 0) system. The resultsare consistent with the configurations of the 411 and 2 E+ states being (3dir) 1 (3d8) 2 and(4so) 1 (3d8) 2 respectively. It is found that the interaction between the 4 II and 2 E+ statescan be well described by the matrix elements of the spin-orbit operator acting betweenthe two states.Chapter 6Laser Intracavity Spectroscopy of CoOPart IHyperfine and Spin-orbit Structure of the 40, Ground State of CoO6.i IntroductionConsiderable advances have been made in the past few years on the spectroscopy ofthe 3d transition metal oxides [47], which reflect their importance in astrophysics andhigh temperature chemistry: the symmetries and bond lengths for the ground states arenow established, as are properties of some of the low-lying excited states. Our knowledgeis far less complete for the spin parameters, such as the spin-orbit coupling and hyperfineconstants, which give the information needed to elucidate the details of the chemicalbonding; this is particularly so for some of the later members of the series. With therecent accurate measurement of the second-order spin-orbit splitting in the X 3 E- state ofNiO by Friedmann-Hill and Field [85], the most significant gap is now the ground state ofCoO. It has not been easy to obtain high resolution spectroscopic data for gaseous CoOand NiO because there are unusually large numbers of low-lying electronic states, so thatthe optical spectra are crowded and perturbed; the analyses reported so far have usedlaser excitation methods combined with a Broida oven [86] or hollow cathode [85] sourcewith a comparatively low effective temperature, to reduce the complexity, combined200Chapter 6. Laser Intracavity Spectroscopy of Co0^ 201with extensive wavelength-resolved fluorescence in order to cope with the perturbations[87, 88].This Part reports the details of the spin structure of the ground state of CoO. Thecobalt oxide molecules were generated in the cavity of a standing-wave dye laser by meansof a microwave discharge, and observed by laser-induced fluorescence. The intracavitynature of the experiment means that much more laser power is available to excite fluo-rescence than would normally be available outside the laser cavity, so that the sensitivityis high, while the spectra automatically contain a sub-Doppler component because ofthe two directions of travel of the laser radiation. The hyperfine structures of the lowJ lines of three of the strongest sub-bands of the "red system", near 6400 A, have beenrecorded. These give the determinable hyperfine parameters of the two lowest-energyspin-orbit components of the X 4/ ground state of CoO, and by inference the values ofthe Frosch and Foley a and b c paramters [17]. The X40 1 Fermi contact parameteris found to be negative and quite large, confirming that the electron configuration forthe ground state is (4sa) 2 (3d8) 3 (3dir) 2 . In a separate series of experiments, wavelength-resolved fluorescence spectra have given the three spin-orbit intervals of the X 401, v 0level to 0.3 cm- ' accuracy. Following the rotational analysis of a pair of sub-bands witha common upper vibronic level, but with the ft = 7/2 and 5/2 components of X 40 1 aslower levels, we have also obtained an accurate value for the lowest spin-orbit interval,11 = 5/2 — 7/2.6.ii Experimental detailsThe intracavity laser system used for these experiments is based, like that of Ref. [66],on a Coherent Inc. model 599-21 single-mode standing-wave dye laser. The body of thelaser is firmly bolted to an optical table, while the output coupler mirror is placed on aChapter 6. Laser Intracavity Spectroscopy of CoO^ 202heavy optical mount and moved back from its normal position to extend the laser cavityby 50 cm. The beamsplitter and the mirrors which direct part of the laser output to thefrequency scanning mechanism are mounted just in front of the output coupler, outsidethe laser cavity. The "tweeter" mirror of the laser is focussed to a point about 1 m.beyond the output coupler and beamsplitter.A fluorescence cell fitted with Brewster-angle windows was placed in the extendedcavity of the laser, and laser-induced fluorescence was collected by a photomultipliertube mounted on top of the cell. CoO molecules, generated by a 2450 MHz microwavedischarge in a flowing mixture of cyclopentadienyl cobalt dicarbonyl vapour (AldrichInc.), argon and nitrous oxide, were passed through the fluorescence cell from the side.We find that a microwave discharge of this type is a plentiful source of CoO molecules,giving much higher concentrations than can be obtained by sputtering; the disadvantagefor laser work is the intensity of the background flame, but with suitable light baffles thesignal-to-noise ratio is very adequate.Although its cavity is approximately doubled in length, the dye laser still gives con-tinuous single-frequency scans of 30 GHz width if sufficient care is taken with the opticalalignment. The CoO spectra were obtained by tuning the frequency of the dye laser bymeans of an external ramp voltage produced by a computer. To obtain Doppler-limitedexcitation spectra, the photomultiplier output was recorded directly. To extract the sub-Doppler component a sine wave modulation at about 70 Hz frequency was added to theramp voltage, with its amplitude chosen to produce a frequency modulation of the laser ofabout twice the expected line width; the component of the fluorescence signal at twice themodulation frequency was demodulated by a lock-in amplifier to give second-derivativeline-shape sub-Doppler signals. This procedure eliminated the Doppler profiles almostentirely: the only indication of their existence is the increased background noise in thevicinity of the sub-Doppler signals. Centre dips, or cross-over resonances between linesChapter 6. Laser Intracavity Spectroscopy of CoO^ 203sharing a common lower state level, are an artifact of this technique just as they are inintermodulated fluorescence [35].Spectra in the 6400 A region were recorded with DCM laser dye. Frequency cali-bration was obtained by splitting the output of the dye laser, with one part going to astabilized 750 MHz free spectral range Fabry-Perot etalon [67], and the other going to auranium-neon hollow cathode lamp whose spectrum was recorded optogalvanically.The dye laser was pumped by a Coherent Inc Innova 90-6 argon ion laser operatingat 514 5 nm wavelength, with typical power of about 2 W. Although the argon ion lasertube deteriorated significantly during the experiments, producing only about 1 W at theend, this was entirely adequate for recording excitation spectra and wavelength resolvedfluorescence spectra of CoO from the intracavity system, using the dyes DCM and R6G(Exciton Inc.)6.iii Hyperfine structure6.iii.A Appearance of the spectraThe strongest bands in the excitation spectrum of CoO, as recorded with DCM laserdye, lie near 6400 A. Rotational analysis [88] shows that they are very strongly red-degraded parallel-polarized sub-bands where the lower states, with ft" = 7/2 and 5/2,can be assigned as the lowest two spin-orbit components of the ground state, X 402.Because of their intensity we have chosen these bands for the work reported in thispaper. Specifically we have studied the two ft = 7/2 — 7/2 sub-bands whose R headslie at 6338 and 6436 A, and the it = 5/2 — 5/2 sub-band whose R head is at 6411 A,reported in Ref. [88]. As yet no sub-bands have been identified in our laser excitationspectra that involve the other two spin-orbit components of the ground state, 40312 and4A 1 1 2 ; presumably this is because these components (which lie 642 cm' and 1000 cm',{I J' F'I(J'F'; J"F") oc (2F' + 1)(2F" + 1)2(6.1)1 F" J"Chapter 6. Laser Intracavity Spectroscopy of CoO^ 204respectively, above 40 712 ) are not sufficiently populated in our discharge system, wherethe rotational distribution corresponds to room temperature.A typical hyperfine pattern is illustrated in Fig. 6.1, where we show the P(92) line ofthe St' = 7/2 — X 4A 7/ 2 sub-band at 6436 A. As expected for case aoj hyperfine coupling[17], the hyperfine widths of the lines decrease with J, and the parameters are suchthat at these intermediate J values the total width is about 0.2 cm -1 . Fifteen hyperfinecomponents appear in the Figure, as well as various centre dips (crossover resonances).Since the only stable isotope of cobalt, 59Co, has I = 7/2, these are readily assigned as theeight "main" hyperfine transitions with OF = AJ = —1 and the seven "satellite" lineswith OF = 0. According to the rules of angular momentum coupling [6], a rotationalline is made up of hyperfine components following the selection rules OF = 0, +1, whoserelative intensities are given byThis equation predicts that the OF = —AJ "satellite" components of an R or P lineshould be considerably weaker than the others, and indeed they are not seen in Fig. 6.1.The relative intensities of the hyperfine components of a rotational line are not wellreproduced by Eq. (6.1) in our intracavity spectra. We find experimentally that thestronger lines are reduced in intensity compared to what would be expected from theweaker lines. Since the output power of the laser does not change noticeably when flu-orescence occurs, this effect presumably results from saturation: in other words opticalpumping in a stronger transition removes a large enough fraction of the molecules fromthe lower level that the fluorescence intensity is no longer proportional to the total con-centration.Patterns containing hyperfine satellite lines, such as those in Fig. 6.1, give direct9-1010-1111-1212-13 F' - F1115520.074 cm -112-12cdFigure 6.1: Hyperfine structure of the P(91) line of the SZ' = 7/2 — X 4A 7/2 band of Co0 at 6436 A;"cd" indicates a center dip or cross-over resonance.cdi10-10 9-96-6I8-98-8cdcd cdL .7-86-7I^5-6e15519.885 cm-1^cn.ocbcl4.,:::20....,cd)07-7IAChapter 6. Laser Intracavity Spectroscopy of CoO^ 206hyperfine combination differences for both the upper and lower states to comparativelyhigh J values. This allows the J-dependence of the hyperfine splittings in the two states tobe measured separately, rather than inferred from the changing "spread" of the OF = AJcomponents; this information is important for molecules in case (a) coupling where spin-uncoupling effects cause a very sizeable variation in the apparent magnetic hyperfinepatterns with rotation even at low J values [88].Figs. 6.2 and 6.3 illustrate the hyperfine patterns in the Q(32) lines of the 6338 Aand 6411 A bands (ft = 7/2 and 5/2); the F assignments are again easily made bycombination differences. In Q lines the hyperfine components with OF = 0 and +1 havecomparable intensity at low J. Because the hyperfine energies go as F(F + 1) withineach rotational level, the hyper fine structure of a rotational line looks very similar to therotational structure of a vibrational band: there are hyperfine R, Q and P "branches",corresponding to the three values of AF and, in the examples shown, the OF = — 1"P branch" forms the equivalent of a rotational P branch head. Although it may looksimilar, as in the present case, the intensity distribution in the hyperfine branches usuallydiffers quite considerably from that in rotational branches because, instead of Eq. (6.1),the appropriate intensity expression [6] isJ'^1 jilI(J'S-2'; J"9") cx (2J' + 1)(2J" + 1)()(6.2)-AI'^q CrThe difference is that Eq. (6.1) reflects the distribution of the intensity among the Fcomponents of a J' — J" rotational line, where the vector coupling is J + I = F, whileEq. (6.2) arises from the projection of the molecule-fixed transition dipole moment ontothe space-fixed axes of the apparatus.EUtoO)Ot—tiIf)r-I1-06-65-4 •4-43-3^5-6^4-5 3-4 2-33-2^2-10-1••7-75-51-2I^1-1 i2-2Figure 6.2: Hyperfine structure of the Q(32) line of the ST = 7/2 — X 407/ 2 band of Co0 at 6338 A.II•V— F"7-66-5I /7-7- F" 6-65-5 1-27-6 6-5 5-4 4-32-3S4-43-2^2-13-44-55-66-7U MFigure 6.3: Hyperfine structure of the Q(32) line of the 1' = 5/2 — X 4A5/2 band of Co0 at 6411 A.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 2096.iii.B Energy level expressionsThe patterns of the electron spin structures of the upper states are not yet understoodin detail, and so far only the two lowest components of the ground state, X 407/ 2 and4A51 2 , have been analysed from high resolution spectra; therefore we have fitted the datato standard case (a) rotational and hyperfine energy level expressions [72], but omittingmatrix elements off-diagonal in a In other words, even though the parallel polarizarionmakes it fairly certain that the two upper electronic states are parts of 40 electronicstates, like the ground state, we have treated all the SZ sub-states independently. Thefact that the S2 sub-states are not truly isolated, but interact through a spin-uncouplingmechanism, produces a large apparent centrifugal distortion correction to the case (a)magnetic hyperfine parameters.The matrix elements that we have used are< JCI I Hrot Hmag. hf s.JSZ >^ (6.3)To d BJ(J +1) — DJ 2 (J + 1) 2 +[h DhJ(J +1)][F(F +1) — I(I + 1) — J(J +1)] CZ2J(J +1)< J —1 QIHmag hfsIJ9> =^ (6.4)—[hDhJ(J +1)]\/J2 S2 2 \AI J F +1)(J + — F)(F J — 1)(F + — J +1)2JV4J 2 — 1where the determinable case (a) magnetic hyperfine parameters are related to the Froschand Foley [17] parameters by= aA (b c)E.^ (6.5)For the 51 = 7/2 upper state of the 6411 A band and the 4 A512 component of the groundstate we needed to add an electric quadrupole hyperfine term to the diagonal element.This was taken directly from Ref. [72], and is given here for completeness:< JO Hquadrupole JC2 >= e2yqo[3522 — J(J 1)]{3W(W + 1) — 41(1 + 1)J(J + 1)} 81(21 — 1)J(J + 1)(2J — 1)(2J + 3)(6.6)Chapter 6. Laser Intracavity Spectroscopy of CoO^ 210where W = F(F + 1) — I(I + 1) — J(J + 1). The energy levels were calculated from amatrix treatment of the expressions (6.3), (6.4) and (6.6), that is, including those parts ofthe rotational and hyperfine Hamiltonians with the selection rules AJ = 0, +1; Aft = 0only.6.iii.0 ResultsA separate least squares run was carried out for each set of f2 sub-states. The dataset for the 52 = 7/2 states consisted of 299 hyperfine transitions from 24 R, Q and Protational lines belonging to the two ft' = 7/2 — 4071 2 sub-bands at 6338 and 6436A, where the J values range from 32 to 122; that for the f2 = 5/2 states contained 73components of six rotational lines from the 6411 A band (52' = 5/2 -4052; J = 1 - 102).The derived parameters are given in Table 6.1, and the assigned hyperfines lines of thethree subbands are given in Appendix F.The rotational constants given in Table 6.1 are close to those obtained from theDoppler-limited data of Ref. [88], although they cannot be compared directly since thereare many rotational perturbations in the higher J rotational levels of the upper states. Aswas found in Ref. [88], the upper state centrifugal distortion parameters are completelyunrealistic because of these perturbations, and they should not be taken literally; theyare needed in order to reproduce the observed line frequencies, but have no physicalmeaning. The ground state centrifugal distortion parameters are not as well determinedas in the combination difference fits of Ref. [88] because of the more limited range of Jin the present data sets; similarly the ground state quadrupole parameter e 2 Qq0 (405 / 2 )is probably also not meaningful.The parameters ph, representing apparent centrifugal distortion corrections to themagnetic hyperfine parameters, are quite small in the ground state, but are up to anorder of magnitude larger in the excited states. As described below, these parametersChapter 6. Laser Intracavity Spectroscopy of CoO^ 211Table 6.1: Rotational and hyperfine constants derived from the red bands of CoOUpper StateBandSr6338 A7/26436 A7/26411 A5/2To 15772.5349 (2)^15535.7869 (21) 15899.3120 (6)B 0.407680 (10)^0.422829 (44) 0.424799 (63)106 D 1.32 (5)^3.27 (24) -6.59 (156)h 0.05664 (12)^0.09715 (47) 0.06616 (40)104 Dh -0.19 (12)^3.08 (21) 7.8 (7)e2Q q 0.0044 (17)Ground State4A7/2 4A5/2To 0 304.321a FIXEDB 0.500639^(12) 0.502879 (31)106D 0.81^(7) 0.68 FIXEDh 0.02914^(13) 0.0403 (3)104 Dh 0.28^(13) 0.42 (56)e201, -0.0014 (19)Q 0.00027 0.00031All values in cm -1 . Error limits in parentheses are twice the standard deviations. aValueobtained from a separate spin-orbit splitting measurement, see Section 6.iv.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 212mostly result from the effects of spin-uncoupling, which have not been explicitly includedin the matrix elements for the rotational and hyperfine structure; their magnitudes canbe estimated by second-order perturbation theory.6.iii.D Interpretation of the hyperfine parametersOne of the main uses of hyperfine parameters in transition metal containing radicalsis to establish the electron configurations. In CoO the hyperfine parameters, combinedwith the bond lengths, allow us to distinguish between the three possible configurations(4so- ) 2 (3d5) 3 (3dir) 2 , (4so-) 1 (3d8)3 (3dr) 2 (3da) 1 and (0,2p7r)3 (4so-) 2 (3d8)3 (3d7r)3 , which allproduce 4,6, electronic states. The first configuration of the three is the logical choice forthe ground state, being an interpolation between the ground configurations of FeO andNiO (o-53ir 2 'A and u 2S4r 2 SE- , respectively); it is expected to lead to a negative Fermicontact parameter because of spin-polarization effects, while the second configurationshould give positive Fermi contact parameters since the unpaired 4sa electron is capableof penetrating into the spinning nucleus.The determinable hyperfine parameters h for the ground state can be rewritten interms of Frosch and Foley's a and (b c), using Eq. (6.5), asX4A : a = 0.02295 cm-1; (b c) = —0.0111 7 cm-1 (6.7)For molecules following case (a) coupling it is not possible to separate the Fermi contactparameter bF = b from the dipolar parameter c unless the matrix elements of theisotropic operator b I • S that are off-diagonal in Sl give observable effects. These matrixelements [72] contain J-dependent factors such that their effects only become noticeableat high J values, as the transition to case (b) coupling caused by the spin-uncouplingoperator becomes important. In the present case the range of J in the data set is not largeenough for bF and c to be separated experimentally. However we can estimate [73, 74]Chapter 6. Laser Intracavity Spectroscopy of CoO^ 213the dipolar parameter c from ab initio calculations of the size of the 3d orbitals, and findit to be very small; therefore the negative sign of (b + c) is only consistent with a negativevalue of bF, or in other words an electron configuration which has no unpaired electronsin the a orbital derived from the cobalt 4s atomic orbital, namely (4so-) 2 (3d6) 3 (3dr) 2 .To estimate c we use the expression [17, 73, 74]1c = (it0/47rhc)(3.WingNILN/2) — E(< 3 cos 2 O — 1 >/), < r -3 >ni).2S iThe angular factors in Eq. (6.8) can be obtained from2[3A2 — 1(1+ 1)] < 3 cos 2 0 — 1^=(21 — 1)(21 + 3)so that, besides the nuclear magnetic moment of 59 Co, only the expectation values <r -3 >ni are needed to obtain c for a particular electron configuration.The logical assignment for the ground state configuration of CoO, o- 2 83r 2 , turns outto give c = 0, since, with the angular factors < 3 cos 2 8 — 1 > cis= —4/7 and < 3 cos 2 0 —1 >d,r = 2/7, from Eq. (6.9), the summation over the three unpaired electrons gives —4/7+ 2(2/7) = 0. Obviously the 3dr orbital, so called, must be principally a mixture ofcobalt 3dr and oxygen 2pr, with some cobalt 4pr. Since neither ab initio calculations[89] nor experiment give us any way of estimating the linear combination coefficients, wecan only say that the value of c is "likely to be small". The individual contributions ofthe b' electron and the two 7t electrons can be estimated, using the value < r -3 > 3d=4.53 x 1025 cm3 , given by Morton and Preston [75], to be 0.004 cm -1 . This number is anupper limit to the absolute value of c.The upper states of the bands reported in this Part of Chapter 6, on the other hand,are consistent with "A states from the other two electron configurations. It can be seenfrom Table 6.1 that the 6411 A (52' = 5/2) and 6436 A (St' = 7/2) bands have roughlythe same upper state B' values, while the other strong band at 6338 A has a smaller B'(6.8)(6.9)Chapter 6. Laser Intracavity Spectroscopy of CoO^ 214value. It seems likely that the upper states of the two long wavelength bands are the52 = 7/2 and 5/2 components of a vibrational level of the (4sa) 1 (3d6)3 (3dir) 2 (3da) 1 4Astate. Two lines of reasoning lead to this assignment.(i) Isotropic hyperfine parameterIf the 6411 and 6436 A bands belong together, the upper state hyperfine parametersfollow from Table 6.1 and Eq. (6.5):2 40 : a = 0.02533 cm-1; (b + c) = 0.03099 cm-1 . (6.10)Now the expected value for the isotropic parameter b for a state of CoO where thereis an unpaired electron in the 4so m.o. can be estimated by scaling the data for VO,assuming that the composition of the 4scr orbital does not change between CoO and VO.The scaling factors are the ratio of the nuclear magnetic moments and the ratio of thevalues of the 4s electron densities at the nucleus, 14 s (0), which can be taken from thecalculations of Ref. [75]. The (48))(18) 2 X4 E- ground state of VO has [64] b = 0.02731cm-1 ; multiplying this by the ratio fINT4s(0)(59Co) / fiNkli4.(0)(51 V) = 1.2802, we expectquartet states of CoO with a single unpaired 4sa electron to have b 0.0349 cm'. Thisnumber is to be compared to the value of (b + c) = 0.0310 cm -1 given in Eq. (6.10).(ii) Ratio of the spin-orbit coupling constant to the hyperfine a parameterThe spin-orbit coupling constant, A, for a multiplet electronic state is related simply tothe electron orbit-nuclear spin coupling constant, a. Specifically, if there is a singleelectron responsible for the orbital angular momentum, the spin-orbit coupling constantis given (in cm -1 ) byA = (gp2B Zeff 14r eohc3)while the hyperfine a parameter is given bya = (110147rhc)(2pBgNitN )11<<11>,> .(6.11)(6.12)Chapter 6. Laser Intracavity Spectroscopy of Coo^ 215Since cop° = c -2 , the two parameters are in the ratioA= gtiBZeff 71 ^,zgNtiN(6.13)in first approximation.Anticipating the result, from Section 6.iv, that the X40 5/ 2 component lies 304.3 cm"above X4 7/ 2 , we find from Table 6.1 that the upper state of the 6411 A band lies 363.5cm' above the upper state of the 6436 A band. The ratio of the a parameters in theupper and lower states should then be approximatelya(2 4A) _ 363.5a(X 4A) — 304.3= 1.19.^ (6.14)Experimentally this ratio, from Eqs. (6.7) and (6.10), is 1.10.Obviously we cannot expect exact agreement because it is clear that the asymmetryin the ground state spin-orbit coupling, described below, implies considerable spin-orbitdistortion of the ground state, while we have no way of assessing the corresponding effectin the upper state. Nevertheless the available evidence points to the Sr = 5/2 upper levelof the 6411 A band being associated with the Sr = 7/2 upper level of the 6436 A band,rather than with that of the 6338 A band.If the other choice is made, the hyperfine parameters of the upper electronic statewould be a = 0.0357 cm" and (b + c) = —0.0105 cm" (from Table 6.1). The Fermicontact parameter would be compatible with the "charge transfer" electron configurationr3o-2 83 7r 3 , but the implied spin-orbit interval AA = —126.8 cm -1 would lead to a muchsmaller value of a, near 0.0096 cm", which is not consistent. In addition the rotationalconstants would differ by more than 0.02 cm", which is not easily explained in terms ofa single electronic state.Given that the 6411 and 6436 A bands are probably associated as spin-orbit compo-nents of a single vibrational band, a logical assignment is that the CV = 7/2 upper levelChapter 6. Laser Intracavity Spectroscopy of CoO^ 216of the 6338 A band is part of the r 3o- 2 83 7r 3 "Ai state, for which (as just indicated) theFermi contact parameter should be small and negative. From the data in Table 6.1 the ahyperfine parameter would be about 0.029 cm", which by Eq. (6.14) is consistent witha 4/ii state of CoO where the spin-orbit intervals are about 380 cm -1 . There is in fact aweak CY = 5/2 band at 6305 A whose upper level lies at 16163 cm' (that is, 390 cm -1above the Sr = 7/2 level) and which was initially thought to be possibly associated withit as part of a second 4 A — 4 ,6, electronic transition. However this band, which appears atthe extreme left of Fig. 1 of Ref. [88], is very much weaker than what would be expectedfor a spin-orbit companion to the 6338 A band and it has a much larger upper staterotational constant (see Section 6.ix); it is therefore unlikely that it is associated withthe 6338 A band.To summarize this section, the two hyperfine parameters a and (b c) that can bedetermined from the S2 = 7/2 and 5/2 components of a 4 ,6, state in case (a) coupling canlead to firm deductions about the electron configurations. In the present case, combiningthe hyperfine evidence with the spin-orbit couplings and the bond lengths, it is clear thatthe ground state of CoO comes from the configuration o-26.3 7r 2 , and that the 6411 and6436 A sub-bands belong to a vibrational band of the transition a- S3r 2 cr 4 0 i 4-* a-2 83 71-24A i , analogous to the orange system [56] of FeO. The upper state of the 6338 A bandpossibly belongs to the charge transfer state r 302 63706.iii.E Centrifugal distortion of the hyperfine h parametersThe apparent centrifugal distortion of the magnetic hyperfine structure, representedby the parameter Dh in Eqs. (6.3) and (6.4) and Table 6.1, arises from spin-uncoupling.It can be understood using arguments similar to those that explain (see Part I of Chapter5) the magnetic hyperfine structure of the B 4 11 state of VO.Consider the value of Dh for the X407 2 component. This arises as a cross-termChapter 6. Laser Intracavity Spectroscopy of Co0^ 217between the Fermi contact interaction and the spin-uncoupling operator. The matrixelement between the 407/ 2 and 405 / 2 spin components in case (a) coupling can be written< 4 0 7/ 2 , JIFillniag .hfs.+ Hrot.I405121JIF > =—V3[J(J + 1) — 35/41 [B 4,17+ 0 1(6.15)where, as before, W = F(F 1) — I(I + 1) — J(J + 1). Application of second orderperturbation theory to this element produces three corrections to the 40 7/ 2 energy. Thefirst has the form 3B 2 [J(J+ 1) — 35/4]/AE(40 7/ 2 — 4 .6, 51 2 ), and is a contribution to theeffective B value of the 7/2 substate [20], originally described by Mulliken. The secondterm is a cross-termAE (2) = 3B b[J(J + 1) — 35/4] W 2J(J 1) AE( 40 7/2 — 4 5/0(6.16)which can be identified with the Dh term of Eq. (6.3). If we disregard the factor of 35/4compared to J(J 1) we findDh =3B b (6.17)AE(407/2 — 4A5/2) •With AE( 40 7/ 2 — 4 A512) = —304.3 cm', B = 0.503 cm -1 and b = —0.0112 cm -1 , thisgivesDh = 0.16 x 10 -4cm-1 ,^ (6.18)which is seen to be well within the error limits of the experimental value in Table 6.1.The third term arising from Eq. (6.15) is a very small correction to the quadrupole term,which is not considered further. Similar arguments would apply to the Dh term of the4 5/ 2 component, but since the error limit on the experimental value is considerablylarger, the comparison is less meaningful.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 2186.iv Spin-orbit structure of the X4 02 ground state of Co0We have recently examined (see Part II) the rotational structure of the red systemof Co0 in the region 6100-6500 A in considerably more detail than Ref. [88]. Thereare about twenty overlapping bands in this part of the spectrum and, with the strongred-degradation and widespread perturbations, analysis was only possible with extensivewavelength-resolved fluorescence experiments. When lines of the more severely perturbed= 7/2 — X4 07/2 sub-bands are excited, many of them give weak emission signals about300 cm' to the red of the exciting line, as illustrated in Fig. 6.4. The patterns of theseweaker lines vary widely: sometimes a single line appears, more usually two lines and, inone instance, three. Since the X4 0, S2" = 5/2— 7/2 interval had been estimated [88] to beof the order of 240 cm", from application of the Mulliken formula (see Eq. 6.15), theseweak lines clearly have to be case (a)-forbidden spin-satellites. The confusing intensitypatterns could then be deciphered by careful comparison of the combination differences,since the B-values for the two lower levels S2" = 5/2 and 7/2, are known [88]. It turnsout that when a single line appears it is usually the R line though at higher J it may bethe Q line. Patterns of two lines are invariably R and Q, as in Fig. 6.4. Eventually itwas established that the spin-orbit interval SI = 5/2 — S2 = 7/2 is about 304 cm".The dispersion of the 0.75 m spectrometer used for these wavelength-resolved fluores-cence measurements is 11 A /mm, so that the accuracy of the determination is at best+0.3 cm". However, after much searching through the Doppler-limited laser excitationspectra, a fragment of a weak parallel-polarized it' = 5/2 — = 5/2 sub-band at 15580cm' was found to have the same upper state rotational combination differences as arather stronger perpendicular-polarized ST = 5/2 — S2" = 7/2 sub-band at 15884 cm'.Least squares fitting of the intervals between corresponding lines, with the rotationalconstants set equal to the known values for the CI" = 7/2 and 5/2 spin components ofR(19.5)Avb\•SZ = 7/2 - X407/2 (v', 0)P(21.5)fZ = 7/2 - X405/2 (v', 0)R(19.5)Q(20.5)Figure 6.4: Resolved fluorescence from selective excitation of the R(191) line of the ft = 7/2 — X 4A7/2 (6338 A)band of CoO; the weak R and Q spin-satellite lines are about 300 cm -1 to the red of the exciting line.Chapter 6. Laser Intracavity Spectroscopy of Ca)^ 220the ground state, established the lowest spin-orbit interval asAE(52" = 5/2 — f2" = 7/2) = 304.321 + 0.007 cm -1^(2a)^(6.19)The assigned lines of these two sub-bands are given as Table 6.2.The common it' = 5/2 upper state is unusual, being a level with an extremely largebut smoothly-varying A-doubling; evidently a nearby E state must be present to causethe large A-doubling. The details of the interacting levels will be reported in Section6.ix.Various other groups of case (a)-forbidden spin satellites could be identified and as-signed similarly from wavelength-resolved fluorescence spectra. An example is given inFig. 6.5. In this Figure we show the spectra obtained from exciting the R(30D lines ofthe 6338 A band (71-30- 2 53 71- 3 4A712 — X 40 7/2 ). Near this J' value one of the A-componentsis involved in an avoided crossing pattern while the other is essentially unperturbed [88].At the frequency of the (v', 1) vibrational band the unperturbed line gives a simple Pand R pattern, while the perturbed component gives in addition a weaker P and R pat-tern 150 cm -1 to the red. The extra pair of lines can be assigned as emission to the40 1 /2 , v = 0 component of the ground state, even though it lies about 100 cm -1 belowwhere it should appear if the spin-orbit intervals all follow Eq. (6.19). The implicationis that the ground state spin-orbit structure is fairly asymmetric, but excitation of otherbands proves that this must be so, because similar weak emissions to the 4A3/ 2 spin-orbitcomponent appear in a consistent location.The results of the wavelength-resolved fluorescence measurements are collected inTable 6.3. As indicated above the monochromator measurements of the spin-orbit andvibrational intervals are probably accurate to ±0.3 cm -1 , while the effective B values areprobably accurate to +0.0008 cm -1 .The degree of asymmetry in the ground state spin-orbit structure can be appreciatedChapter 6. Laser Intracavity Spectroscopy of CoO^ 221Table 6.2: Assigned rotational lines of the SY = 5/2 - S2" = 5/2 band at 15580 cm -1 andthe 5/2 - 7/2 band at 15884 cm -1 (*=blended).J R= 5/ 2 - = 5/2Q P8.59.515580.26* 15580.46*15579.50*10.5 15577.935 15578.329 15560.35011.5 15576.45* 15577.00* 15557.087 15557.37712.5 15574.75* 15575.500 15553.820 15554.22013.5 15572.807 15573.801 15550.330 15550.86214.5 15570.61* 15571.962 15546.600 15547.32915.5 15568.165 15569.961 15542.64* 15543.64*16.5 15567.776 15539.79417.5 15565.493 15535.79718.5 15531.627= 5/2 - = 7/23.5 15880.3854.5 15882.40*5.5 15886.609 15881.371 15876.9186.5 15880.107 15880.193 15874.8497.5 15885.742 15878.787 15872.611 15872.7098.5 15884.978 15877.073 15877.224 15870.180 15870.2829.5 15884.045 15875.258 15875.464 15867.565 15867.71610.5 15873.230 15873.534 15864.749 15864.95311.5 15881.650 15871.020 15871.419 15861.736 15862.02112.5 15880.193 15868.601 15869.148 15858.512 15858.91213.5 15878.580 15865.95* 15866.691 15855.62314.5 15876.815 15863.080 15864.073 15851.44815.5 15874.869 15859.989 15861.306 15847.569 15848.56416.5 15872.785 15858.367 15844.78217.5 15870.579 15855.289 15840.84918.5 15868.245 15852.07619.5 15848.74420.5 15845.28521.5 15841.72722.5 15838.08623.5 15834.36524.5 15830.58725.5 15826.739R(30.5)14860.64 cm -1P(32.5)14797.00 cm-1(b) Unperturbed R(30.5) line excitedmtwitiktR(30.5) P(32.5)(a) Perturbed R(30.5) line excited14860.39 cm -10 = 7/2 — X 40 1/2(v',0)R(30.5) P(32.5)14701.43 cm -1 14636.88 cm -1wywooiwi'44Chapter 6. Laser Intracavity Spectroscopy of Co0^ 222Figure 6.5: Resolved-fluorescence patterns from the (a) perturbed and (b) unperturbedR(302) lines of the 6338 A band of CoO; the extra R and P lines in (a) are perturba-tion-induced Cr = 7/2 — X 40 1 /2 emission lines.Chapter 6. Laser Intracavity Spectroscopy of Ca)^ 223Table 6.3: Results from wavelength-resolved fluorescence measurements of the groundstate of Co0X4A, v = 0ft 7/2 5/2 3/2 1/2To 0 304.321 643.03 1001.88Be f f 0.50058a 0.50288a 0.5028 0.5058X4A, v = 1C2 7/2 5/2To 850.19 1155.27Be f f 0.4977 0.4985All parameters in cm'. aConstants quoted from Ref. [88].from Table 6.3. The rotationless parts of the diagonal elements for a 4 LS, state in case (a)coupling are given by [62]To(IA7/2)To(46,5/2)To(413/2)To( 4 A1/2)= To + 3A + 2A + -7/ — 11B= To + A — 2A — -77 — 3B= T0 —A-2A+h-FB= T0 — 3A + 2A — s7/ + B(6.20)Converting these to the form of spin-orbit parameters rather than sub-state origins, wehaveTo = 488.8 2 , A = 166.2 1 , A = 7.32, i = —1.1883 cm-1 .^(6.21)It can be seen that the second-order spin-orbit parameter A is about a twentieth of thefirst-order parameter A. If there were no first-order spin-orbit interaction, or in otherwords if this were a 1 E state, this value of A would cause a second-order splitting of4A :.-_-' 30 cm -1 between the 4 E312 and 4-1/2 components.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 2246.v ConclusionThis study of the ground state of CoO, using wavelength-resolved laser excitationand sub-Doppler intracavity laser-induced fluorescence, has given the details of the hy-perfine structure for the lowest two spin-orbit components, X 4071 2 and X405 / 2 , andlocated the other two components, X 4 A3/2 and X40 1 / 2 , though with lower accuracy.The hyperfine structure is consistent with the X 401 ground state coming from the elec-tron configuration (480) 2 (3c/6)3 (3c/70 2 . Some deductions can be made about the electronconfigurations of the upper states of the "red" system of CoO: the upper state of the6411 A and 6436 A sub-bands is probably another 4A i state, from the configuration(4so-) 1 (3d6)3 (3chr) 2 (3da) 1 , while the 52' = 7/2 upper level of the 6338 A band appears tobelong to the charge transfer state (0,2pr)3 (4so-) 2 (3d8)3 (3dir)3 4 Ai.Wavelength-resolved fluorescence has located the X 403/2 and X4 0 1 / 2 componentsso to give a direct measurement of the spin-orbit structure of the ground state. Theobservation of the X 40 112 component in a ST = 7/2 — = 1/2 emission is interestingsince this is a highly forbidden ASI = 3 transition. But it is suspected from rotationalanalysis of the 6338 A sub-band [88] that the 52 = 7/2 upper state is perturbed by anunseen 4 E 1 12 sub-state so that it contains some of the character of an 52 = 1/2 spin-orbitcomponent. Interactions between spin-orbit sub-states differing by three units of ft areseldom observed; it is believed that various unidentified ft = 5/2 and 3/2 sub-states mustact as intermediate states in this third-order spin-orbit interaction.The spin-orbit structure of the X40 i state is found to be fairly asymmetric; the threespin-orbit intervals are measured to be 304.3, 338.7 and 358.8 cm -1 respectively. It ispossible that the spin-orbit asymmetry is caused by interactions between the X4Ai andan unseen 2 ,6, state from the same (43(7) 2 (3d6) 3 (3thr) 2 electron configuration. Since nosuch low-lying 20 states have yet been observed in CoO, we must reserve final judgementChapter 6. Laser Intracavity Spectroscopy of Ca)^ 225on the true cause of the asymmetry of the spin-orbit structure of the ground state.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 226Part IIRotational Analysis of Some Sub-bands of the Red System of Gaseous CoO6.vi IntroductionThe electronic bands of cobalt oxide, CoO, in the red and near infrared were firstreported by Howell and Rochester [91] and later by Malet and Rosen [92] about halfa century ago. No rotational analysis of any of these bands was performed until 1987,when Adam et al [88] rotationally analyzed three comparatively strong sub-bands ofthe red system. The rotational work of Ref. [88], combined with the hyperfine andspin-orbit study of Part I of this Chapter has well characterized the structure of the(480)2(36)3(3c/7 )2 4 A iLA ground state, but the structure of the upper states of the redsystem is by no means fully understood. In our continuing efforts to understand theexcited states of CoO, we have recently investigated the rotational structures of some ofthe remaining sub-bands of the red system.In Part II, we report an analysis of 16 CoO sub-bands in the region 15500 — 16500cm -1 from the laser excitation spectra taken with our intracavity laser spectrometer.The high sensitivity of the intracavity experiments enabled us to detect even the veryweak rotational lines without using a modulation method. Doppler-limited spectra of the16 sub-bands in our wavelength region have been recorded; rotational analysis of thesesub-bands indicates that both parallel and perpendicular transitions are present, withthe lower states being only the v=0 levels of the X 40 71 2 and X 4A 51 2 spin-orbit sub-states. The upper states of most of the sub-bands are perturbed, both vibrationally androtationally; the extensive perturbations result in a very complex upper state energy levelChapter 6. Laser Intracavity Spectroscopy of Co0^ 227structure, reminiscent of that of Fe0 [56]. The present study shows that there possiblyexist three separate 4 6, electronic states, three 4(1) states and one 411 state within theenergy interval 15500 — 16500 cm -1 relative to the ground state in CoO.6.vii ExperimentalThe Co0 source and the fluorescence-based intracavity laser spectrometer has beendescribed in detail in Part I. Dopper-limited spectra were obtained by collecting thetotal fluorescence of Co0 with a photomultiplier tube (PMT) and directly recording theelectric current from the PMT as a function of laser frequency. Our laser excitationspectra cover most of the 15500 — 16500 cm -1 region with a resolution of about 0.03cm-1 . The laser dyes used in this wavelength region were R6G and DCM (Exciton Inc.);the circulating radiation power inside the laser cavity is typically one watt, which is morethan adequate for Doppler-limited experiments.Absolute frequency calibration of the Co0 spectra was provided by iodine absorptionspectra whose line frequencies were taken from Gerstenkorn and Luc's atlas [43]. Toobtain absolute rotational and lower-state vibrational assignments of the Co0 sub-bands,wavelength-resolved fluorescence spectra were also recorded for selected rotational linesby passing the laser-induced fluorescence through a 0.75 m spectrometer (Spex model1702) and recording the dispersed fluorescence with a microchannel-plate intensified arraydetector (PAR model 1461). The resolution of the dispersed spectra is limited by thecharacteristics of the array detector to about 0.2 cm-1.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 2286.viii Appearance of the spectrumThe red system of CoO is a complex group of bands in the wavelength region 5700 —7000 A. The wavenumbers of the principal sub-band heads, as observed by laser exci-tation, have been listed in Ref. [88]. At low resolution, the major sub-bands are seenas forming several vibrational progressions extending throughout the red region withintervals of about 500-700 cm -1 .The sub-bands which we have investigated lie in the region 6060— 6450 A (or 15500 —16500 cm-1 ). The laser excitation spectra in the low frequency end of this region aredominated by three comparatively strong sub-bands; these are the 6338, 6436 and 6411A bands analyzed in Part I and Ref. [88]. At the high frequency end of our wavelengthregion, two fairly strong overlapping sub-bands are easily identified near 6127 and 6151A ; another weaker but obvious sub-band is located near 6221 A. The energy positionsof the 6127 and 6221 A sub-bands seem to indicate that they are the next members ofthe vibrational progressions that include the two 40 71 2 — X4 A 7/2 sub-bands at 6338 and6436 A. High resolution studies, as we discuss below, identify 13 other weak sub-bands,in addition to the six obvious sub-bands mentioned above, in the 15500 — 16500 cm -1region. Descriptions of the individual sub-bands and their upper levels now follow.6.ix Rotational analysis of the CoO sub-bands6.ix.A The 4 0712 — X 4 07/ 2 sub-bandsThe strongest features in the laser excitation spectrum of CoO are bands with Q' =Q" 7/2; sub-bands coming from the X 40 5 / 2 sub-state are also observed, but withmuch less intensity; no sub-bands from the X403/2 and X 40 112 sub-states or vibrationalChapter 6. Laser Intracavity Spectroscopy of CoO^ 229hot bands have been observed in absorption. This is easy to understand from the Boltz-mann factors. As was shown in Part I of this Chapter, the lowest unobserved state,4A3/ 2 , v 0, lies 643 cm -1 above 4A 7/2 , v 0; assuming an effective temperature of 350K, the population of the 4031 2 state would be only 7% of that in 4 A 71 2 . Therefore thelaser excitation spectrum (which is essentially an absorption spectrum) is understand-ably dominated by bands coming from the X40 7/ 2 and X 4 05/ 2 sub-states. Previousanalysis of the 6338, 6436 and 6411 A bands in Ref. [88] and Part I of this Chapter hascharacterized the structure of the X 40 i ground state very well; our discussions of thenew sub-bands of CoO will mainly focus on the structures of the excited states.Figs. 6.6 and 6.7 illustrate the low J structures of the two strong 40 7/ 2 — X40 71 2sub-bands near 6127 and 6151 A. Rotational assignments of these two subbands werestraightforward to make using the known lower state combination differences [88], thoughthe overlapping of the 6151 A band with a 40 51 2 —X42 5/ 2 band (see Fig. 6.7) complicatedthe analysis a bit. Both sub-bands can be quickly identified as parallel-polarized bandssince the intensities of the Q branches decrease very rapidly with J and can only befollowed to J ti 20.5, while the R and P branches can be followed to J 40.5. Theupper states of the 6127 and 6151 A bands apparently have much smaller rotationalconstants than the ground state since the R branches of both sub-bands turn aroundat the very first rotational lines. Like almost all other known CoO sub-bands, the 6127and 6151 A sub-bands have heavy perturbations in their upper states. Figs. 6.8 and6.9 show the upper state energy levels of the two sub-bands, suitably scaled to magnifythe details, plotted against J(J + 1). It is clear from these two pictures that there aretwo different local perturbations acting in both upper states. The first perturbation,occurring at low J, is fairly small in magnitude and seems to be caused by an unseenorbitally nondegenerate state such that one A-component of the SY = 7/2 sub-state suffersan avoided crossing, while the other A-component remains unperturbed. The second3.55.5 7.5 9.5^11.5 R(J)Q(J)P(J)13.53.5 4.5 5.5^6.5^7.54.58.5 5.54.40t4 v‘emvii" ""Owisv4vtots +,04) I 1^I^I^I^1^I 1^I^I I16322.0 16320.0 16318.0 16316.0 16314.0 16312.0 16310.0Figure 6.6: The low J structure of the 407/2 — X407/2 subband of Co0 near 6127 A.*16249.307 cm-1I IA,^*^i„,rgko, *AL* wop 4,,Aus 1 , 014R 4 15 1^I^I^i^Plii 1T1 .555 8.510.5 14.5x405/2^Q 2.5 6.5Figure 6.7: The low J structure of the 40712 —X4 A7/2 subband of Coo near 6151 A. The 40512—X405/2subband, with an R head at 6152 A, appears also in this wavelength region. * indicates a line belongingto the 6127 A band.46,7/2 — X407/2^Q 3.5IIIIIIIIR I I1 VA1J.*/ fr^444)v)/I^II^II^I^i1^11i^I^IIII^I I^II^11 I^I 14.5 III5.54A512co16259 -16257 -16255 .^0I^r500^J(J+ 1)^1000 1500E/cm- 1Chapter 6. Laser Intracavity Spectroscopy of Co0^ 232E/crril - 16326 -CV=7/26127 A band16322 -16318 -10^ 1000^J(J+ 1)^2000Figure 6.8: The upper state energies of the 6127 A band of CoO, plotted against J(J+ 1).A quantity of 0.408 x J(J + 1) — 6.0 x 10' x [J(J+ 1)1 2 has been subtracted to magnifythe scale.Figure 6.9: The upper state energies of the 6151 A band of CoO, plotted against J(J+ 1).A quantity of 0.402 x J(J + 1) — 6.0 x 10-7 x [J(J+ 1)1 2 has been subtracted to magnifythe scalP16314Chapter 6. Laser Intracavity Spectroscopy of Coo^ 233perturbation, occuring at higher J, is apparently caused by an orbitally degenerate statewith reasonably large A doubling. The two local perturbations acting in the upper stateof the 6127 A sub-band are relatively well-behaved and enough extra lines have beenidentified to give approximate rotational constants for the two perturbing states and themagnitudes of the perturbation matrix elements:A) To = 16311.2 cm", Beff = 0.419, H12 = 0.3 cm";B) To = 16257.5 cm', Bell = 0.461, H12 = 2.8 cm".It is quite clear, judging from the term values and rotational constants of the two per-turbing states, that they must be different. Similarly the upper state of the 6151 A bandis perturbed by two other different electronic states.Besides the two strong fr = 7/2 — f2" = 7/2 sub-bands discussed above and thetwo previously analyzed sub-bands (at 6338 and 6436 A), there are two further ST =7/2 —C2" = 7/2 sub-bands in the 15500-16500 cm" region. These are the comparativelyweak 6221 A band and an even weaker sub-band near 6388 A; the lower state combinationdifferences of these two sub-bands confirm that they both come from the v = 0 level ofthe X40 7/2 sub-state.The 6221 A sub-band is comparatively clear, and free from overlapping by other sub-bands. The hyperfine widths of its low J rotational lines are very wide (, 0.3 cm"for P(6.5)), rather like those of the 6436 A sub-band (which may imply that the 6221and 6436 A bands are related). The branch structure of this sub-band behaves regularlyup to J' = 15.5 where extra lines start to appear; still, the rotational assignments arestraightforward to make using the known lower state combination differences. The upperstate is severely perturbed as can be seen in Fig. 6.10 which illustrates the scaled upperstate energy levels plotted against J(J + 1). It appears that the two A-components ofthe St' = 7/2 excited state are respectively perturbed near J = 15.5 and 25.5. It canbe argued that these perturbations could be caused by a single 4 E 1 / 2 state whose two15652E/cmi l15651 -il'=7/2, 6388 A band15650 -15649Chapter 6. Laser Intracavity Spectroscopy of Co0^ 234Figure 6.10: The upper state energies of the 6221 A band of CoO, plotted against J(J+1).A quantity of 0.410 x J(J + 1) — 6.0 x 10' x [J(J+ 1)] 2 has been subtracted to magnifythe scale.0 500 J(J+1) 1000Figure 6.11: The upper state energies of the 6388 A band of CoO, plotted against J(J+1).A quantity of 0.405 x J(J + 1) — 6.0 x 10' x [J(J+ 1)] 2 has been subtracted to magnifythe scale.Chapter 6. Laser Intracavity Spectroscopy of Co0^ 23512-components are widely separated so that they would cross the upper state of Fig.6.10 at quite different places. One interesting aspect of the perturbations in the 6221A sub-band is the effect of the perturbations on the hyperfine widths of the rotationallines. It was found that all the perturbed rotational lines have smaller widths than theunperturbed lines with the same quantum numbers; this is clearly demonstrated by thewidths of the perturbed and unperturbed R(14.5), R(15.5) and R(16.5) lines shown inFig. 6.12. The decreased linewidth due to perturbations indicates that the perturbingstate must have very different hyperfine widths from the Sr = 7/2 sub-state.The 6388 A sub-band is the weakest f2' = 7/2 — 12" = 7/2 sub-band that we havebeen able to assign in the 15500 — 16500 cm -1 region; it is unfortunately overlappedby the high J lines of the strong 6338 A band and lines of two other weak sub-bands.Rotational assignments of this sub-band were eventually made after extensive wavelength-resolved fluorescence studies had identified most of the rotational lines in the crowded15600 — 15650 cm -1 region. The 6388 A band was found to consist of only an R and aP branch, which can be followed to J' = 29.5. Fig. 6.11 shows the upper state energylevels of this sub-band, suitably scaled, plotted against J(J + 1). Clearly the f2' = 7/2excited state is perturbed by an unseen electronic state near J = 23.5; the perturbingstate has a larger effective B constant and an unresolved A-doubling.Effective rotational constants have been derived for the newly observed Co0 excitedstates by the method of least squares. Particular attention has been paid to the selec-tion of the rotational lines for the fits since all known Co0 sub-bands are perturbed tovarying degrees. Only the most obviously unperturbed rotational lines in each sub-bandwere selected, and the line frequencies were fitted to the standard formula [20] with thelower state constants fixed to values given in Ref. [88]. The derived effective rotationalconstants for the 12 = 7/2 sub-states are given in Table 6.4, and the rotational lines ofthe four new 12' = 7/2 —St" = 7/2 sub-bands are listed in Appendix G. Table 6.4 also listsChapter 6. Laser Intracavity Spectroscopy of Co0^ 236a) R(14.5) b) R(14.5) 0.1 cm-1a) R(15.5)^b) R(15.5) 1 0.1 cm- ' 1a) R(16.5) b) R(16.5) 1 0.1 cm- ' 1Figure 6.12: Comparisons of the widths of the (a) perturbed and (b) unperturbed ro-tational lines of the 6221 A band of CoO, showing the effect of perturbations on thehyperfine structures.Chapter 6. Laser Intracavity Spectroscopy of Coo^237Table 6.4: Constants derived for the upper states of the red system of Co0 (in cm -1St' - 52" Ahead /A To Bet f h Jmax Remarks7/2 - 7/2 6127 16319.025 0.40837 0.061 45.5 Intense. Perturbed atJ' = 26.5 and 33.5.7/2 - 7/2 6151 16256.73 0.4029 0.064 37.5 Intense. Perturbed atJ'= 21.5 and 31.5.7/2 - 7/2 6221 16072.9 0.4101 0.097 32.5 Medium intensity.Perturbed.7/2 - 7/2 6338 15772.513 0.40531 0.05664 47.5 Intense. See Ref. [88]and Part I7/2 - 7/2 6388 15650.80 0.4075 0.067 30.5 Weak; Q branch notobserved.7/2 - 7/2 6436 15535.77 0.4224 0.0972 27.5 Intense. See Ref. [88]and Part I5/2 - 5/2 6142 16581.79 0.414 34.5 Very weak. Large A-doubling.5/2 - 5/2 6152 16556.86 0.4275 0.04 29.5 Medium intensity.Perturbed at high J.5/2 - 5/2 6195 16443.1 0.418 0.067 28.5 Weak. Perturbed atlow J.5/2 - 5/2 6305 16162.45 0.426 0.055 27.5 Medium intensity.5/2 - 5/2 6373 15991.49 0.4117 34.5 M. intensity. Both A-components perturbed.5/2 - 5/2 6411 15899.295 0.4250 0.06616 21.5 Medium intensity. SeeRef. [88] and Part I.5/2 - 5/2 6418 15884.64 0.409 18.5 Fragment. Same upperstate as 6294 A band.5/2 - 7/2 6294 15884.64 0.409 25.5 Weak. One A-component perterbed.7/2 - 5/2 6154 16553.63 0.406 35.5 Very weak.7/2 - 5/2 6370 15999.84 0.423 34.5 Weak. Perturbed athigh J.9/2 - 7/2 6186 16162.29 0.3946 0.054 27.5 Very weak. Notperturbed.9/2 - 7/2 6314 15836.4 0.42 0.065 24.5 Weak. Manyperturbations.9/2 - 7/2 6295 15881.79 0.4134 0.086 27.5 Weak. Not perturbed.Chapter 6. Laser Intracavity Spectroscopy of Co0^ 238the approximate hyperfine constants, h, for most of the upper states; they were derivedfrom the partially resolved hyperfine structures (see Fig. 6.13) or the widths of the lowJ rotational lines. Briefly, this is done as follows.It was shown in Part I that the hyperfine structures of the ground state and the othertwo known excited electronic states of Co0 can be well described by the formalism ofcase (a/3). In first approximation, the hyperfine energies of a case (a) electronic state aregiven by the diagonal matrix elements of the nuclear magnetic hyperfine Hamiltonian ofEq. (6.3). If the centrifugal distortion correction terms are neglected for the very low Jlevels, we have, from Eq. (6.3), the following approximate expression for the hyperfinetransition energies:v^+ J' (J i + 1) + 52'h' F' (F' + 1) — J' (J' 1) — I (I + 1) 2J'(J' + 1)— F"(F" +1) — J"(J" +1) — I(I + 1)^(6.22)1[T,C B"J"(J" +1) + f2"h" 2J"(J" + 1)where I, the nuclear spin, is 7/2 for 59Co and F, the total angular momentum quantumnumber, takes values between Fmin and Fmax (in increments of 1) with Frn ir, and Firia„given byFmin = I J — I I,^Fmas = J I.^(6.23)From Eq. (6.22) we obtain the energy separation, vi — v2 , between two hyperfine transi-tions, 1 and 2, within a given rotational line asF'(F +1) —^+1)— v2 =^12J'(J' + 1)cill h,, Fi" (Fi" + 1) — FAPT + 1)2J"(J" 1) (6.24)Using Eq. (6.24), one can calculate the approximate h constants for the upper statesfrom the partially resolved hyperfine structures of the low J rotational lines of the 4 07/2 —X4A 7/ 2 sub-bands, shown in Fig. 6.13, and the known lower state h constant (see TableChapter 6. Laser Intracavity Spectroscopy of CoO^ 2394A4LA7/2 X A7/216311.2387 .2756 .3191 cm -1F"=r+1=^7 8 9I^I^I P (5 .5 )F"-4A7/2 _x4A7/216254.3273 .3614 .4020 cm -16 7 81^1^IR( 1 0.5)Q(4.5)Figure 6.13: Partially resolved hyperfine structures of the P(5.5) line of the 6127 A band(top) and the Q(4.5) line of the 6151 A band (bottom).Chapter 6. Laser Intracavity Spectroscopy of CoO^ 2406.1). In cases where no individual hyperfine transitions are resolved, one can still use Eq.(6.24) to calculate the upper state h constants by simply replacing Fi. and F2 with Fmaxand &in) 111 - V2 with the hyperfine width of a given rotational line. Since almost allthe red sub-bands of CoO were found to have the high F components of the low J linesat the high frequency side, v(Fni„x ) — v(Fm in ) is almost always positive.6.ix.B The 4A512 — X4A 5/2 and the perpendicular sub-bandsAlthough no S-2' = 3/2 — f2" = 3/2 or 1/2 — 1/2 sub-bands have been observed inour laser excitation spectra, a large number of weak Sr = 5/2 — f2" = 5/2 sub-bands (atotal of seven), along with some perpendicularly polarized sub-bands, have been iden-tified in the region 15500 — 16500 cm'. The significance of these sub-bands is thatthe 405/2 - X4051 2 sub-bands provide the energy positions of the 12 = 5/2 spin-orbitcomponents of the excited 40 states, and possibly therefore the important upper statespin-orbit coupling data, while the perpendicular sub-bands may imply that 4 (1) or 4IIelectronic states are also present. Unfortunately extensive upper state perturbations inthese sub-bands have distorted the energy level structure to the extent that the deter-mined spectroscopic constants may not retain their true meanings. Nevertheless, theirobservations greatly increase the number of excited states of CoO known so far.The strongest f2' = 5/2 — ft" = 5/2 sub-band in our wavelength region is the 6411A band, analyzed in Ref. [88] and Part I. Six other sub-bands were also identifiedas f2' = 5/2 — 12" = 5/2 sub-bands from their parallel-polarized nature and the factthat they all originate from the v = 0 level of the X 405 1 2 sub-state. Three of thesub-bands, with R heads near 6152, 6305 and 6373 A show reasonable intensities inour excitation spectra and are relatively easy to assign rotationally from their lowerstate combination differences. The R head regions of the 6152 and 6305 A bands areshown in Fig. 6.7 and Fig. 6.14 respectively; as can be seen, both sub-bands areEcocoto.017) I I15.5^;16.5^16.5^117.5 - 5/2 - X4 A7/2 ^12.5 13.5II 21.5 22.5,15.5^ 16.5^ = 9/2 - X427/2 1^7.5Figure 6.14: The R head region of the 40512 — X4 A5/2 subband of Co0 near 6305 A. Some rotationallines belonging to the 6294 (ST = 5/2 — X 407/ 2 ) and 6295 A (n, 9/2 — X 4 6. 7/ 2 ) bands occur also inthis region.QPRQ1.114.5n^ii4A5/2 - x4A5/21110.51^11 - 2.5Chapter 6. Laser Intracavity Spectroscopy of CoO^ 242overlapped by other sub-bands, making branch analysis more difficult. The remainderof the 12' = 5/2 — 52" = 5/2 sub-bands are extremely weak and heavily overlappedby other stronger sub-bands. As a result, their branch structures were impossible torecognize at the beginning of the spectral analysis, and could only be picked out aftermost of the strong rotational lines had been assigned and extensive wavelength-resolvedfluorescence studies performed. Six weak perpendicularly polarized sub-bands have alsobeen identified in the region 15500 — 16500 cm -1 ; these sub-bands are extremely difficultto recognize in the low resolution spectra. From the high resolution spectra, it wasfound that five of the sub-bands have 012 = +1, since the R lines are stronger thanthe corresponding P lines, and that one sub-band has .6,52 = —1, with a stronger Pbranch. All of the sub-bands have relatively strong Q branches. The lower states of theseperpendicular sub-bands were identified as the X 4 07/ 2 and X405/ 2 (v" = 0) spin-orbitsub-states from the lower state combination differences; it was subsequently establishedthat three of the Afl = +1 sub-bands are It' = 9/2 — X4 07/2 , two are Cr = 7/2 — X4 .6. 5 / 2and the All = —1 sub-band is ST = 5/2 — X4 712 .Attempts to derive the upper state rotational constants from the 9' = 5/2 —5 -2" = 5/2and the perpendicular sub-bands have been made by the method of least squares. Becauseof the limited number of rotational lines that have been observed for these weak sub-bands, and the extensive upper state perturbations (which we shall discuss below), theusable data from each sub-band are very limited and consequently the accuracy of thederived constants does not reflect the accuracy of the spectral data. The determinedrotational constants for the upper states, for what they are worth, are given in Table 6.4,and the assigned lines of the 52' = 5/2 — 52" = 5/2 and the perpendicular sub-bands arelisted in Appendix G.Extensive perturbations occur in the upper states of virtually all the 52' = 5/2 —52" =5/2 and the perpendicular sub-bands. These are best illustrated by the upper stateChapter 6. Laser Intracavity Spectroscopy of Coo^ 243energy level plots shown in Figs. 6.15 — 6.22. Since most of the upper levels were foundto lie very close together in groups of two, it is appropriate to discuss them in pairs.The 5-2' = 5/2 and 9/2 upper levels shown in Fig. 6.15 are interesting. Althoughthe origins of these two sub-states lie only 3 cm -1 apart, they do not appear to interactwith each other. Rather the 52' = 5/2 level seems to be perturbed by a distant lower-lying orbitally nondegenerate state with a larger B constant, which causes the smoothlyvarying A-doubling, while the Sr = 9/2 level appears to be totally unperturbed. Thefr = 5/2 upper state was found to be shared by the 52' = 5/2 — X 405 / 2 sub-band near6418 A and the 52' = 5/2 — X4071 2 sub-band near 6294 A ; this was identified from theidentical upper state energy plots of the two sub-bands, as indicated in Fig. 6.15. In ourlaser excitation spectra, the 6294 A band appears to be stronger than the 6418 A band,which suggests that the 52' = 5/2 — X 4 071 2 sub-band is a spin-allowed 4 11 5 12 — X407i2transition while the 52' = 5/2 — X 4 A 5 / 2 sub-band is its spin-satellite. The significanceof the fact that the 6418 and 6294 A sub-bands share a common upper state is that itallowed the spin-orbit interval between the 52" = 7/2 and 5/2 components of the groundstate to be accurately measured, as described in Part I.The two upper levels shown in Fig. 6.16 are believed to interact with each otherthrough a J-dependent mechanism. The operator responsible for this interaction is theL-uncoupling operator —2BJ • L, which has selection rules AS/ = AA = +1. At low J,the matrix elements of —2BJ • L are small compared to the separation of the two 52 sub-states, and their effect is equivalent to producing a correction to the effective B value ofeach sub-state. At higher J values, the matrix elements are large compared to the stateseparation, and the result is an apparent large centrifugal correction to the rotationalenergy of an individual sub-state, which causes the overall curvatures of the two energylevel plots. Besides the interaction with the 52' = 7/2 level, the 52' = 5/2 level also suffersthree different local perturbations; this is clearly evident in the magnified energy levelFigure 6.15: The upper state energies of the 6294, 6295, and 6418 A bands of CoO, plotted against J(J + 1). Aquantity of 0.415 x J(J + 1) — 6.0 x 10-7 x [J(J + 1)1 2 has been subtracted to magnify the scale.158750 400^J(J+1) 80015890Figure 6.16: The upper state energies of the 6370 and 6373 A bands of CoO, plotted against J(J+ 1).A quantity of 0.415 x J(J + 1) — 6.0 x 10' x [J(J + 1)1 2 has been subtracted to magnify the scale.i^I I^1^I1200 J(J+1)0^400^80016010 ^E/cmil l16000-Chapter 6. Laser Intracavity Spectroscopy of CoO^ 246plot shown in Fig. 6.17. The first avoided crossing, occurring at low J, affects only oneof the two A-components, and thus may be responsible for the A-doubling. The other twoavoided crossings occur at relatively high J, one in each of the two A-components. Sinceno perturbation-induced extra lines have been observed near the avoided crossings, it isimpossible to characterize the perturbing states. Returning to the interaction betweenthe 52' = 5/2 and 7/2 levels shown in Fig. 6.16, the 52' = 5/2 — X 40 5 / 2 sub-band (6373A) was found to be much stronger than the 52' = 7/2 — X 4 05 /2 sub-band (6370 A) in thelaser excitation spectra; this may imply that the 52' = 5/2 — 52" = 5/2 sub-band is a case(a) spin-allowed 4051 2 — 40 5 / 2 sub-band that carries the oscillator strength, while theSr = 7/2 — f2" = 5/2 sub-band is merely a spectator sub-band which borrows intensityfrom the nearby 5/2 — 5/2 sub-band.Fig. 6.18 shows the scaled upper state energies of the 6186 and 6305 A sub-bands,plotted against J(J + 1). Although the origins of the 52' = 9/2 and 5/2 levels areessentially identical, they do not appear to be interacting with each other, a situationrather similar to the one in Fig. 6.15. In fact, the straight lines of the two energylevel plots would indicate that both levels suffer no perturbations at all. The fact thatthe Sr = 9/2 upper level is connected to the X 40 7/ 2 lower state while Cr = 5/2 tothe different X405/ 2 lower state is also consistent with the assessment that there is nointeraction between the two upper levels.The energies of the pair of upper levels illustrated in Fig. 6.19 seem to indicate thatthey are the next members of the vibrational progressions which include the two levelsshown in Fig. 6.16. If so, the same J-dependent interaction mechanism, that appearsto be acting in Fig. 6.16, should also be present here. Though this is not very evidentfrom the energy plot of the 52' = 7/2 level, the curvature of the energy level plot of the5-2 1 = 5/2 sub-state possibly indicates the presence of a J-dependent perturbation comingfrom the L-uncoupling operator —2BJ • L. The obvious main difference between Figs.0^400^80015987^1^i^1^1^1^►^1^1^i^►^1^11200 J(J+1)Figure 6.17: A magnified plot of the upper state energies of the 6373 A band. A quantity of0.415 x J(J + 1) — 6.0 x 10' x [J(J + 1)1 2 has been subtracted.16175El cm- 1 -n' 5/2, 6305 A band16165 -16155 -161450■fr 9/2, 6186 A bandi400^J(J+1) 800Figure 6.18: The upper state energies of the 6186 and 6305 A bands of CoO, plotted against J(J+ 1).A quantity of 0.415 x J(J + 1) — 6.0 x 10' x [J(J + 1)] 2 has been subtracted to magnify the scale.16565E/cmC 116555-CV-5/2, 6152 A band16545 -n1 7/2, 6154 A band016535 I^I^I 1^14001^I^18001^1^11200 J(J+1)Figure 6.19: The upper state energies of the 6152 and 6154 A bands of CoO, plotted against J(J+ 1).A quantity of 0.415 x J(J + 1) — 6.0 x 10-7 x [J(J + 1)] 2 has been subtracted to magnify the scale.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 2506.16 and 6.19 is that the relative positions of the 52' = 5/2 and 7/2 levels are reversedsuch that the 5/2 level lies higher than the 7/2 level in Fig. 6.19. This means that the5-2' = 5/2 state must have a larger vibrational interval than the 7/2 state. The fact thatthe Si' = 5/2 level has a smaller slope than the 52' = 7/2 level in Fig. 6.16, but a largerslope in Fig. 6.19 is consistent with the two levels interacting through the L-uncouplingoperator, because the effect of the interaction is to raise the effective rotational constantof the higher level and to lower that of the lower level. Therefore the true rotationalconstants of the 52' = 5/2 and 7/2 levels must be very similar, about 0.415 cm -1 . Therelative intensities of the 6152 and 6154 A sub-bands resemble those of the 6370 and 6373A bands, that is the 6152 A (52' = 5/2 — X40 5 / 2 ) sub-band is fairly strong, while the6154 A (S2' = 7/2 — X 40 5/2 ) sub-band is very weak and may have borrowed intensityfrom the 5/2 — 5/2 sub-band.The last pair of levels which we have investigated is the 52' = 9/2 levels shown in Fig.6.20. It appears that a J-dependent interaction, such as has been discussed in Figs.6.16 and 6.19, is occurring between the top and the bottom levels, causing the curvaturesof the energy level plots. Since the rotational lines going to the bottom level were foundto be about twice as broad as the lines going to the top two levels, and therefore havelarger hyperfine widths, it may be that only the bottom level is actually an 52 = 9/2level, while the true identities of the top levels are 52 = 7/2, in order for the J-dependentinteraction through the L-uncoupling operator to become possible between the top andbottom levels. The irregularly varying A-doubling in the 52' = 7/2 level may be causedby interaction from a higher-lying nondegenerate state which only perturbs the lowerA-component. Small local perturbations are also found to occur in the bottom SF = 9/2level, with extra rotational lines appearing randomly from J' = 16.5 to 23.5 in the 6314A band.Finally Figs. 6.21 and 6.22 illustrate the energy level plots of two relatively isolated15840800400158300 J(J+1)Figure 6.20: The upper state energies of the 6314 A band of CoO, plotted against J(J+ 1). A quantityof 0.415 x J(J + 1) — 6.0 x 10-7 x [J(J 1)1 2 has been subtracted to magnify the scale.cnChapter 6. Laser Intracavity Spectroscopy of CoO^ 252ST = 5/2 upper states. It is seen from the two pictures that both upper levels suffer manylocal perturbations. In Fig. 6.21, extra rotational lines are induced at both low J andhigh J; this indicates that at least two different perturbations are present. It seems thatone of the perturbing states might be orbitally nondegenerate, and therefore responsiblefor the smoothly increasing A-doubling. Similarly, the irregular A-doubling in the upperstate of Fig. 6.22 appears to be caused by another orbitally nondegenerate state whichperturbs the upper A component only.6.x Results and discussionRotational analysis of the sub-bands of the red system of CoO has revealed a verycomplex upper state energy level pattern, which is best appreciated in terms of the energylevel diagram shown in Fig. 6.23, where the sub-states are categorized according to theirSi values and labelled using a modification of Field's notation [85].At first sight, the level diagram of Fig. 6.23 shows no clear associations between thethree types of ft sub-states; in other words, the sub-states appear to act independently asin case (c) coupling. Close examination indicates that although the spin-orbit interactionin CoO is large enough to cause massive level interactions between different Si sub-states, it is not large enough to destroy the separate identities of the electron orbitaland spin quantum numbers, A and S, so that the levels are still organized according tocase (a)-type energy patterns. Consequently we have given the identities of the upperstates of the parallel-polarized sub-bands as 40 sub-states, and those of the Ail = 1perpendicular sub-bands as 4 11 sub-states, in accordance with the case (a) selection rulesfor electronic transitions from a 4 ,6, lower state. As for the identity of the Si = 5/2level at 15884.6 cm -1 , we have temporarily assigned it as the [15.8] 4 11 5/ 2 sub-state sincethe spin-allowed [15.8] 4 11 5 / 2 — X4 0 7/ 2 sub-band is found to be stronger than the spin500^J(J+ 1)0 100016445 ^E/cmi l .16444 -16581.5 ^0 J(J+1) 50016582.5fl'=5/2, 6142 A bandE/cmi lChapter 6. Laser Intracavity Spectroscopy of Coo^ 253Figure 6.21: The upper state energies of the 6195 A band of CoO, plotted against J(J+1).A quantity of 0.418 x J(J +1) — 6.0 x 10' x [J(J + 1)1 2 has been subtracted to magnifythe scale.Figure 6.22: The upper state energies of the 6142 A band of CoO, plotted against J(J+1).A quantity of 0.414 x J(J +1) — 6.0 x 10' x [J(J+ 1)1 2 has been subtracted to magnifythe scale.Chapter 6. Laser Intracavity Spectroscopy of Coo52 = 9/2^= 7/2^9 = 5/216581.7916556.86[15.6]405/2, v + 1 16553.6316500 [15.5] 46, 5 / 2 , v + 116443.1[1 5. 7]407/ 2 , v + 116319.03E/cm-1 [15.6 ] 1 6, 7/ 2 , v + 1 16256.73[ 1 6.1]4 09/ 2 , v 16162.4516162.29[15.5] 407/ 2 , v + 116072.916000 [15.6] 405 1 2 , v 15999.8415991.49[15. 88]4 4)9/2 , v [ 1 5 . 5 ] 405/2, 15899.2915881.7915836.4[15.7]40 7/ 2 , v15884.64[15.8] 4 115/ 2[15.83 ]4 09/ 2 , v15772.51[1 5. 6 ] 407/2 , v15650.80[15.5 ] 407/2 , v 15535.7715500300^-X4A5/ 27 304.321X40 71 2 , v = 00254Figure 6.23: Energy level structure of the red system of Co0Chapter 6. Laser Intracavity Spectroscopy of Co0^ 255satellite [15.8] 4 1I51 2 — X405 1 2 sub-band. Some SI sub-states are not assigned since theyare believed to be just 'spectator' states that have borrowed oscillator strength from thenearby states by interactions. Absolute vibrational numberings of the upper states arenot available at the present time, since no isotope experiments have been performed forCoO. To indicate the relations between the different vibrational levels more clearly, wehave given the lowest level of a particular electronic state the arbitrary quantum numberv.To understand the energy level pattern of Fig. 6.23 in detail, the SI sub-states must beassigned to their various electronic states, which requires that two relations be establishedfirst: one is the vibrational relation between electronic sub-states of the same type, theother is the relation between sub-states with different ft values. Let us first examine thevibrational connection between the 4071 2 levels, since they give rise to the most prominentsub-band features in the Co0 spectrum. The vibrational frequencies of the excited statesof Co0 are expected to be in the range of 500 — 700 cm -1 , thus spin-orbit sub-states ofthe same type which are separated by this amount can be potentially related as successivevibrational levels of the same electronic state. Judging from the energy separations ofthe 40 7/ 2 sub-states, it appears that the levels at 15535.8, 15650.8 and 15772.5 cm -1(see Fig. 6.23) must belong to three different electronic states, which we have labelled asthe [15.5] 40, [15.6] 4 0, and [15.7] 40 states, and that the levels at 16072.9, 16256.7, and16319.0 cm -1 represent single vibrational excitations with respect to these three 407/ 2sub-states. This vibrational linkage of the 40 7/ 2 levels is very strongly supported by thehyperfine structures of the individual sub-states. It can be seen in Table 6.4 that thevibrationally-related 40 7/ 2 sub-states have very similar hyperfine parameters, h; this isexactly as expected for different vibrational levels of the same electronic sub-state. Thusit seems highly likely that three different 46, excited states are present in the red systemof CoO, with vibrational separations of 537 cm -1 ([15.5] state), 606 cm -1 ([15.6] state)Chapter 6. Laser Intracavity Spectroscopy of CoO^ 256and 547 cm -1 ([15.7] state) respectively.Next we examine the 405 1 2 sub-states. The 405 1 2 at 15899.29 cm-1 and the 4L.712 at15535.77 cm -1 are almost certainly related since their effective rotational constants fitthe Mulliken formula [20]Bef f = B (1 + 2/3E/AA) (6.25)reasonably well. From the Bell constants given in Ref. [88] and Eq. 6.25, we obtainB([15.5] 40) = 0.4263 cm -1 . Taking AA as the separation of the 7/2 and 5/2 spin-orbitcomponents, we find, again from Eq. 6.25,Bei f ( 4 A5/2) — Bef f( 4 A7/2) = —2B 2 /AA = 0.0010 cm-1which should be compared to the difference of the measured Bef f constants for the15899.29 and 15535.77 cm-1 levels of 0.0026 cm -1 . With perturbations so widespread,this agreement is not unacceptable. In addition, the rotational and hyperfine analysisof Part I also provided strong evidence that the 15899.3 cm -1 level is the SI = 5/2component of the [15.5] 4A state. Therefore given this assignment and the vibrationalfrequency of the [15.5] state given earlier, the next vibrational level of the [15.5] 405/2component is expected to lie near 16437 cm -1 , which is very close to the observed levelat 16443.1 cm -1 .It is unfortunately not possible to identify the other S2' = 5/2 levels with the samecertainty as the [15.5] 4A 512 levels because of the lack of accurately measured molecularconstants and the presence of extensive perturbations. However we can base some spec-ulations on the relative positions of the observed levels. It is seen in Fig. 6.23 that the405 12 level at 15991.49 cm -1 is separated from the [15.6] 407/ 2 , v level by ,--, 340 cm -1 ,which is fairly close to the corresponding spin-orbit interval of the [15.5] 4 0 state. It istherefore tempting to assign the 15991.49 cm -1 level as belonging to the [15.6] 40 state,on the assumption that the spin-orbit coupling in the [15.6] 4 0 state is essentially causedChapter 6. Laser Intracavity Spectroscopy of Coo^ 257by the same unpaired 3db electron as in the [15.5] 40 state. The 52 = 5/2 level at 16162.45cm" does not appear to be the spin-orbit component of [15.7] 40 state since it lies toofar above the [15.7] 40 71 2 , v level and has a much larger Bef f constant (see Table 6.4).Available information also indicates no obvious connections between the 16162.45 cm"level and the other two 4 A states. Therefore, for the lack of spectroscopic data, this levelmust remain unassigned.The other 52 = 5/2 sub-state which we have assigned is the 16556.9 cm' level; it isassigned as the v + 1 vibrational level of the [15.6] 40 5 / 2 sub-state. This assignment isessentially based on the fact that the 16556.9 cm" level suffers the same J-dependentinteraction with the nearby 52 = 7/2 sub-state as the [15.6] 40512 , v level, as describedin Section 6.ix.B, and the separation between the two [15.6] 405 / 2 vibrational levels isnot too different from the separation between the corresponding [15.6] 40 7/ 2 levels. Thelast and unassigned It = 5/2 vibrational level is the 16581.8 cm -1 level. This extra 52sub-state gives rise to a very weak parallel-polarized sub-band at 16278 cm -1 , and it mayhave obtained its oscillator strength by interaction with the nearby [15.6] 405/ 2 level.The energy level pattern of the 4 4) electronic states of Co0 is very fragmentary ascan be seen in Fig. 6.23. The three 52 = 9/2 sub-states at 15836, 15882 and 16162cm' are undoubtedly the spin-orbit components of three different 4 4) electronic states,which we have labelled as [15.83] 4 0, v, [15.88] 4 4), v and [16.1] 4 4), v. The two unassigned52 = 7/2 levels at 15999.8 and 16553.6 cm -1 may be successive vibrational levels ofa 4 4) state since they were observed as perpendicularly polarized 52' = 7/2 — X 405/ 2sub-bands. However their strong interactions with the nearby 405/ 2 levels make thisassignment somewhat tentative, since they may have lost their true meanings. No other 52components belonging to the three 4 4) states have been observed in the red system of CoO,so that our understanding of these electronic states must be based on the informationfrom the 4 4)9/ 2 sub-states at the present time.Chapter 6. Laser Intracavity Spectroscopy of CoO^ 258Table 6.5: Approximate spectroscopic constants for the excited states of CoOaStates we AA B h(^= 3/2)[15.5]40[15.6] 40[15.1 40[15.8]4 11[15.841 0[15.88] 4 0[16.1] 405376065473673400.420.410.410.410.420.410.400.0970.0660.0570.0650.0860.054allote: spin-orbit coupling constants, AA, are approximated as the separa-tions between the E = 3/2 and 1/2 components; hyperfine parameters, h, arecalculated from the hyperfine widths of the low J lines.To summarize this Section, fragments of sub-states belonging to three 4 0, three 4 0and one 4 11 excited states have been identified from rotational analysis of the red systemof CoO in the region 15500 — 16500 cm -1 ; the derived spectroscopic parameters for theseelectronic states are summarized in Table 6.5.It was shown in Part I that evidence from accurately measured hyperfine parametersand spin-orbit data can lead to deductions about the electron configurations of the X 4A,[15.5] 42 and [15.7] 4 A states. Specifically the configurations for these electronic stateshave been assigned as:X46,: (4so-)2(36)3(3d7)2^[15.5] 4 0 :^(4sa) 1 (3d8) 3 (3dir) 2 (3dcr) 1[15.7] 4 0 : (0, 2p71- )3 (4sa) 2 (3d6) 3 (3thr) 3 .It is very likely the other observed 46, state, [15.6] 40, also comes from one of the lattertwo electron configurations since both give rise to more than one 4 ,6, state. Electronicstates from the (4scr) 1 (3c/(5) 3 (3d7r) 2 (3da) 1 electron configuration should all possess largehyperfine structures, similar to that of the [15.5] state, since an unpaired 4so electronChapter 6. Laser Intracavity Spectroscopy of Co0^ 259is capable of penetrating into the spining nucleus and producing large Fermi contactnuclear hyperfine interactions. Since the [15.6] 1 0 state shows relatively small hyperfinestructure and has similar spectroscopic parameters to the [15.7] 4 0 state (see Table 6.5),it is logical to assign it to the other configuration, (0, 2p7r) 3 (43(7) 2 (36) 3 (3dr) 3 .Since complete molecular constants have not been measured for the new 4 4) and 4 11states, definite deductions about their electron configurations are not possible. However,some speculations may be made based on the estimated hyperfine parameters of theobserved 409/2 sub-states given in Table 6.5.The determinable hyperfine constant h for a given 1 sub-state is related to the Froschand Foley parameters according to Eq. (6.5). Thus if the parameter a for the 40 statesis given the same value as that measured for the [15.5] 40 state (see Eq. 6.10), we havefrom Eq. (6.5):3^3h = 0.025 x 3+ -2 (b + c) = 0.075+ -2 (b + c). (6.26)Upon substituting the h constants of the 4 09/ 2 sub-states given in Table 6.5 into Eq.(6.26), we obtain negative (b + c) constants for the [15.83] and [15.9] 10 states, and asmall positive (b+ c) constant (,--, 0.01 cm -1 ) for the [15.88] 4 0 state. The negative (b+ c)parameters would have to result from spin-polarization, and are only consistent with the[15.83] and [15.9] 40 states coming from electron configurations where no unpaired 4so-elelctron is present. One possible candidate is the (0, 2/371 - )3 (4so- ) 2 (36) 3 (3thr) 2 (3da) 1configuration, where an oxygen-centred 2p7r electron has been promoted to the metal-centered 3do orbital. A total of 32 electronic states can be derived from this electron con-figuration, including five 4 0 and six 4 II states; thus it is possible that both the [15.83] 1 0and [15.9] 40 excited states come from this configuration.It is not certain if the small positive (b+c) constant for the [15.88] 4 0 state represents apositive Fermi contact interaction since this (b+c) value is considerably smaller than thatChapter 6. Laser Intracavity Spectroscopy of CoO^ 260of the (4.sa) 1 (3dS) 3 (3c/r) 2 (3da) 1 [15.5] 40 state (see Eq. 6.10). Obviously, more accuratehyperfine and spin-orbit data are needed before further discussion of the configurationscan be attempted.6.xi ConclusionThis rotational study of the sub-bands of the red system of CoO, using Doppler-limited laser excitation spectroscopy, has mapped out the upper state energy levels inthe 15500 — 16500 cm -1 region. 18 vibrational sub-states belonging to three 40, three4 4) and one 4 11 states have been located. 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Rosen, Nature.156, 570 (1945).Appendix AMatrix Elements of the Nuclear Electric Quadrupole Hamiltonian inCase (ap) Coupling< SE; BlIFI ik I SE; JS 2 I F >e 2 Qqo [351 2 — J(J + 1)]{3R(J)[R(J) + 1] — 4J(J + 1).1 (I + 1)} —^ (A.1)8/(2/ — 1)J(J + 1)(2J — 1)(2J + 3)< SE; .11 -2I FlikISE; J —1,S2I F >= e2 Qq0 311[R(J) + J + 1b, 0 2 — fi2 P(J)Q(J) 2J (2J — 2)(2J + 2)1(2/ — 1)V4J 2 — 1(A.2)< SE; J9/Flik ISE; J— 2,C2/F>e 2 Qqo 3 VRJ — 1 ) 2 — 92V2 — 92)P(J)Q(J)P(J — 1 )C0 — 1 ) 4/(2/ — 1)4J(J — 1)(2J — 1).1(2J — 3)(2J + 1)< A ± 2; SE; JS2 + 2, IFIil (Q 2 ) I A; SE; JS1 I F >= e 2 Qq2N1J(J + 1) — S-2(C2 + 10(J + 1) — (SI + 1)(9 + 2)x 3R(J)[R(J) + 1] — 4J(J + 1).1(.1 + 1) 16/(2/ — 1)J(J + 1)(2J — 1)(2J + 3)= (A.3)(A.4)< A ± 2; SE; J — 1, ft ± 2, 'FIJI (Q 2 ) I A; SE; JS2 I F >= T e2 Qq2 [R(J) + J + 1]P(J)Q VW J(J + 1) — 51(11 ± 1)16/(2/ — 1)(J — 1)J(J + 1)V4J 2 — 1x V(J T SI — 1)(J + 9 — 2)^ (A.5)266Appendix A: Matrix Elements of Nuclear Electric Quadrupole Hamiltonian^267where< A + 2; SE; J — 2, S2 ± 2, I F 1 H(Q 2 ) I A; SE; J12 I F >= e2Qq2 P(J)Q(J)P(J — 1)Q(J — 1)\/(J T Ci)(J + 52 — 1)321(2/ — 1)J(J — 1)(2J — 1) \ /(2J — 3)(2J + 1)x \/(J T n — 2)(J T C2 — 3)R(J) = F(F + 1) — J(J + 1) — I(I + 1)P(J) = V(F—I+J)(F+J+I+1)Q(J) = V(J+I—F)(F—J+I+1).(A.6)(A.7)Appendix BDerivation of the Matrix Elements of the Electron Spin-orbit and NuclearMagnetic Hyperfine Hamiltonian for the 1 11/311 Complex from the SirConfiguration of NbNThe matrix elements of the electron spin-orbit and nuclear magnetic hyperfine Hamil-tonian for the Sir 1 111 3 11 complex can be calculated from the Slater determinant wavefunctions using spherical tensor methods. The Slater determinant wave functions of the1 11 state and the three spin components of the 3H state are written as1 n >^162+7r- ;^(ce,3 — Oa) >1 3111 > = 13152 + 7r -^+ Oa) >1 311 2 > = 1 5 2+ ; as >3110 >^62-1-7- ; 00 >(B.1)(B.2)(B.3)(B.4)where 1(52+7- > is the electronic part of the wave function, while a and /3 are the twospin functions, respectively with m s = f 2. The function 16 2+r - > represents just oneof the two components of the H state, namely the 11+ component; the II- component isgiven as 1.52- 7r+ >. Therefore each spin-orbit component of the 1 11/311 complex is strictly268Appendix B: Derivation of Matrix Elements for the I/ 3 II Complex of NbN^269represented by two Slater determinant wave functions.i Electron spin -orbit interactionIn spherical tensor form, the matrix elements of the spin-orbit Hamiltonian,fiso = E aziz •can be obtained in case (a) coupling by expanding the scalar product:<^S'E';^F I ilso I 7 I A; SE; All F > = E(-1) 4x E < S'52' I Tql (Si) I SE >< 77' A' I Tl q (aili) I 77 A > .^(B.5)Evaluating the matrix element of the operator n( .4i ) by means of the Wigner-Eckarttheorem, we obtain< 77' ; S'E'; JSZIF I Ilso I i1A; SE; Jar F > = E(-1) 4 (-1) siS' 1 Sx E < s' II T 1 (.4i) s >< 77 ,A ,^> .^(B.6)—E' q ELet the primed state be the 1 11 state, where S' = E' = 0. Since we are taking theunprimed state as the 'II state, we obviously have A' = A = 1, or in other words q = 0.Therefore the only non-vanishing matrix element in Eq. (B.6) isS'= 0 1 S=1< Sr 1 1I1 I ilso I Sr 31q >_—E' = 0 0 E = 0x E <^0 11 TT%) II S 1 >< 0 1 1 Tc!(ailiz) I 77 A >= — 1^< 821 I ad,'^> + ^< irT I a ir iz 171-T >)2 (as 4. (B.7)Appendix B: Derivation of Matrix Elements for the 1 11/ 311 Complex of NbN^270where we use the expressions [16]< S —^II S >< S — 1117' 1 (.42) II S > 1 .\/(2S — 1)(2S + 1) 21 / 28+1 2 V S(2S — 1) • (B.8)The sign of the matrix element given in Eq. (B.7) must be watched since it could introducea parity error for the interacting levels if it is simply ignored. The parameters a s and a,are the spin-orbit parameters of the S and 7r electrons respectively.The matrix elements within the 3 11 state can be derived similarly, and the results are1 \<^3I2 fiso Sr 311 i > =^< (57 3110I fiso 1 67^> = (as — 2 air) (B.9)The parameter (a s — -la ir ) can be identified as the experimental spin-orbit constant AAof the 311 state.ii The nuclear spin-electron orbit interactionThe matrix elements of the operatorE aii • iiin the nuclear magnetic hyperfine Hamiltonian can be written according to Eq. (2.63),which gives the matrix elements of the scalar product of two commuting tensor operators,in the form< A'; S'E'; J'S2' IF I E TV)^IyA; SE; JS2I F >= (-1)-14.1+' IF J I < I II^(i) II I >1 I Jx > < 77 /A i ; "' II naiii) II ijA; JS2 > 8ss,SED. (B.10)Appendix B: Derivation of Matrix Elements for the 1 H/311 Complex of NbN^271Clearly the non-zero matrix elements have to be diagonal in S and E; this means thatno matrix elements of the E i ail • li operator exist between singlet and triplet states.Note that the coefficient a i is not the same as the electron spin-orbit interaction param-eter discussed in Section i. The first reduced matrix element in Eq. (B.10) is given byEq. (2.56) of the text, while the second reduced matrix element, upon projection of theoperator Ti(aii i ) from the space-fixed axis system back to the molecule-fixed axis system,becomes< 77'A'; Jilt' II E 1.,9 1 *(w)T1 q (a ii i )II yA; JSt > = (-1)J1-° '.1(2J + 1)'x \/(2J' + 1) E^J^1 J <^I Tl q (aiii) I r/A > .q a/ q(B.11)Thus Eq. (B.10) reduces to< A'; SE; JIT IF I E 711 (i) • naili) I riA; SE; JSZIF > (- 1) .7+ 1+F I F j' I^+ 1)(2/ + 1) V(2J 1)(2,/' + 1)1 I Jx (-1) .P-°^ji 1 J^<^Vq(aiii) I i7A >^(B.12)—I/ 0 52where q takes the value 0, only, because A' = A. Upon substitution for the 3-j and6-j symbols with the standard formulae, we obtain the diagonal and off-diagonal matrixelements as< 45.2 7rT; JSZIF I E T 1 (i) • T l (a,i,) I 62± 7rT; JSZIF >^fiR(J) 2J(J + 1) (2(18 — a ir) (B.13)< (52± 7r; JSZIF I E TV) • T i (a ii i )1452± 7rT; J — 1, SIIF >^V p^p(j),9(j)2/V4J2 — 1x (2a 6 — a ir )^(B.14)Appendix B: Derivation of Matrix Elements for the 1 11/311 Complex of NbN^272where R(J), P(J) and Q(J) are given by Eq. (A.7). Eqs. (B.13) and (B.14) suggest that,in the single configuration approximation, the experimental hyperfine parameters aA ofthe 1 11 and 3H states should be the same, and equal to the coefficient (2a s — a„).iii The Fermi contact and dipolar interactionsFollowing the method used in Section ii, the matrix elements of the Fermi contactand dipolar Hamiltonians can be shown to reduce to the following expression:<^S'E'; J'n'IF I E bFiTi(i) • Ti(ei) — E gliBgntinr -3 VIT)T 1 (i) • T i (.4i, C2)IF J I'^,^I n A ; SE; JS2IF > = (-1)J+I+F^V I(I + 1)(2I + 1)1 I Jf^ S'^SxV(2J 1)(2,P + 1)E (-1) .P-41 '^(-1)s, -^1^— /' 1 Jq S2^—E q Ex E < s'II T 1 ( 4 i)^>< y'A'S' I bFilriAS >^ji 1^7(1 E^s'^s^1 2 1qq ^;17;2^ q E q2x E < s'il Ti(ei) s > VklgitBgnit. < y'A' ri3 Tqt (C)Ii/A >1^(B.15)where the first term inside the square bracket represents the Fermi contact interaction,and the second term contains the dipolar interaction term. We now evaluate the Fermicontact interaction first, which, in the absence of spin polarization, would be zero. FromEq. (B.15), the Fermi contact term may be written as a product of four terms:<^S'E'; XS/ 1/F HFC IyA; SE; JSIIF > = AxBxCxD^(B.16)Appendix B: Derivation of Matrix Elements for the 1 11/311 Complex of NbN^273whereA = (-1)J+/+F F ji I V/(/ + 1)(2/ + 1).\/(2J + 1)(2J' + 1) (B.17)/1 I JS' 1 SB = (-1) si-E1^(B.18)E' q E( J'^ )C = (_1).r-O'1 J (B.19)--1/' q S2D = < S' 11 T 1 (.41) II S > bF1-1- < S' 11 T 1 (.i 2 ) II S > bF2^(B.20)bFl = < A 1 A 2 A 1 bFl 1 A i A 2 A >bF2^< Adt2 A 1 bF2 I A 1 A 2 A > .^(B.21)withThe parameters bp l and bF2 are the Fermi contact parameters for the 1st (6) and 2nd(r) electrons respectively.First we calculate the matrix elements diagonal in J, where the A term is alwaysA = F(F + 1) — I(I + 1) — J(J + 1) v/(J(J + 1)(2J + 1)2J(J + 1)= W(FIJ) /^2J(J + 1)\/(J(J + 1)(2J + 1).Considering all possible combinations of spin quantum numbers, we have the followingsituations.(1) S' = S = 1, E' = E, q = 0B= (_i)i_E ^l 1 1 )^E— E 0 E^V6(J 1 J^C2 C = (-1)J-°^90 ti )^^J(J + 1)(2J + 1)Appendix B: Derivation of Matrix Elements for the 1 1-1/311 Complex of NbN^2741D2(S + 1)(2S + 1),,^,^1A/1_^LF2y0F1 OF2) = u2such that< 311, J521 HFcI 3 11,^>. AxBxCxDC2EW(FIJ) [1 , L^LF2u42J(J + 1) 2(B.22)(2) S' = S = 1, E' = E + 1, q = +1B = (-1)1-E1^= \ S(S + 1) — E(E + 1) 1^1^112 C = (-1)J -- '41J J(J + 1) — Q(52 ± 1) 2J(J + 1)(2J + 1)such that1D2(S + 1)(2S + 1) ^L^V6ILVi(UF1 OF) =^F1 UF2)< 311, J5/ ± lif/Fc 1 311, J52 >. AxBxCxD— W(FIJ) J(J +1) —^+1)VS(S + 1) — E( +1)4J(J +1)x^+ bF2)1, (B.23)(3)^S' = S — 1 = 0, = E = 0, q = 0,^= 52B = (-1)°( 0 1^1 )= 1—0 0^0C = (-1)J-1J 1 J,—St 0 5/ VJ(J+ 1)(2J + 1)D — 1 \ (2S — 1)(2S + 1) bFi + 1 (2S + 1) uF2 = —^— uO IL^LF2)2^ 2 1 S(2S — 1)^2Appendix B: Derivation of Matrix Elements for the 1 11/311 Complex of NbN^275which gives<111±,./IfiFcl3IIi,J>= AxBxCxD(4)^S' = S - 1 = 0,B = (-1)0^0^1W(FIJ)±=[1 o„ n.^L ^ (B.24)- 0F21S2' = +12J(J + 1)E' = 0, q = -E1^1) -2= +1,0^q E^0C = (-1)J-1 ( J 1^J )= ENJ(J + 1) - S2(52 + 1)2J(J + 1)(2J + 1)T1 q = -E SID = 1 \ 1(2S - 1)(2S + 1) bFl ^1 U. + (2S + 1) L^AL ^\F 2 = - Y)F1 - uF2 )2^S 21 S(2S - 1) 2which gives< 1 1-1 ± , JillFc I 3IVor 2, J>= AxBxCx D^ 1WFIJ) 1)V/J(J + 1) - f/(Ct ± 1) {-2 (bFi - bF2)]=E2\ .^(B.25)12- J(J Next we evaluate matrix elements < 1 H, J - llilFcril, J >; the A and D terms aregiven by(F+I+J+1)(I+J-F)(F+J-I)(F+I-J+1) _Y(FIJ)_V^ 4J^ 2V7D = 1 \ 1(2S - 1)(2S + 1) bFi +^b1 (2S + 1) ,F2 — — A(UL F1 - LOF2),2^S^21 S(2S - 1)^2There are three non-vanishing matrix elements:(1) E' = E = 0, q = 0, C2' = -42 = +1B = (-1)° (0 1 1 )0 00 Nig1= -C = (-1)J-2 J - 1 1 J+1 0 +1= (J + 1)(J - 1) J(2J - 1)(2J + 1)<111±,J-11f/Fc131q,J>= AxBxCxD(bFl — bF2)] ,— Y (FIJ) 1 J 2 — 1 [ 12J^4J2 — 1 [2(B.26)(2) E' = E + 1 = 0, q = +1B = (-1)° 0 1 1^1o 1 —1 ) = 0(J — 1 1 J—1 1 0J(J —1)2J(2J — 1)(2J + 1)C = (-1)' =1(3)^E' = E — 1 = 0, q = —1B0^1^1 1= —= (-1)° (0^—1^1)— 1 1 JC = (- 1) j-2 (J—1 —1 2(J + 1)(J + 2)= 1 2J(2J — 1)(2J + 1)Appendix B: Derivation of Matrix Elements for the 1 11/311 Complex of NbN^276such thatsuch that< 1 111 , J — 11 ilFc 1 311, J > = AxBxCxDY(FIJ) J(J — 1)^ {l i^L F2\=^ (1. - u)] )2J \ 2(4J 2 — 1) 2 i(B.27)such that< 1 11 1 ,J —1111Fc131q,J>= AxBxCxDY (F J) (J10 + 2) {1 , L^L \uk Fi — uF2)] •2J \ 2(4J2 - 1)^2Similarly there are three < 1 11, J I HFC 1 3H, J — 1 > matrix elements; they can beshown to be(1)^< 1 11±, J I fiFc 1 311, J — 1>(B.28)30 ( 1 2 1+(-1)° 3`1^0 0 02 13^1 0 -1c(20 + (- 1)+ 1015 ( 1cQ+1 (B.32)Appendix B: Derivation of Matrix Elements for the 1 11/3 11 Complex of NbN 277Y(FIJ) J2 - 1 {1 _^- uF2)]2(B.29)2J^4J2 - 1(2) <1111, J I 11Fc I 311, J - 1 >Y(FIJ) J(J^1 ) 1(bFl — bF2 ) ][2(B.30)2J^\ 2(4J2 - 1)(3) < 1 11± , J I ii-Fc I 311, J - 1 >Y(FIJ) (J - 1)(J - 2) [1(^UF2)12J^\ 2(4J2 - 1)^2 •(B.31)The dipolar interaction matrix elements are given by the q 2 = 0 component of thesecond term of Eq. (B.15). As can be seen, the expression for the dipolar term is verysimilar to that for the Fermi contact term with the difference mainly coming from anextra 3-j symbol. If the summation over q in the Fermi contact expression is written asb(Q_ 1 Qo Q+1 ) where b represents the Fermi contact parameter, the dipolar term canbe expressed as(1 V36-1) ^(31^2 1c1Q--1-1 0 1where c is the dipolar parameter given byc = 3 DiBg7ifir, < 77'A' I r i-3 Tqt (C) I q A > .Upon substitution for the 3-j symbols, expression (B.32) becomes2 „ 1 „^„ 1 ,„^„-^+ -3 c(Vo - -3 cY-Fi = cYo - -3 cVV-i Q+1).3(B.33)(B.34)Appendix B: Derivation of Matrix Elements for the 1 11/311 Complex of NbN^278Obviously the second term on the right side of Eq. (B.34) is indistinguishable from theFermi contact term b(Q_ 1 Qo Q+1), while the first term represents matrix elementsdiagonal in SI (q 0). Therefore the dipolar interaction gives rise to three additionalmatrix elements between 'II and 3 11 states whose expressions parallel those of i/Fc. givenby Eq. (B.24), (B.26) and (B.29), i.e.< ln± , j //dip< in±, J — 1 I fl dip 1 3I1> ± 11W(FIJ) [1(B.35)2J(J + 1)^2> < 1 111 , JI Hdip 1 3I1 >Y(FIJ) ^J 2 — 1 [1— c2)] . (B.36)(c12J^4J2 — 1 L2Besides the dipolar interaction term, the second term of Eq. (B.15) also gives rise toa hyperfine A-doubling term because of the q ±2 components. The matrix elementcan be shown to be< A + 2; SE 1; J — 1St ± 1 /F1 —^gitBgrign r -3 .0Mr(i) • 711 (.4i, 0 (2) )dY(FIJ) 1A; SE; </f2 IF>= 4J-V4J2 — 1(S 1) — E( + 1)x NAJ C/)(J^— 1)^ (B.37)where d is the measurable hyperfine A-doubling parameter.Appendix CLine Frequencies of the Vibrational Bands of the B 3 (I) — X30 andC3II — X36, Transitions of NbN in cm -1279Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^280JB3$ 2R- X341(171)Q^PB3 $ 2J- X34 1( 2 9Q2 )P14 16110.983 16096.212 9 16040.37615 16111.816 16096.064 10 16039.30816 16112.642 16095.906 11 16038.20817 16113.455 16095.741 12 16037.10418 16095.558 13 16048.716 16035.98719 16115.053 16095.367 14 16048.577 16034.86020 16115.836 16095.167 15 16048.427 16033.73521 16116.608 16094.960 16 16048.259 16032.59022 16117.369 16094.742 17 16048.083 16031.43923 16118.120 16094.502 18 16047.898 16030.27624 16118.861 16094.262 19 16047.702 16029.10725 16119.594 16094.013 20 16047.493 16027.92226 16120.313 16093.754 21 16047.278 16026.72627 16121.023 16093.478 22 16047.051 16025.52528 16121.721 16093.198 23 16046.812 16024.31329 16122.409 16092.908 24 16046.564 16023.08430 16123.087 16092.600 25 16046.305 16021.84931 16123.754 16092.289 26 16046.035 16020.60332 16124.409 16091.968 27 16045.756 16019.34933 16125.053 16091.632 28 16045.465 16018.08434 16125.688 16091.289 29 16045.163 16016.80735 16090.939 30 16044.851 16015.52236 16126.922 16090.570 31 16044.531 16014.22037 16090.195 32 16044.198 16012.91938 16128.116 33 16043.858 16011.59839 16051.116 34 16043.500 16010.27240 16049.723 35 16043.137 16008.93741 16048.341 36 16042.76342 16046.918 37 16042.37643 16045.526 38 16041.95839 16041.550Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^281JB3 2 - X 34 1 (3 9R^Q3)P JB3C - X341(49 4)R^Q^P10 16011.935 13 15952.66011 16012.793 14 15952.50512 16013.644 15 15952.343 15937.81813 16014.475 16000.845 16 15952.168 15936.67614 16015.300 16000.695 17 15951.981 15935.52715 16016.115 16000.536 15985.932 18 15951.783 15934.36216 16016.916 16000.360 15984.779 19 15951.575 15933.18517 16017.707 16000.187 15983.632 20 15951.352 15931.98618 16018.493 15999.993 15982.473 21 15972.412 15951.121 15930.80919 16019.257 15999.792 15981.297 22 15973.137 15950.880 15929.59220 16020.016 15999.578 15980.114 23 15973.846 15950.625 15928.38421 16020.765 15999.356 15978.911 24 15950.359 15927.14622 16021.502 15999.119 15977.713 25 15975.267 15950.082 15925.90523 16022.229 15998.873 15976.498 26 15975.901 15949.794 15924.65424 16022.942 15998.616 27 15976.588 15949.495 15923.38725 16023.644 15998.349 15974.032 28 15977.238 15949.198 15922.11326 16024.332 15998.074 15972.781 29 15977.882 15948.876 15920.83927 16025.018 15997.786 30 15978.512 15948.542 15919.54128 16025.687 15997.485 31 15979.128 15948.203 15918.23129 16026.346 15997.181 32 15979.734 15947.84530 16026.994 15996.858 33 15980.334 15947.47931 16027.630 15996.524 34 15947.10432 16028.254 15996.180 35 15946.70633 16028.869 15995.82534 16029.464 15995.46035 16030.060 15995.08436 16030.638 15994.69437 16031.205 15994.29638 16031.761 15993.89539 16032.307 15993.46240 16032.836 15993.036Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^282JB3C - X34 1 (59 5)R^Q^P JB& - X 34,(6,R^Q6)P10 15894.866 15 15854.70111 15893.787 16 15854.51212 15892.689 17 15854.31313 15891.569 18 15854.10014 15918.388 19 15853.87515 15903.786 20 15853.64016 15919.958 15903.592 21 15853.39217 15920.725 15903.404 15887.046 22 15853.13118 15921.475 15903.197 15885.880 23 15852.86119 15922.222 15902.983 15884.702 24 15852.57820 15922.956 15902.754 25 15852.28221 15923.678 15882.311 26 15851.978 15827.11922 15924.386 15902.266 15881.102 27 15851.655 15825.85123 15925.086 15902.003 28 15851.323 15824.57024 15925.772 15901.729 29 15850.979 15823.25625 15926.445 15901.447 30 15880.231 15850.634 15821.96826 15927.109 15901.150 31 15880.816 15850.264 15820.64827 15927.760 15900.842 32 15881.384 15849.874 15819.32128 15928.384 15900.519 33 15881.943 15849.482 15817.97029 15929.021 15900.182 34 15882.488 15849.075 15816.61430 15929.634 15899.845 35 15883.021 15848.664 15815.24931 15930.241 15899.492 36 15848.22632 15930.809 15899.125 38 15884.53333 15931.406 15898.748 39 15885.01334 15931.986 15898.35835 15932.528 15897.95736 15933.065 15897.54437 15933.601 15897.11638 15934.115 15896.67939 15934.608 15896.23140 15935.100 15895.77141 15895.29442 15894.81943 15894.31744 15893.79945 15893.281Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^283JB32 - X34 1(79R^Q7)P JB3CR- X34 1 (8 9Q8 )P13 15818.584 15805.262 12 15754.28514 15819.380 13 15754.11615 15820.155 15804.912 14 15753.93716 15820.886 15804.713 15 15753.74017 15821.625 15804.482 16 15753.53218 15822.346 15804.282 17 15753.30519 15804.063 18 15753.07720 15823.773 15803.823 19 15752.836 15734.87721 15824.458 15803.560 20 15752.565 15733.69222 15825.139 15803.290 15782.380 21 15773.056 15752.284 15732.45723 15825.805 15803.005 15781.145 22 15773.706 15751.996 15731.22224 15826.452 15802.710 15779.900 23 15774.344 15751.69325 15827.075 15802.397 15778.641 24 15774.969 15751.37526 15827.675 15802.066 15777.378 25 15775.583 15751.04927 15801.697 15776.092 26 15776.181 15750.70528 15801.434 15774.766 27 15776.762 15750.34629 15801.063 15773.539 28 15777.332 15749.97330 15800.690 29 15777.887 15749.58831 15800.305 30 15778.426 15749.19932 15799.903 31 15778.953 15748.782 15719.56033 15799.493 32 15779.465 15748.360 15718.19834 15799.071 33 15779.955 15747.918 15716.81934 15747.453 15715.42135 15746.988 15714.00436 15746.507Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^284JB3 2 - X 34 1 (1,R^Q0)P JB32 - X341(291)R^Q^P12 17142.623 12 17086.640 17073.91013 17143.411 13 17087.431 17073.70614 17144.186 14 17088.181 17073.49015 17144.942 15 17088.941 17073.26016 17145.680 16 17089.667 17073.01017 17146.405 17 17090.355 17072.74818 17147.121 17128.417 17110.671 18 17091.072 17072.46819 17147.813 17128.130 17109.437 19 17091.736 17072.17420 17148.497 17127.827 17108.179 20 17092.425 17071.86521 17149.159 17127.509 17106.843 21 17093.078 17071.537 17050.98122 17149.805 17127.175 17105.527 22 17093.716 17071.196 17049.66223 17150.440 17126.829 17104.212 23 17094.324 17070.840 17048.33124 17151.057 17126.465 17102.859 24 17094.927 17070.466 17046.98125 17151.659 17126.085 17101.486 25 17095.529 17070.078 17045.61526 17152.246 17125.690 17100.113 26 17096.082 17069.674 17044.23627 17152.815 17125.279 17098.718 27 17096.638 17069.255 17042.84228 17153.373 17124.853 17097.312 28 17097.181 17068.81929 17153.910 17124.412 17095.904 29 17097.703 17068.369 17040.00730 . 17154.434 17123.955 17094.453 30 17098.212 17067.901 17038.56531 17154.938 17123.483 17093.011 31 17098.718 17067.417 17037.10832 17155.428 17122.997 17091.521 32 17099.181 17066.919 17035.63533 17155.902 17122.490 17090.066 33 17099.636 17066.404 17034.14734 17156.360 17121.974 17088.576 34 17100.113 17065.87435 17156.802 17121.437 17087.029 35 17100.503 17065.32636 17157.231 17120.885 36 17100.906 17064.76837 17157.642 37 17101.30638 17158.038 38 17063.59139 17062.985Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^285JB3 2 - X 34 1 (3 9R^Q2)P JB32 - X341(49R^Q3)P6 17025.410 12 16949.5157 17026.273 13 16948.3408 17027.128 14 16960.696 16947.13610 17028.763 15 16960.455 16945.93111 17029.568 17017.882 16 16960.195 16944.69512 17030.351 17017.690 17 16959.918 16943.44313 17031.128 17017.479 18 16959.625 16942.20414 17031.865 17003.638 19 16959.320 16940.93115 17032.616 17017.027 20 16958.994 16939.63516 17033.326 17016.765 17001.202 21 16958.653 16938.33317 17034.028 17016.503 16999.953 22 16980.547 16958.293 16937.00518 17034.714 17016.212 16998.701 23 16981.138 16957.919 16935.66119 17035.386 17015.925 16997.437 24 16981.713 16957.528 16934.31220 17036.041 17015.601 16996.148 25 16982.266 16957.122 16932.94121 17036.679 17015.267 16994.832 26 16982.810 16956.695 16931.55222 17037.300 17014.917 16993.513 27 16983.354 16956.259 16930.14523 17037.908 17014.554 16992.175 28 16983.840 16955.810 16928.73824 17038.496 17014.171 16990.825 29 16984.335 16955.334 16927.29425 17039.068 17013.775 16989.456 30 16984.810 16954.842 16925.83926 17039.626 17013.361 16988.074 31 16985.261 16954.336 16924.37427 17040.164 17012.931 16986.674 32 16985.705 16953.813 16922.88028 17040.689 17012.486 16985.261 33 16986.123 16953.270 16921.38329 17012.025 16983.840 34 16986.531 16952.718 16919.85930 17041.685 17011.548 16982.379 35 16986.925 16952.147 16918.32831 17042.162 17011.052 16980.919 36 16987.292 16951.561 16916.77932 17042.617 17010.542 37 16950.95433 17043.056 17010.017 38 16950.32934 17043.472 17009.476 39 16949.68035 17043.888 17008.908 40 16949.02436 17044.279 17008.33537 17044.642 17007.74638 17045.01339 17045.355B3C - X 34 1 ( 7, 6 )Q 16787.27516786.97616786.65816786.32316785.97416785.59316785.21016784.80116784.38016783.93216783.46916782.95316782.53916782.00816781.44416780.91916780.35516779.15916778.534P16768.25716766.95916761.57616758.77516757.345J^R16171819202122232425262728 16810.05829 16810.48930 16810.89731 16811.30232 16811.6723435Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^286B32 - X341(59 4)J^R^Q 13 16917.43514 16918.143 16903.72415 16918.854 16903.45916 16919.554 16903.18717 16920.225 16902.90518 16920.897 16902.60819 16921.530 16902.28820 16922.155 16901.95621 16922.766 16901.60422 16923.358 16901.23723 16923.933 16900.85424 16924.496 16900.45225 16925.033 16900.03726 16925.558 16899.60027 16926.067 16899.15028 16926.557 16898.68429 16927.030 16898.19730 16927.485 16897.69731 16927.924 16897.17132 16928.343 16896.64333^16896.08734 16929.135 16895.51835 16929.504 16894.93136 16929.8568302 - X34 1 (6, 5 )J^Q ^P14 16846.13515^16845.88217 16845.32118^16844.99519^16844.68120 16844.35621 16823.90422^16843.606 16822.57523 16843.20624 16819.85725^16842.373 16818.47426 16841.92827^16841.481 16815.66528 16840.97829^16840.48130 16839.966 16811.30231 16809.822Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^2871213141516542.05316541.90016541.73316541.55716530.15916529.014JB3 2 - X 30 1 (8,R^Q7)P12 16740.831 16 16541.36513 16741.538 17 16541.16314 16742.224 16714.824 18 16559.777 16540.95315 16742.903 19 16560.546 16540.72716 16712.365 20 16561.296 16540.49117 16744.167 16727.166 21 16562.032 16540.24018 16726.832 16709.833 22 16562.761 16539.98019 16745.358 16726.482 16708.530 23 16563.479 16539.70720 16726.112 16707.225 24 16564.180 16539.42121 16746.510 16725.720 16705.898 25 16564.874 16539.12422 16747.018 16725.314 16704.548 26 16565.551 16538.81723 16724.890 27 16566.216 16538.49524 16748.033 16724.439 16701.791 28 16566.870 16538.15725 16748.507 16723.973 16700.383 29 16567.511 16537.81526 16748.967 16698.957 30 16568.140 16537.45927 16722.986 16697.513 31 16568.754 16537.09028 16722.465 16696.052 32 16569.359 16536.71130 16750.587 16721.369 33 16569.948 16536.31331 16691.555 34 16570.526 16535.90732 16720.184 35 16571.092 16535.48833 16751.607 16719.561 36 16571.644 16535.05634 16751.904 16718.921 37 16572.18735 16752.179 38 16572.709 16534.15739 16573.229 16533.69240 16573.730 16533.209 16493.658B3 3 - X34 2 (o, o) 41 16574.219 16532.718 16492.18742 16574.693 16532.213 16490.710J R Q P 43 16575.158 16531.697 16489.2166 16536.779 44 16531.169 16487.6977 16542.644 16535.702 45 16530.625 16486.1818 16542.549 16534.617 46 16530.071 16484.6409 16542.443 16533.519 47 16529.50410 16542.326 16532.410 48 16528.92111 16542.196 16531.291 49 16528.338Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^2884445464816480.96716480.40316479.83716437.75616436.21916431.543B313J- x342(1,1)Q^P6 16486.923 49 16429.9667 16492.744 16485.841 50 16428.3608 16492.648 16484.7619 16492.539 16483.669 B303 - X34 2 (2, 2 )10 16492.417 16482.56211 16492.283 16481.447 J R Q P12 16492.138 16480.326 7 16442.55813 16491.982 16479.177 8 16442.46314 16491.812 16478.025 9 16442.346 16433.51915 16491.634 10 16442.222 16432.41716 16491.439 16475.681 11 16453.848 16442.083 16431.30117 16491.234 16474.483 12 16454.681 16441.934 16430.17218 16491.015 16473.280 13 16455.494 16441.771 16429.02919 16490.785 16472.064 14 16456.299 16441.597 16427.87320 16490.539 16470.836 15 16457.089 16441.409 16426.70521 16490.285 16469.595 16 16457.876 16441.210 16425.52822 16490.017 16468.343 17 16458.633 16440.998 16424.33523 16489.734 16467.081 18 16440.773 16423.13224 16489.441 16465.804 19 16460.129 16440.535 16421.91525 16489.138 16464.517 20 16460.857 16440.284 16420.68226 16488.820 16463.220 21 16461.569 16440.019 16419.44227 16488.486 16461.904 22 16462.272 16439.744 16418.18928 16488.146 16460.581 23 16462.960 16439.453 16416.92529 16487.787 16459.245 24 16463.636 16439.157 16415.64830 16487.421 16457.876 25 16464.299 16438.841 16414.35631 16487.044 16456.537 26 16464.950 16438.511 16413.05432 16486.646 16455.164 27 16465.585 16438.174 16411.73833 16486.242 16453.778 28 16466.210 16437.824 16410.41034 16485.826 29 16466.819 16437.453 16409.07035 16485.398 30 16467.417 16437.083 16407.71736 16484.953 31 16468.001 16436.689 16406.35037 16484.496 32 16468.573 16436.285 16404.97438 16484.029 33 16469.130 16435.870 16403.58639 16483.556 34 16469.674 16435.439 16402.18040 16483.064 16443.747 35 16470.205 16434.999 16400.76542 16482.038 36 16470.724 16434.54743 16481.510 16439.265 37 16471.227 16434.078Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^28938 16471.722 16433.601 34 16418.706 16384.67039 16472.195 16433.106 35 16419.218 16384.21740 16472.660 16432.591 36 16419.715 16383.74641 16432.078 37 16420.201 16383.26642 16431.543 38 16420.682 16382.77443 16430.998 39 16421.128 16382.26444 16430.434 40 16421.569 16381.73645 16429.867 41 16422.003 16381.19946 16429.300 42 16422.423 16380.65447 16428.665 43 16422.816 16380.09744 16379.525JB3$3R- X 342(39Q3)PJB33 - X 342(49 4)R^Q7 16391.9958 16400.664 16391.890 16384.090 4 16346.0449 16401.524 16391.772 16383.003 5 16346.94410 16402.367 16391.646 16381.897 6 16347.83411 16403.192 16391.504 16380.784 7 16348.713 16340.95812 16404.022 16391.349 16379.656 8 16349.568 16340.85113 16404.829 16391.182 16378.514 9 16350.42214 16405.619 16391.004 16377.360 10 16351.262 16340.59415 16406.401 16390.810 16376.193 11 16352.079 16340.45116 16407.170 16390.605 16375.014 12 16352.889 16340.29517 16407.922 16390.385 16373.815 13 16353.686 16340.12318 16408.661 16390.153 16372.618 14 16354.467 16339.93519 16409.396 16389.909 16371.398 15 16355.238 16339.74120 16410.109 16389.652 16370.167 16 16355.990 16339.52821 16410.815 16389.382 16368.925 17 16356.732 16339.30222 16411.497 16389.097 16367.667 18 16339.06723 16412.174 16388.799 16366.397 19 16358.178 16338.81124 16412.833 16388.489 16365.125 20 16358.881 16338.54225 16413.481 16388.165 16363.824 21 16359.567 16338.26626 16414.115 16387.831 16362.513 22 16360.244 16337.97127 16414.743 16387.478 16361.195 23 16360.899 16337.66428 16415.344 16387.118 16359.864 24 16361.545 16337.34429 16415.936 16386.743 16358.512 25 16362.180 16337.01130 16416.519 16386.354 26 16362.79531 16417.091 16385.951 16355.787 27 16363.40132 16417.642 16385.536 16354.407 28 16363.99333 16418.189 16385.109 16353.015 29 16364.574Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^2903031323316365.12516365.68616366.22616366.739B33 - X 34 2 (2, 1)234 16367.250 17471.57435 16367.745 3 17472.50236 16368.224 17473.41845 17474.296 17468.421B3 3 - X 34 2 (i,o) 6 17475.167 17468.3067 17476.025 17468.180J Q P 8 17476.871 17468.0418 17526.537 9 17477.684 17467.882 17459.0589 17526.378 10 17478.490 17467.70510 17526.213 11 17479.272 17467.507 17456.72211 17526.016 12 17480.037 17467.295 17455.55912 17525.807 17513.980 13 17480.788 17467.068 17454.31813 17525.584 17512.767 14 17481.519 17466.817 17453.09814 17525.339 17511.542 15 17482.234 17466.555 17451.84415 17525.078 16 17482.928 17466.26516 17524.800 17509.026 17 17483.608 17465.966 17449.30017 17524.499 17507.746 18 17484.265 17465.65418 17524.193 17506.457 19 17484.913 17465.314 17446.69619 17523.861 17505.145 20 17485.530 17464.950 17445.36620 17503.783 21 17486.142 17464.583 17444.01521 17523.153 17502.460 22 17486.726 17464.201 17442.63722 17522.770 17501.097 23 17487.296 17463.783 17441.25623 17522.366 17499.713 24 17487.841 17463.355 17439.85224 17521.952 17498.315 25 17488.365 17462.90925 17521.516 17496.893 26 17462.45126 17521.061 17495.460 27 17489.388 17461.96427 17520.605 17494.008 28 17489.857 17461.47228 17520.114 17492.536 29 17490.322 17460.972 17432.56429 17519.605 17491.062 30 17490.758 17460.429 17431.07830 17489.548 31 17491.180 17459.86931 17518.544 17488.033 32 17491.580 17459.299 17427.98632 17517.978 33 17491.967 17458.71233 17517.399 34 17492.338 17458.10234 17516.815 35 17492.685 17457.48335 17516.204 36 17456.84036 17515.559 37 17493.329 17456.17637 17514.92138 _1_7514.260Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^2912324252617368.48117369.00517369.50617369.98017345.24917344.80817344.33817322.98217321.57417320.14117318.689JB3 3 - X342(39R^Q2 )P6 17416.225 27 17370.452 17317.2157 17417.071 28 17370.894 17315.7348 17417.904 30 17371.6889 17418.714 17408.96310 17419.498 17408.786 B304 - X303(1, 1 )11 17408.58612 17408.375 J R Q P13 17421.779 8 16800.20914 17422.496 9 16799.10615 17407.614 10 16797.98616 17407.321 11 16796.86118 17425.191 17406.660 12 16820.395 16795.72121 17427.026 17405.597 13 16821.214 16794.56522 17427.594 17405.190 17383.769 14 16807.216 16793.39723 17382.377 15 16822.810 16807.022 16792.22424 17428.685 17404.354 17380.976 16 16823.593 16806.819 16791.03525 17429.203 17379.532 17 16824.357 16806.600 16789.83226 17403.413 17378.107 18 16806.368 16788.61527 17430.182 19 16806.125 16787.39229 17431.078 20 16805.869 16786.14921 16805.597 16784.894B303 - x3A.(4,3) 22 16805.316 16783.62523 16805.021 16782.348J R Q P 24 16804.712 16781.0566 17356.766 25 16804.388 16779.7509 17359.252 26 16804.05611 17360.796 17349.149 27 16803.70812 17361.524 17348.915 28 16803.34513 17362.257 17348.675 29 16802.97414 17362.962 17348.424 30 16802.58316 17347.848 31 16802.18717 17347.534 32 16801.77318 17347.204 33 16801.349 16768.85219 17366.220 17346.853 17328.449 34 16800.908 16767.43220 17366.815 17346.483 17327.105 35 16800.455 16766.00021 17367.392 17346.084 17325.747 36 16799.988 16764.55822 17367.951 17324.375 37 16799.509 16763.095Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^29238 16799.017 16761.625 34 16782.721 16748.457 16715.17339 16798.509 16760.138 35 16783.229 16747.991 16713.72540 16797.992 16758.643 36 16783.718 16747.510 16712.27041 16797.457 16757.132 37 16784.199 16747.018 16710.80742 16796.912 38 16784.664 16746.510 16709.32343 16796.351 39 16785.115 16745.986 16707.83244 16840.016 16795.777 16752.525 40 16785.551 16745.453 16706.32645 16795.191 16750.973 41 16785.974 16744.906 16704.80346 16840.780 16794.591 16749.390 42 16786.383 16744.341 16703.26347 16841.140 16793.976 16747.801 43 16786.774 16743.764 16701.73348 16841.481 16793.351 16746.199 44 16787.154 16743.172 16700.17149 16841.816 16792.708 16744.592 45 16787.520 16742.568 16698.58950 16842.132 16792.056 16742.960 46 16787.863 16741.951 16697.00151 16842.435 16791.382 16741.314 47 16788.202 16741.314 16695.40052 16842.727 16790.707 16739.664 48 16788.524 16740.671 16693.78653 16842.995 16738.000 49 16788.829 16740.013 16692.15954 16843.253 50 16789.118 16739.334 16690.51951 16789.396 16738.645B3 4 - X 3A3 (2, 2) 52 16789.653 16737.94453 16737.226J R Q P14151616754.95816754.75916754.547 JB304R- X 3A3 (3, 3)Q^P17 16754.322 13 16716.05618 16754.085 16736.431 14 16716.84019 16753.833 16735.198 15 16717.60820 16753.569 16733.954 16 16718.36721 16753.290 16732.697 17 16719.107 16701.55722 16752.999 16731.425 18 16719.837 16701.31023 16752.694 16730.147 19 16720.550 16701.05124 16752.377 16728.850 20 16721.253 16700.78125 16752.046 16727.538 21 16721.929 16700.492 16680.02326 16751.700 16726.221 22 16722.607 16700.189 16678.74827 16778.779 16751.342 16724.884 23 16723.268 16699.877 16677.45728 16779.386 16750.973 16723.534 24 16723.912 16699.548 16676.16029 16779.976 16750.587 16722.175 25 16724.540 16699.207 16674.84230 16780.551 16750.188 16720.800 26 16725.158 16698.854 16673.51831 16781.114 16749.777 16719.410 27 16725.757 16698.484 16672.17532 16781.663 16749.351 16718.010 28 16726.348 16698.098 16670.82233 16782.200 16748.910 16716.596 29 16726.920 16697.702 16669.455Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^29330 16727.478 16697.290 16668.073 30 16673.786 16643.77931 16728.023 16696.866 16666.679 31 16674.312 16643.34032 16728.553 16696.427 16665.271 32 16674.827 16642.88633 16729.070 16695.973 16663.847 33 16675.314 16642.41934 16729.570 16695.506 34 16675.794 16641.93535 16730.057 16695.026 35 16676.261 16641.43636 16730.527 16694.530 36 16676.707 16640.92537 16730.987 16694.019 37 16677.143 16640.39838 16731.425 16693.496 38 16677.563 16639.85639 16731.855 16692.959 39 16677.96940 16732.270 16692.407 40 16678.35841 16691.841 41 16678.73542 16733.050 16691.259 16650.430 42 16679.093 16596.95843 16733.423 16690.665 16648.870 43 16679.436 16595.37944 16733.776 16647.297 44 16679.765 16593.79045 16734.116 16645.715 45 16592.20346 16734.443 16644.110 46 16680.378 16590.57847 16734.753 16642.492 48 16587.30048 16735.046 16640.87049 16735.325 B3gP4 - X343 (5 9 5)50 16735.58551 16735.837 J R Q PJB3 4 - X343(49R^Q4)P1516171816594.49816594.26416594.02716593.75015 16648.651 19 16593.47616 16664.906 16648.425 20 16593.179 16573.91717 16665.639 16648.187 21 16592.877 16572.64518 16666.347 16647.932 22 16592.553 16571.36619 16667.046 16647.663 23 16615.329 16592.217 16570.06620 16667.731 16647.383 16627.998 24 16615.936 16591.867 16568.75421 16668.404 16647.086 16626.733 25 16616.531 16591.500 16567.43122 16669.061 16646.776 16625.457 26 16617.110 16591.118 16566.08923 16669.701 16646.449 16624.168 27 16617.674 16590.723 16564.73124 16670.331 16646.113 16622.856 28 16618.221 16590.312 16563.36225 16670.943 16645.759 16621.544 29 16618.754 16589.886 16561.98526 16671.541 16645.393 16620.210 30 16619.271 16589.44527 16672.128 16645.011 16618.861 31 16619.774 16588.99228 16672.692 16644.614 16617.500 32 16620.257 16588.52029 16673.248 16644.204 16616.124 33 16620.733 16588.034Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^29434 16621.188 16587.532 38 16566.870 16529.62935 16621.627 39 16567.215 16529.01436 16622.051 40 16528.40337 16622.462 41 16567.85338 16622.856 43 16568.42639 16623.234 44 16568.68540 16623.596 45 16568.93041 16623.940 16544.215 46 16569.16342 16624.272 48 16569.55943 16624.588 16541.051 49 16569.73044 16624.88345 16625.164 B304 - x- 343 (79 7)46 16536.19047 16625.679 16534.536 J Q P15 16481.23048 16625.91449 16626.136 16 16480.96250 16626.333 17 16480.67851 16626.519 18 16480.36519 16480.062 16462.001B3 4 - X3 A3 (6 9 6) 20 16479.727 16460.71921 16479.374J R Q 22 16479.005 16458.10617 16538.662 23 16478.623 16456.76618 16538.385 24 16478.217 16455.41719 16538.095 25 16477.795 16454.05121 16537.459 26 16477.35523 16536.779 28 16476.42524 16560.334 16536.413 29 16475.93125 16560.903 16536.033 30 16475.42326 16561.462 16535.636 31 16474.89427 16561.985 16535.219 32 16474.350 16443.98828 16562.522 16534.789 33 16473.781 16442.46329 16563.030 16534.343 34 16473.19830 16563.516 16533.882 35 16472.590 16439.39331 16563.994 16533.405 36 16471.963 16437.82432 16564.456 16532.913 37 16471.320 16436.21933 16532.410 38 16470.65334 16565.325 16531.883 39 16469.960 16433.02335 16565.733 16531.340 40 16469.241 16431.36836 16566.123 16530.787 41 16468.493 16429.69437 16566.507 16530.215 42 16427.983Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^295JB3 4 - X 3A3 (8, 8)R^Q^P JB34 - X 343 (1, o)R^Q^P19 16416.769 11 17853.19520 16435.781 16416.347 12 17853.970 17829.29621 16415.860 13 17854.714 17828.08022 16415.344 14 17855.445 17826.83723 16414.753 15 17856.162 17840.378 17825.57624 16414.050 16 17856.856 17840.089 17824.30425 16437.756 16413.411 17 17857.541 17823.02326 16438.096 16412.787 16388.422 18 17858.197 17839.473 17821.71227 16438.403 16386.837 19 17858.845 17839.117 17820.39028 16438.686 16411.497 16385.245 20 17859.464 17838.755 17819.03529 16438.949 16410.815 16383.638 21 17860.070 17838.378 17817.67130 16439.157 16410.113 16382.015 22 17860.664 17837.982 17816.28931 16439.393 16409.396 16380.358 23 17861.228 17837.562 17814.88232 16439.557 16408.661 16378.676 24 17837.126 17813.47433 16407.869 16376.961 25 17862.304 17836.677 17812.02334 16439.801 16407.056 16375.223 26 17862.820 17836.207 17810.57835 16439.883 16406.206 16373.447 27 17863.312 17835.72036 16405.321 16371.651 28 17863.794 17835.21837 16404.399 16369.810 29 17864.250 17834.68938 16403.439 16367.937 30 17864.689 17834.15039 16402.438 16366.028 31 17865.109 17833.58340 16401.393 16364.073 32 17865.511 17833.00341 16400.297 33 17865.888 17832.40834 17866.253 17831.80235 17831.15636 17866.937 17830.50337 17867.228 17829.83238 17867.517 17829.13839 17828.43140 17868.035 17827.70341 17868.267 17826.95742 17826.18943 17825.40244 17824.60145 17823.763Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^296JB3 4 - X 343 (6,Q5 )P JB3 4 - X 3A3 (7, 6)R^Q^P18 17512.165 13 17477.88819 17529.006 17510.821 14 17478.53020 17528.591 17509.446 15 17479.150 17463.93721 17528.157 17508.057 16 17479.745 17463.58322 17527.701 17506.645 17 17480.318 17463.206 17447.05023 17527.226 17505.219 18 17480.867 17462.812 17445.69824 17526.730 17503.771 19 17481.396 17462.392 17444.33025 17526.213 17502.295 20 17481.900 17461.948 17442.93626 17525.676 17500.802 21 17482.385 17461.479 17441.52127 17525.113 17499.285 22 17482.839 17460.987 17440.09328 17524.529 17497.755 23 17483.281 17460.480 17438.63029 17523.932 17496.199 24 17483.693 17459.94630 17523.314 17494.624 25 17484.081 17459.38831 17522.666 17493.029 26 17484.447 17458.80632 17521.999 17491.413 27 17484.789 17458.203 17432.56433 17521.311 17489.771 28 17485.111 17457.573 17430.99834 17520.605 17488.115 29 17485.404 17456.930 17429.39035 17519.882 17486.440 30 17485.677 17456.259 17427.78036 17519.132 17484.736 31 17485.919 17455.559 17426.14037 17518.354 17483.014 32 17486.142 17454.836 17424.48238 17517.561 17481.270 33 17486.342 17454.090 17422.78539 17516.748 17479.506 34 17486.510 17453.324 17421.08040 17515.909 17477.723 35 17486.655 17452.524 17419.35041 17515.051 17475.920 36 17486.767 17451.702 17417.57642 17514.178 17474.089 37 17486.864 17450.856 17415.78343 17513.275 17472.245 38 17486.937 17449.98844 17512.352 17470.380 42 17446.26345 17511.412 48 17486.21547 17509.45748 17508.44549 17507.413Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^297JB304R- X343 (o, 1)Q^P JB304R- X343 (1,Q2)P21 15824.736 17 15799.49022 15824.575 18 15800.33523 15848.213 15824.415 15801.598 19 15801.173 15781.45224 15849.017 15824.233 15800.441 20 15802.011 15781.30525 15849.824 15824.052 15799.267 21 15802.836 15781.14226 15850.634 15823.862 22 15803.658 15780.97427 15851.416 15823.664 23 15804.453 15780.79928 15852.198 15823.460 24 15805.237 15780.613 15756.95129 15852.969 15823.256 25 15806.044 15780.425 15755.78230 15853.742 15823.036 26 15806.834 15780.224 15754.59831 15854.504 15822.803 27 15807.616 15780.020 15753.40832 15822.565 28 15808.385 15779.805 15752.21233 15822.329 29 15809.131 15779.584 15751.01134 15822.080 30 15779.357 15749.79435 15821.818 31 15779.119 15748.57836 15821.552 32 15778.875 15747.35437 15821.280 33 15778.62238 15821.001 34 15778.35739 15820.712 35 15778.08640 15820.417 36 15777.80937 15777.52838 15815.609 15777.234 15739.83839 15816.285 15776.933 15738.55940 15816.954 15776.622 15737.26541 15817.612 15776.304 15735.96142 15818.263 15775.980 15734.67643 15818.907 15775.66044 15775.30645 15774.96946 15774.59747 15774.227Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^298JB3 4 - X33 (2,R^Q3)P JB3 ,4R- X 303(39Q4)P16 15754.619 17 15693.44917 15755.431 18 15693.30818 15756.304 15737.668 19 15712.642 15693.153 15674.63319 15757.131 15737.516 20 15713.463 15692.98820 15737.355 21 15714.265 15692.811 15672.33021 15737.194 22 15715.053 15692.62822 15737.015 23 15715.830 15692.43623 15736.836 24 15716.602 15692.234 15668.83324 15736.644 25 15717.361 15692.029 15667.65125 15736.444 26 15718.118 15691.817 15666.48326 15736.239 27 15718.860 15691.585 15665.26927 15736.020 28 15719.605 15691.35028 15735.796 15708.358 29 15691.10629 15735.562 15707.152 30 15690.85830 15735.325 15705.936 31 15690.59431 15735.076 15704.709 32 15690.32532 15734.817 15703.490 33 15690.04733 15734.552 15702.245 34 15689.76034 15734.279 15700.993 35 15689.46435 15734.005 15699.728 36 15689.15736 15733.710 15698.469 37 15688.84737 15733.416 15697.200 38 15688.52838 15733.098 39 15688.19539 15732.782 15694.637 40 15687.85840 15772.565 15732.457 15693.336 41 15687.51241 15773.204 15732.131 15692.029 42 15687.15542 15773.835 15731.790 15690.723 43 15686.78743 15774.456 15731.439 15689.402 44 15686.41544 15775.066 15731.08345 15775.660 15730.71446 15776.265 15730.33747 15776.844Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^299JB34 - X 343 ( 41R^Q5)P JB3 4 - X 343 (5,R^Q6)P18 15666.814 15648.399 18 15621.04519 15667.623 15648.233 19 15621.820 15602.56720 15668.405 15648.055 20 15602.38221 15669.198 15647.871 15627.513 21 15623.377 15602.18022 15669.988 15647.679 22 15624.121 15601.97523 15670.729 15647.476 15625.216 23 15624.883 15601.772 15579.63124 15671.487 15647.268 15624.015 24 15625.629 15601.54325 15672.230 15647.052 15622.852 25 15626.311 15601.320 15577.26126 15646.823 15621.629 26 15627.070 15601.077 15576.06327 15673.691 15646.589 27 15627.777 15600.843 15574.83028 15674.427 15646.338 28 15600.568 15573.61729 15646.084 29 15600.29730 15645.820 30 15600.01531 15645.546 31 15599.69032 15645.263 32 15599.39133 15644.983 33 15599.08534 15644.681 34 15598.81235 15644.370 35 15598.47936 15644.045 36 15598.14837 15643.716 37 15597.80238 15643.379 38 15597.44539 15643.034 39 15597.07940 15642.676 40 15596.70141 15642.307 41 15596.31442 15595.91743 15595.52044 15595.11045 15594.67746 15594.239Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^300JB3 4 - X303 (6,R^Q7)JBV4 - X313(7,8)Q17 15573.364 21 15504.81318 15574.137 22 15504.51620 15575.664 15555.567 23 15504.20521 15576.414 15555.368 24 15503.91422 15577.149 15555.144 25 15503.57423 15577.880 15554.918 26 15503.23224 15578.605 27 15502.87925 15579.303 15554.436 28 15502.50826 15554.180 29 15502.12427 15580.688 15553.912 30 15501.72728 15553.629 31 15501.31629 15553.341 32 15500.87230 15553.044 33 15500.44031 15552.73832 15552.42333 15552.09134 15551.75435 15551.40536 15551.04937 15550.67838 15550.29739 15549.90640 15549.49441 15549.09042 15548.66944 15547.79945 15547.338Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^301JC311o, - X30 1 (0, o)R^Q^P JC317of - X 34,(o, o )R^Q^P4 17898.238 5 17914.7455 17897.207 6 17915.692 17902.7836 17909.071 17896.169 8 17917.570 17900.6897 17910.019 17895.126 9 17918.498 17908.569 17899.6318 17910.949 17894.077 10 17919.420 17908.498 17898.5689 17911.880 17901.952 17893.014 11 17920.334 17908.419 17897.49610 17912.802 17901.879 17891.950 12 17921.239 17908.334 17896.41811 17913.711 17901.798 17890.878 13 17922.138 17908.240 17895.33212 17914.614 17901.710 17889.800 14 17923.028 17908.140 17894.24113 17915.509 17901.615 17888.712 15 17923.911 17908.032 17893.13814 17901.511 17887.618 16 17924.789 17907.916 17892.03115 17917.280 17901.401 17886.516 17 17925.655 17907.794 17890.92016 17918.154 17901.284 17885.406 18 17926.520 17907.664 17889.80017 17919.020 17901.159 17884.286 19 17927.370 17907.527 17888.67318 17919.877 17901.025 17883.158 20 17928.217 17907.382 17887.53719 17920.728 17900.886 21 17929.056 17907.230 17886.39420 17921.571 17900.735 22 17929.886 17907.071 17885.24421 17922.406 17900.582 23 17930.710 17906.905 17884.08422 17923.231 17900.419 17878.596 24 17931.524 17906.73023 17924.049 17900.247 17877.435 25 17932.333 17906.54824 17924.862 17900.071 17876.266 26 17933.132 17906.35925 17925.655 17899.884 27 17933.927 17906.162 17879.38926 17926.464 17899.688 17873.908 28 17934.705 17905.956 17878.19327 17927.246 17899.489 17872.719 29 17935.482 17905.747 17876.99328 17928.024 17899.278 17871.523 30 17936.250 17905.522 17875.78529 17928.798 17899.063 17870.316 31 17937.008 17905.29630 17929.560 17898.839 17869.104 32 17937.759 17905.058 17873.35331 17930.314 17898.618 17867.890 33 17938.506 17872.12032 17931.061 17898.366 17866.658 34 17939.242 17904.568 17870.87933 17931.799 17898.118 17865.423 35 17939.967 17904.309 17869.63734 17932.529 17897.861 17864.184 36 17940.684 17904.050 17868.38535 17933.248 17897.596 17862.934 38 17903.485 17865.86436 17933.928 17897.323 17861.713 39 17903.193 17864.58737 17934.705 17897.044 17860.41038 17935.360 17896.757 17859.13739 17936.046 17857.85240 17856.563Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^302JC3H., - X34 1 ( 1 9R^Q1 )P JC311of - X3A 1 (i, 1)R^Q^P7 17844.453 5 17858.044 17847.1978 17853.253 17844.386 17836.500 6 17858.970 17846.1529 17854.154 17844.300 17835.429 7 17859.894 17852.000 17845.10510 17855.042 17844.205 17834.345 8 17860.808 17851.933 17844.04411 17855.923 17844.097 17833.257 9 17861.705 17851.851 17842.97812 17856.787 17843.984 17832.158 10 17862.601 17851.759 17841.89813 17857.652 17843.859 17831.048 11 17863.490 17851.660 17840.82414 17858.501 17843.724 17829.931 12 17864.367 17851.550 17839.72215 17859.339 17843.580 17828.804 13 17865.232 17851.435 17838.61716 17860.169 17843.427 17827.666 14 17866.089 17851.308 17837.50717 17860.990 17843.263 17826.521 15 17866.937 17851.171 17836.38018 17861.798 17843.089 17825.355 16 17867.777 17851.025 17835.25019 17862.601 17842.904 17824.195 17 17868.605 17850.867 17834.11420 17863.384 17842.710 17823.023 18 17869.421 17850.703 17832.96421 17864.167 17842.507 17821.835 19 17870.232 17850.527 17831.80222 17864.935 17842.292 17820.640 20 17871.033 17850.347 17830.64023 17865.693 17842.066 17819.431 21 17871.828 17850.154 17829.46624 17866.435 17841.830 17818.212 22 17872.605 17849.951 17828.27725 17867.162 17841.589 17816.984 23 17873.365 17849.738 17827.08526 17867.882 17841.330 17815.748 24 17874.132 17849.515 17825.87327 17814.500 25 17849.283 17824.66428 17869.310 17813.242 26 17875.623 17849.044 17823.45129 17870.005 17840.498 27 17876.353 17848.791 17822.21830 17870.687 17840.199 17810.698 28 17877.076 17848.531 17820.97631 17871.353 17839.882 29 17877.789 17848.265 17819.72032 17872.015 17839.576 30 17878.486 17847.980 17818.46033 17839.236 31 17879.171 17847.691 17817.17334 17838.888 32 17847.394 17815.90735 17838.517 33 17847.078 17814.61936 17838.16637 17837.78938 17837.396Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^303JC3110, - X341(2,Q2 )P JC311of - X 341(2 , 2 )P7 17778.007 17771.185 9 17777.0548 17777.886 17770.090 10 17775.9319 17777.748 17768.988 11 17774.78510 17777.599 17767.851 12 17773.62311 17777.427 17766.710 13 17772.44312 17777.240 17765.546 14 17771.25113 17777.041 15 17770.05214 17776.818 16 17768.82215 17776.585 17 17767.58316 17776.332 18 17766.33217 17776.070 19 17765.05818 17775.78319 17775.48620 17775.16421 17774.83522 17774.48423 17774.11724 17773.72625 17773.31826 17772.89627 17772.45128 17771.99829 17771.51330 17771.00931 17770.48832 17769.95133 17769.38235 17768.19337 17766.890Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^304JC3II1 - X 34 2 (o, o )R^Q^P JC3111R- X 34 2 (1,Q1)P7 17065.037 4 16999.3608 17065.925 17057.011 5 17009.125 16998.3029 17066.798 17056.894 17047.983 6 17010.017 17003.144 16997.23710 17067.660 17056.768 17046.864 7 17010.914 17003.046 16996.14811 17068.509 17056.626 17045.729 8 17011.787 17002.933 16995.06012 17069.344 17056.471 17044.588 9 17012.646 17002.807 16993.95213 17070.167 17056.304 17043.431 10 17013.494 17002.674 16992.82914 17070.976 17056.123 17042.260 11 17014.324 17002.522 16991.69515 17071.772 17055.930 17041.078 12 17015.142 17002.355 16990.54416 17072.555 17055.724 17039.884 13 17015.937 17002.174 16989.38217 17073.325 17055.505 17038.676 14 17016.742 17001.980 16988.20218 17074.081 17055.272 17037.453 15 17017.511 17001.772 16987.01219 17074.825 17055.028 17036.218 16 17018.275 17001.553 16985.80920 17075.555 17054.770 17034.974 17 17019.021 17001.317 16984.59121 17076.273 17054.499 17033.714 18 17001.068 16983.35422 17076.975 17054.214 17032.442 19 17000.802 16982.11123 17077.666 17053.918 17031.157 20 17000.531 16980.85524 17078.341 17053.606 17029.861 21 17021.871 17000.23925 17079.001 17053.286 17028.551 22 17022.546 16999.93526 17079.654 17052.949 17027.227 23 17023.187 16999.61327 17080.292 17052.600 17025.891 24 17023.853 16999.27928 17080.914 17052.237 17024.544 25 17024.488 16998.93029 17081.524 17051.861 17023.188 26 17025.107 16998.56930 17082.121 17051.473 17021.810 27 17025.710 16998.19331 17082.704 17051.071 28 17026.300 16997.80132 17050.656 29 17026.876 16997.40733 17083.831 17050.228 30 17027.437 16996.98234 17084.376 17049.785 31 17027.984 16996.55235 17084.900 17049.332 32 16996.10936 17085.414 17048.864 33 17029.031 16995.64337 17048.385 34 16995.16838 17047.884 35 17030.017 16994.68039 17047.378 36 17030.498 16994.174 16958.83037 17030.951 16993.656 16957.33538 17031.404 16993.132 16955.83939 16992.580 16954.32040 17032.23041 16954,249Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^305JC31-11R- X3 A,(2, 2)Q^P J03112 - X 3 A3 (o, o )R^Q^P5 16938.635 20 17392.9466 16937.565 21 17391.6847 16951.129 16936.476 22 17390.3998 16951.982 16935.370 23 17435.718 17389.1309 16952.815 16934.251 24 17436.400 17387.83110 16953.637 16933.116 25 17437.071 17386.52411 16954.441 16931.965 26 17437.726 17410.972 17385.20312 16955.227 16930.799 27 17438.368 17410.625 17383.87013 16955.995 16929.619 28 17438.996 17410.263 17382.52514 16956.747 16942.096 16928.423 29 17439.612 17409.897 17381.16815 16957.484 16941.858 16927.208 30 17440.214 17409.512 17379.79716 16958.208 16941.605 16925.976 31 17440.804 17409.115 17378.41117 16958.913 16941.335 16924.733 32 17441.378 17408.705 17377.01318 16959.611 16941.048 16923.469 33 17441.940 17408.281 17375.60619 16960.276 16940.746 16922.193 34 17442.488 17407.843 17374.18120 16960.931 16940.427 16920.890 35 17443.023 17407.393 17372.74521 16961.572 16940.093 16919.589 36 17443.544 17406.926 17371.29822 16962.188 16939.741 16918.269 37 17444.052 17406.453 17369.83923 16939.377 16916.927 38 17444.545 17405.965 17368.36424 16938.993 16915.570 39 17445.025 17405.461 17366.88025 16938.596 16914.190 40 17445.492 17404.944 17365.37726 16938.175 16912.805 41 17445.945 17404.415 17363.87027 16937.741 16911.396 42 17446.381 17403.873 17362.34328 16937.292 16909.978 43 17446.806 17403.315 17360.79729 16936.824 16908.553 44 17447.218 17402.743 17359.25230 16936.345 16907.089 45 17447.612 17402.159 17357.68931 16935.840 16905.615 46 17447.999 17401.564 17356.11232 16935.324 16904.127 47 17448.364 17400.953 17354.52533 16934.792 48 17448.71734 16934.251 16901.106 49 17449.05035 16933.668 50 17449.38836 16898.008Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^306JC311R- X343(1,1)Q^P JC3/12R- X 343 (2,Q2)P20 17343.069 10 17302.341 17281.95021 17341.841 11 17303.05122 17340.524 12 17303.732 17279.47623 17385.411 17339.189 13 17304.395 17290.81824 17386.030 17361.445 17337.842 14 17305.055 17290.492 17276.89525 17386.578 17361.065 17336.476 15 17305.662 17275.58526 17387.212 17360.673 17335.099 16 17289.755 17274.23027 17387.778 17360.257 17333.708 17 17306.826 17289.349 17272.87028 17388.334 17359.835 17332.298 18 17307.355 17288.925 17271.46629 17388.874 17359.387 17330.878 19 17307.861 17288.470 17270.04230 17389.397 17358.930 17329.442 20 17308.357 17287.99431 17389.906 17358.460 17327.991 21 17308.824 17287.49032 17390.430 17357.976 17326.524 22 17309.261 17286.96033 17390.877 17357.475 17325.041 23 17309.666 17286.40834 17391.341 17356.958 17323.548 24 17310.048 17285.82635 17391.788 17356.427 17322.041 25 17310.404 17285.214 17260.97836 17392.223 17355.883 17320.517 26 17310.728 17284.573 17259.37337 17392.641 17355.325 17318.980 27 17311.023 17283.914 17257.74638 17393.041 17354.750 17317.428 28 17311.283 17283.214 17256.08539 17393.427 17354.160 17315.862 29 17311.531 17282.490 17254.40440 17353.555 17314.282 30 17311.726 17281.734 17252.69141 17352.951 17312.687 31 17311.898 17280.955 17250.94942 17352.302 32 17312.046 17280.142 17249.17543 17351.653 33 17312.160 17279.288 17247.38044 17350.984 34 17312.240 17278.421 17245.54845 17350.304 35 17312.283 17277.505 17243.68146 17349.615 36 17276.568 17241.78547 17348.915 37 17275.585 17239.85738 17274.574 17237.89039 17273.52741 17271.324Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^307C3112J- X 343(3 1 3)Q^P JC3112R- X 343 (1, o)Q^P12 17273.328 17261.727 19 18377.43713 17273.021 17260.498 20 18395.718 18376.03714 17272.725 17259.178 21 18416.929 18395.29115 17272.353 17257.871 22 18417.460 18394.841 18373.20316 17271.999 17256.542 23 18394.363 18371.74217 17271.630 17255.185 24 18418.453 18393.873 18370.27318 17271.195 17253.806 25 18418.917 18393.355 18368.77419 17270.769 17252.402 26 18419.371 18392.830 18367.26120 17270.289 17250.972 27 18419.802 18392.283 18365.72621 17269.775 17249.523 28 18420.214 18391.699 18364.16922 17269.251 17248.035 29 18420.601 18391.111 18362.59723 17268.672 17246.513 30 18420.968 18390.500 18361.00124 17268.054 17244.975 31 18421.316 18389.861 18359.39825 17267.488 17243.343 32 18421.642 18389.214 18357.76926 17266.604 17241.785 33 18421.951 18388.539 18356.11927 17265.873 17239.992 34 18422.236 18387.847 18354.44728 17265.154 17238.281 35 18422.498 18387.138 18352.74829 17264.419 36 18422.748 18351.03930 17263.663 37 18422.988 18385.654 18349.31431 17262.886 38 18423.171 18384.882 18347.56032 17262.074 39 18423.359 18384.08933 17261.236 40 18423.523 18383.27934 17260.365 41 18382.44135 17259.464 42 18381.59736 17258.53337 17257.566 03 112 - X 343 (2, 1)38 17256.55839 17255.507 J R Q P6 18312.19441 17253.3127 18311.03410 18317.055 18307.34011 18316.738 18306.05212 18329.020 18316.393 18304.74213 18329.613 18316.021 18303.37114 18330.202 18315.617 18302.03115 18330.726 18315.19316 18331.222 18314.730 18299.19717 18331.693 18314.232 18297.73618 18332.142 18313.707 18296.242Appendix C: Line Frequencies of B - X and C - X Vibrational Bands of NbN^30819 18332.550 18313.148 18294.72620 18332.941 18312.563 18293.16921 18333.304 18311.955 18291.59522 18333.609 18311.312 18290.00023 18333.898 18310.638 18288.33924 18334.152 18309.937 18286.65425 18334.373 18309.183 18284.95726 18334.562 18308.408 18283.19527 18334.714 18307.599 18281.44528 18334.818 18306.755 18279.67929 18334.910 18305.884 18277.81030 18334.966 18304.972 18275.93931 18304.027 18274.03532 18303.044 18272.09333 18302.031 18270.11934 18300.982 18268.11135 18334.662 18299.882 18266.05836 18334.482 18298.753 18263.98037 18334.263 18297.583 18261.85138 18334.001 18296.367 18259.70139 18333.697 18295.117 18257.48640 18293.816 18255.24441 18292.478 18252.95542 18291.09543 18289.66244 18288.18345 18286.65446 18285.00347 18283.47048 18281.81850 18278.47451 18276.80652 18275.04253 18273.26354 18271.470Appendix DLine Frequencies of the Charge Transfer Bands of NbN in cm -1309Appendix D: Line Frequencies of the Charge Transfer Bands of NbN^310Ho - X 34 1 (v',4) near 18285 cm - 1J^R^Q^PHo - X 34 i (e 1 4) near 18298 cm-1J^R^Q^P7 18282.457 4 18297.3718 18281.803 5 18296.9399 18281.050 18272.980 6 18296.43010 18280.236 18271.255 7 18295.83211 18290.087 18279.314 9 18294.398 18286.34912 18289.972 18278.312 18267.572 10 18293.562 18284.59513 18277.246 18265.589 11 18292.617 18282.79014 18289.546 18276.087 18263.541 12 18303.234 18291.595 18280.87615 18289.175 18274.859 18261.384 13 18290.503 18278.88616 18288.730 18273.509 18259.197 14 18302.705 18289.289 18276.79017 18288.183 18272.093 18256.865 15 18302.308 18288.01318 18287.585 18270.576 18254.419 16 18301.832 18286.654 18272.35019 18286.884 18268.984 18251.963 17 18301.269 18285.191 18270.01720 18286.087 18267.304 18249.408 18 18283.634 18267.57221 18285.201 18265.533 18246.748 19 18299.867 18282.011 18265.04922 18284.232 18263.683 18244.001 20 18299.020 18280.30023 18261.734 18241.179 21 18298.101 18278.47424 18259.701 22 18297.083 18276.570 18256.93725 18257.578 23 18295.968 18274.582 18254.06426 18255.369 24 18272.486 18251.09527 18253.066 25 18293.472 18270.296 18248.03528 18250.667 26 18268.031 18244.86329 18248.181 27 18241.63030 18245.601 28 18263.19731 18242.943 29 18260.64432 18240.193 30 18257.99431 18255.24434 18246.386Appendix D: Line Frequencies of the Charge Transfer Bands of NbN34^18426.386^18394.49235^18424.472^18391.66636 18388.75337^18385.76731118363.49418359.76518355.94518352.050112J- X 343 (v', 3) near 18432 cm -1R^Q^P14 18412.138 38 18382.703 18348.07215 18438.722 18424.007 18410.185 39 18379.55016 18438.533 18422.904 18408.126 40 18376.34517 18421.712 18406.074 41 18373.04818 18437.974 18420.483 18403.920 42 18369.70319 18437.578 18419.201 18401.67920 18437.162 18417.837 18399.429212218436.72818436.13018416.38418414.89418397.05818394.678 Ho - X 34,(v', 2) near 18441 cm -123242518435.44518434.75718433.97818413.33318411.72018410.18518392.16918389.63818387.021J R Q P91018446.617 18437.57818436.728 18427.65826272818433.18318432.23818408.21618406.46818404.59518384.34918378.78911121318446.71618446.61718446.43718435.79218434.88418433.76518425.85118423.97018421.98029 18402.658 18375.941 14 18446.144 18432.591 18419.90630 18400.664 18373.035 15 18445.750 18417.75731 18398.605 18369.994 16 18445.246333418363.87118360.722171818444.59918443.83018428.43918426.83935 18357.490 19 18425.02536 18354.20537 18350.851Ho - X 3z1,(e, 2) near 18443 cm -1112 - X 343 (v', 2) near 18442 cm -1 J R Q P8 18440.771J R Q P 91018449.37318449.60718440.17718439.48018431.87818430.28926 18413.935 18389.71827 18411.869 18386.875 11 18449.765 18438.729 18428.59128 18409.652 18383.910 12 18449.835 18437.888 18426.83929 18434.757 18407.351 18380.787 13 18449.826 18436.972 18425.02530 18433.263 18404.960 18377.574 14 18449.724 18435.972 18423.12131 18431.675 18402.482 18374.189 15 18449.521 18434.884 18421.13032 18429.998 18399.904 18370.712 16 18449.187 18433.693 18419.04233 18428.240 18397.236 18367.160 17 18432.378 18416.849Appendix D: Line Frequencies of the Charge Transfer Bands of NbN^312Ho - X 3 4,(v', 3) near 18488 cm-1 Ho - X 3 4 1 (v', 3) near 18499 cm -1J R Q P J R Q P7 18480.078 1 18501.0318 18485.904 18478.613 2 18501.8189 18485.285 18477.060 5 18498.25310 18484.560 18475.449 6 18497.857 18492.32611 18483.790 18473.716 7 18497.351 18490.91612 18471.955 8 18496.787 18489.46613 18494.804 18482.042 9 18496.157 18487.91214 18468.255 10 18495.46015 18494.552 18479.948 18466.271 12 18493.839 18482.85916 18494.318 18478.798 18464.205 13 18492.93917 18493.988 18477.578 18462.080 14 18491.957 18479.16218 18493.595 18476.269 18459.869 15 18490.916 18477.19719 18493.125 18474.930 18457.572 16 18489.78520 18492.576 18473.445 18455.220 17 18488.580 18473.05821 18491.957 18471.917 18452.789 18 18487.302 18470.86122 18491.236 18470.316 18450.250 19 18468.61523 18490.449 18468.615 18447.693 20 18484.540 18466.27124 18489.611 18466.837 18445.001 21 18483.037 18463.86925 18488.673 18465.001 18442.232 22 18461.38926 18487.595 18463.078 23 18479.787 18458.82427 18461.062 24 18500.854 18478.068 18456.18328 18485.285 18458.970 25 18499.950 18476.269 18453.48429 18484.015 26 18498.957 18474.359 18450.68730 18482.654 18454.519 27 18497.857 18472.408 18447.78731 18452.164 28 18496.707 18470.316 18444.82832 18479.613 29 18495.482 18468.169 18441.77933 18477.988 18447.165 30 18494.134 18465.974 18438.66234 18444.496 31 18492.708 18463.648 18435.44532 18491.236 18461.24233 18489.61134 18487.912Appendix ELine Frequencies of the B 4 11 — X 4 E- (1,0) Band of VO in cm -1313Appendix E: Line Frequencies of the B411 - X 4 E- (1, 0) Band of VO^314Nil^T R42 TS43B4 115 1 2 - X 4E- (1 1 0)S R43^S Q42^R Q43 Rp42 0p43 0 04213000+ 13000+ 13000+ 13000+ 13000+ 13000+ 13000+ 13000+4^616.600 605.1125^617.783 604.4156^618.894 603.6377^620.033 602.791 596.0438^621.078 601.885 602.367 594.0889^622.250* 600.909 601.347 592.08310^623.103 599.884 600.257 590.02011^624.040 598.799 599.113 587.89412^624.925 597.653 597.917 585.71213^625.756 596.450 596.667 583.47614^626.544 626.351 610.245 610.456 595.187 595.379 581.16715^627.282 627.102 609.955 610.133 593.862 594.042 578.812 578.99516^627.784 627.970 609.791 609.603 592.660 592.475 576.576 576.38617^628.406 628.606 609.393 609.191 591.227 591.028 574.10618^628.969 629.181 608.945 608.719 571.59419^629.473 608.450 608.186 588.211 587.947 569.02320^629.921 607.889 607.601 586.619 586.337 566.39321^630.310 607.277 606.964 584.974 584.664 563.72722^630.646 606.610 606.266 583.273 582.926 560.97923^630.924 605.885 605.512 581.513 581.167 558.19124^631.140 605.112 604.700 579.704 579.301 555.37025 604.269 603.835 577.832 552.40526 603.372 602.911 575.903 575.444 549.49627 602.408 601.885* 573.914 573.554* 546.46528 601.404 600.909 571.817* 571.365 543.40729 600.334 599.794 569.776 569.247 540.28730 599.214 598.642 567.621 567.047 537.07631 598.035 597.429 565.386 533.85032 596.793 596.157 563.11833 595.481 594.831 560.78334 593.438 558.38835 591.983 555.92536 590.482 553.42537 588.918 550.88038 587.285 548.18139 585.618 545.50940 583.843 542.79941 582.034 539.965*.blendedAppendix E: Line Frequencies of the B411 - X 4 E- (1, 0) Band of VO^315B4 1131 2 - X 4 E- (1,o)N" SR37 Ss33 RQ32 RR33 Q p32 Q Q33567813546.29313547.35513548.38513549.3389 13550.248 13529.69810 13551.095 13539.351 13528.51811 13551.926 13539.121 13527.21312 13552.650 13538.853 13525.92413 13553.321 13538.525 13538.309 13524.539 13524.32814 13553.941 13553.765 13538.141 13537.952 13523.101 13522.92915 13554.499 13554.328 13537.703 13537.517 13521.618 13521.44916 13554.810 13555.045 13537.024 13537.197 13519.897 13520.05817 13555.235 13536.437 13536.642 13518.26418 13555.582 13535.800 13536.085 13516.57019 13555.866 13535.099 13535.335 13514.81320 13534.327 13534.613 13512.97721 13533.489 13533.809 13511.07422 13532.586 15533.086* 13509.12323 13531.627 13507.08724 13530.593 13505.03625 13529.505 13502.82826 13555.961 13528.339 13500.61127 13555.699 13527.102 13498.31028 13555.370 13525.831 13495.94129 13554.993 13524.466 13493.51330 13554.557 13523.035 13491.02831 13554.018 13521.545 13488.46232 13553.425 13519.976 13485.83433 13552.793 13518.36734 13552.070 13516.66835 13551.24936 13550.44337 13549.49638 13548.54639 13547.51340 13546.41541 13545.268*=blendedAppendix E: Line Frequencies of the B4II - )0E - (1, 0) Band of VO^316B4 11,12 - X 4E- (1,0)N"^Q R23^QQ22^Phi 23^p£ 22^0 p23^(302213000+ 13000+ 13000+ 13000+ 13000+ 13000+6 469.4677 468.5428 467.4939 475.976* 466.40410 475.722 465.248 455.451*11 475.409 464.028 453.096*12 475.057 462.742 450.62913 474.550 474.784 461.393 461.615 448.13014 474.029 474.190 459.982 460.168 445.53315 473.360 473.549 458.494 458.691 442.900 443.03816 472.826 472.627 457.169 457.003 440.330 440.15417 472.044 471.825 455.602 455.422 437.506 437.33318 471.182 454.003 434.62819 470.263 452.365 431.65520 469.189 450.710 428.62421 468.071 449.017 425.49822 466.888 447.316 422.28723 465.614 445.598 419.00024 464.267 443.897 415.62925 462.838 442.204 412.18026 461.322 440.534 408.63627 459.695 438.903 405.04628 458.025 437.293^412.980* 401.34829 456.271 435.769^411.262* 397.57930 454.418 409.376* 393.71731 452.526 407.353* 389.82132 450.489 405.184* 385.79733 448.422 402.861* 381.72934 446.284 400.402* 377.51035 444.065 373.28836 441.934* 368.95137 439.666* 364.57738 437.384* 360.037**=blended *=perturbation-inducedAppendix E: Line Frequencies of the B4 11 - X4 E - (1,0) Band of VO^317B4 11_ 1 1 2 - X 4 E- (1, o)N" PQi2 P R13 0(213 0P12 N p13 NO123 13408.8254 13408.534 13406.904*5 13408.263 13406.147*6 13407.951 13404.685 13405.352* 13395.574*7 13407.622 13404.025 13404.462* 13393.293*8 13407.245 13403.264 13403.705* 13390.930*9 13406.812 13402.377 13402.803 13388.52710 13406.327 13401.399 13401.783 13386.04111 13405.796 13405.526 13400.320 13400.665 13383.50112 13405.180 13404.891 13399.118 13399.373 13380.90413 13404.524 13404.288 13397.800 13398.04314 13403.797 13403.612 13396.406 13396.599 13375.461 13375.68915 13403.003 13402.803 13394.818 13395.026 13372.671 13372.84916 13401.935 13402.144 13393.348 13393.138 13369.962 13369.73717 13400.982 13401.180 13391.558 13391.35318 13399.948 13400.165 13389.665 13389.438 13363.96419 13398.839 13399.118 13387.648 13387.390 13360.83920 13397.631 13397.936 13385.519 13385.249 13357.63821 13396.370 13396.663 13383.263 13382.97922 13394.961 13395.315 13380.904 13380.615*23 13393.486 13393.865 13378.423 13378.125*24 13391.929 13392.343 13375.83625 13390.284 13373.12626 13388.527 13370.31027 13386.681 13367.38628 13384.732 13364.35129 13382.690 13361.18530 13380.55831 13378.37732 13376.03333 13373.640*34 13371.09535 13368.44836 13365.70137 13362.90338 13359.99339 13356.98940 13353.866*.blended - -Appendix FAssigned Hyperfine Lines of the 6338, 6436 and 6411 A Sub-bands of Co0in cm'318Appendix F: Hyperfine Line Frequencies of Three Sub-bands of CoO^31912' =R Branch7/2 - Ii" = 7/2, 6338 A BandQ Branch P BranchJ" F' - F" Frequency J" F' - F" Frequency J" F' - F" Frequency3.5 8 - 7 15774.7868 3.5 7 - 6 15771.1911 4.5 7 - 7 15766.68763.5 7 - 6 15774.7678 3.5 6 - 5 15771.1413 4.5 6 - 6 15766.62753.5 6 - 5 15774.7507 3.5 5 - 4 15771.0979 4.5 5 - 5 15766.57683.5 5 - 4 15774.7348 3.5 4 - 3 15771.0608 4.5 4 - 4 15766.53413.5 4 - 3 15774.7215 3.5 3 - 2 15771.0299 4.5 3 - 3 15766.50003.5 3 - 2 15774.7076 3.5 7 - 7 15771.1460 4.5 2 - 2 15766.47473.5 2 - 1 15774.6952 3.5 6 - 6 15771.1026 4.5 1 - 1 15766.45754.5 9 - 8 15774.7600 3.5 5 - 5 15771.0657 4.5 7 - 8 15766.65514.5 8 - 7 15774.7430 3.5 4 - 4 15771.0349 4.5 6 - 7 15766.59924.5 7 - 6 15774.7273 3.5 3 - 3 15771.0104 4.5 5 - 6 15766.55214.5 6 - 5 15774.7127 3.5 2 - 2 15770.9920 4.5 4 - 5 15766.51394.5 5 - 4 15774.6997 3.5 1 - 1 15770.9791 4.5 3 - 4 15766.48374.5 4 - 3 15774.6886 3.5 6 - 7 15771.0575 4.5 2 - 3 15766.46224.5 3 - 2 15774.6796 3.5 5 - 6 15771.0269 4.5 1 - 2 15766.44954.5 2 - 1 15774.6709 3.5 4 - 5 15771.0026 4.5 0 - 1 15766.44515.5 10 - 9 15774.5478 3.5 3 - 4 15770.9844 5.5 8 - 8 15764.82465.5 9 - 8 15774.5325 3.5 2 - 3 15770.9725 5.5 7 - 7 15764.78325.5 8 - 7 15774.5185 3.5 1 - 2 15770.9667 5.5 6 - 6 15764.74705.5 7 - 6 15774.5060 3.5 0 - 1 15770.9667 5.5 5 - 5 15764.71575.5 6 - 5 15774.4935 4.5 8 - 7 15770.3282 5.5 4 - 4 15764.68965.5 5 - 4 15774.4830 4.5 7 - 6 15770.2927 5.5 3 - 3 15764.66895.5 4 - 3 15774.4746 4.5 6 - 5 15770.2614 5.5 2 - 2 15764.65325.5 3 - 2 15774.4668 4.5 5 - 4 15770.2339 5.5 8 - 9 15764.79925.5 9 - 9 15774.5072 4.5 4 - 3 15770.2103 5.5 7 - 8 15764.76085.5 8 - 8 15774.4960 4.5 3 - 2 15770.1902 5.5 6 - 7 15764.72745.5 7 - 7 15774.4863 4.5 2 - 1 15770.1743 5.5 5 - 6 15764.69905.5 6 - 6 15774.4772 4.5 8 - 8 15770.2958 5.5 4 - 5 15764.67575.5 5 - 5 15774.4694 4.5 7 - 7 15770.2645 5.5 3 - 4 15764.65765.5 4 - 4 15774.4626 4.5 6 - 6 15770.2372 5.5 2 - 3 15764.64485.5 3 - 3 15774.4577 4.5 5 - 5 15770.2136 5.5 1 - 2 15764.6370Appendix F: Hyperfine Line Frequencies of Three Sub-bands of Co0^3206338 A Band, Continued.R Branch^Q Branch^P Branch6.5 11 - 10 15774.1498 4.5 4 - 4 15770.1939 6.5 9 - 9 15762.78426.5 10-9 15774.1362 4.5 3 - 3 15770.1781 6.5 8 - 8 15762.75306.5 9 - 8 15774.1234 4.5 2 - 2 15770.1662 6.5 7 - 7 15762.72516.5 8 - 7 15774.1120 4.5 1 - 1 15770.1580 6.5 6 - 6 15762.70076.5 7 - 6 15774.1013 4.5 7 - 8 15770.2322 6.5 5 - 5 15762.67976.5 6 - 5 15774.0917 4.5 6 - 7 15770.2091 6.5 4 - 4 15762.66206.5 5 - 4 15774.0832 4.5 5 - 6 15770.1894 6.5 3 - 3 15762.64786.5 4 - 3 15774.0758 4.5 4 - 5 15770.1736 6.5 9 - 10 15762.76356.5 10 - 10 15774.1155 4.5 3 - 4 15770.1618 6.5 8 - 9 15762.73456.5 9 - 9 15774.1045 4.5 2 - 3 15770.1540 6.5 7 - 8 15762.70876.5 8 - 8 15774.0953 4.5 1 - 2 15770.1500 6.5 6 - 7 15762.68636.5 7 - 7 15774.0866 5.5 9 - 8 15769.2890 6.5 5 - 6 15762.66736.5 6 - 6 15774.0790 5.5 6 - 5 15769.2150 6.5 4 - 5 15762.65186.5 5 - 5 15774.0725 5.5 5 - 4 15769.1954 6.5 3 - 4 15762.66206.5 4 - 4 15774.0674 5.5 4 - 3 15769.1791 6.5 2 - 3 15762.63127.5 12 - 11 15773.5661 5.5 3 - 2 15769.1653 8.5 11 - 11 15758.15807.5 11 - 10 15773.5537 5.5 9 - 9 15769.2634 8.5 10 - 10 15758.13677.5 10 - 9 15773.5423 5.5 8 - 8 15769.2392 8.5 9 - 9 15758.11837.5 9 - 8 15773.5316 5.5 7 - 7 15769.2176 8.5 8 - 8 15758.10127.5 8 - 7 15773.5220 5.5 6 - 6 15769.1982 8.5 7 - 7 15758.08597.5 7 - 6 15773.5132 5.5 5 - 5 15769.1818 8.5 6 - 6 15758.07257.5 6 - 5 15773.5053 5.5 4 - 4 15769.1678 8.5 11 - 12 15758.14297.5 5 - 4 15773.4983 5.5 3 - 3 15769.1569 8.5 10 - 11 15758.12368.5 13 - 12 15772.7963 5.5 2 - 2 15769.1487 8.5 9 - 10 15758.10598.5 12 - 11 15772.7851 5.5 4 - 5 15769.1541 8.5 8 - 9 15758.09038.5 11 - 10 15772.7747 5.5 3 - 4 15769.1454 8.5 7 - 8 15758.07648.5 10 - 9 15772.7649 5.5 2 - 3 15769.1403 8.5 6 - 7 15758.06438.5 9 - 8 15772.7562 6.5 10 - 10 15768.0475 8.5 5 - 6 15758.05418.5 8 - 7 15772.7481 6.5 9 - 9 15768.0275 8.5 4 - 5 15758.0459Appendix F: Hyperfine Line Frequencies of Three Sub-bands of CoO^3216338 A Band, Continued.R Branch^Q Branch^P Branch11.5 16 - 15 15769.3681 6.5 8 - 8 15768.009511.5 15 - 14 15769.3589 6.5 7 - 7 15767.993311.5 14 - 13 15769.3505 6.5 6 - 6 15767.979211.5 13 - 12 15769.3426 6.5 5 - 5 15767.967111.5 12 - 11 15769.3352 6.5 4 - 4 15767.957011.5 11 - 10 15769.3284 6.5 3 - 3 15767.948811.5 10 - 9 15769.3220 6.5 10 - 9 15768.068311.5 9 - 8 15769.3161 6.5 3 - 4 15767.940512.5 17 - 16 15767.8518 7.5 11 - 11 15766.647112.5 16 - 15 15767.8433 7.5 10 - 10 15766.630412.5 15 - 14 15767.8355 7.5 9 - 9 15766.614812.5 14 - 13 15767.8279 7.5 8 - 8 15766.600112.5 13 - 12 15767.8210 7.5 7 - 7 15766.588412.5 12 - 11 15767.8144 7.5 6 - 6 15766.577112.5 11 - 10 15767.8083 7.5 5 - 5 15766.568212.5 10 - 9 15767.8024 7.5 4 - 4 15766.56037.5 11 - 10 15766.66508.5 12 - 11 15765.07678.5 12 - 12 15765.06188.5 11 - 11 15765.04698.5 10 - 10 15765.03338.5 9 - 9 15765.02108.5 8 - 8 15765.00998.5 7 - 7 15764.99998.5 6 - 6 15764.99128.5 5 - 5 15764.9837Appendix F: Hyperfine Line Frequencies of Three Sub-bands of Ca)^3220' = 7/2R Branch- 0" = 7/2, 6436 A BandP BranchJ" F' - F" Frequency J" F' - F" Frequency7.5 12 - 11 15538.0709 8.5 11 - 11 15522.41757.5 11 - 10 15538.0418 8.5 10 - 10 15522.37617.5 10 - 9 15538.0149 8.5 9 - 9 15522.33857.5 9 - 8 15537.9900 8.5 8 - 8 15522.30427.5 8 - 7 15537.9674 8.5 7 - 7 15522.27387.5 7 - 6 15537.0471 8.5 6 - 6 15522.24707.5 6 - 5 15537.0290 8.5 5 - 5 15522.22387.5 5 - 4 15537.0133 8.5 11 - 12 15522.40237.5 11 - 11 15538.0243 8.5 10 - 11 15522.36257.5 10 - 10 15537.9992 8.5 9 - 10 15522.32617.5 9 - 9 15537.9760 8.5 8 - 9 15522.29317.5 8 - 8 15537.9550 8.5 7 - 8 15522.26397.5 7 - 7 15537.9362 8.5 6 - 7 15522.23837.5 6 - 6 15537.9197 8.5 5 - 6 15522.21667.5 5 - 5 15537.9055 8.5 4 - 5 15522.19858.5 13 - 12 15537.5764 9.5 12 - 12 15520.07428.5 12 - 11 15537.5512 9.5 11 - 11 15520.03978.5 11 - 10 15537.5277 9.5 10 - 10 15520.00788.5 10 - 9 15537.5060 9.5 9 - 9 15519.97878.5 9 - 8 15537.4860 9.5 8 - 8 15519.95258.5 8 - 7 15537.4679 9.5 7 - 7 15519.92908.5 7 - 6 15537.4516 9.5 6 - 6 15519.90858.5 6 - 5 15537.4370 9.5 12 - 13 15520.06108.5 12 - 12 15537.5360 9.5 11 - 12 15520.02768.5 11 - 11 15537.5139 9.5 10 - 11 15519.99688.5 10 - 10 15537.4936 9.5 9 - 10 15519.96878.5 9 - 9 15537.4749 9.5 8 - 9 15519.94358.5 8 - 8 15537.4580 9.5 7 - 8 15519.92108.5 7 - 7 15537.4431 9.5 6 - 7 15519.90158.5 6 - 6 15537.4297 9.5 5 - 6 15519.88489.5 14 - 13 15536.9243 10.5 13 - 13 15517.57599.5 13 - 12 15536.9023 10.5 12 - 12 15517.54639.5 12 - 11 15536.8817 10.5 11 - 11 15517.51899.5 11 - 10 15536.8626 10.5 10 - 10 15517.4938Appendix F: Hyperfine Line Frequencies of Three Sub-bands of CoO^3236436 A Band, Continued.R Branch P Branch9.5 10 - 9 15536.8448 10.5 9 - 9 15517.47069.5 9 - 8 15536.8284 10.5 8 - 8 15517.44999.5 8 - 7 15536.8137 10.5 7 - 7 15517.43159.5 7 - 6 15536.8004 10.5 13 - 14 15517.56389.5 13 - 13 15536.8891 10.5 12 - 13 15517.53539.5 12 - 12 15536.8697 10.5 11 - 12 15517.50909.5 11 - 11 15536.8515 10.5 10 - 11 15517.48489.5 10 - 10 15536.8349 10.5 9-10 15517.462410.5 8 - 9 15517.442610.5 7 - 8 15517.425010.5 6 - 7 15517.4096(1 ' = 5/ 2 - (1" = 5/2,Q Branch6411 A BandP BranchJ" F' - F" Frequency JI/ F' - F" Frequency2.5 6 - 5 15594.4353 5.5 8 - 8 15587.59572.5 5 - 4 15594.3838 5.5 8 - 9 15587.57042.5 4 - 3 15594.3391 5.5 7 - 8 15587.54862.5 3 - 2 15594.3013 5.5 6 - 7 15587.52992.5 2 - 1 15594.2699 5.5 5 - 6 15587.51382.5 6 - 6 15594.3665 5.5 4 - 5 15587.50102.5 5 - 5 15594.3265 5.5 3 - 4 15587.49102.5 4 - 4 15594.2934 5.5 2 - 3 15587.48382.5 3 - 3 15594.2665 5.5 1 - 2 15587.47982.5 2 - 2 15594.2469 6.5 9 - 9 15585.71972.5 1 - 1 15594.2337 6.5 8 - 8 15585.70452.5 5 - 6 15594.2577 6.5 7 - 7 15585.68992.5 4 - 5 15594.2359 6.5 6 - 6 15585.67712.5 3 - 4 15594.2210 6.5 5 - 5 15585.66512.5 2 - 3 15594.2121 6.5 9-10 15585.69952.5 1 - 2 15594.2106 6.5 8 - 9 15585.68543.5 7 - 6 15593.8482 6.5 7 - 8 15585.67313.5 6 - 5 15593.8192 6.5 6 - 7 15585.66223.5 5 - 4 15593.7931 6.5 5 - 6 15585.6530Appendix F: Hyperfine Line Frequencies of Three Sub-bands of CoO^3246411 A Band, Continued.Q Branch P Branch3.5 4 - 3 15593.7697 6.5 4 - 5^15585.64553.5 3 - 2 15593.74963.5 2 - 1 15593.73343.5 7 - 7 15593.80413.5 6 - 6 15593.78133.5 5 - 5 15593.76123.5 4 - 4 15593.74433.5 3 - 3 15593.73083.5 6 - 7 15593.73723.5 5 - 6 15593.72323.5 4 - 5 15593.71243.5 3 - 4 15593.70513.5 2 - 3 15593.70093.5 1 - 2 15593.70054.5 8 - 7 15593.12444.5 7 - 6 15593.10574.5 6 - 5 15593.08854.5 5 - 4 15593.07314.5 4 - 3 15593.05924.5 3 - 2 15593.04774.5 8 - 8 15593.09204.5 7 - 7 15593.07734.5 6 - 6 15593.06444.5 5 - 5 15593.05324.5 4 - 4 15593.04334.5 7 - 8 15593.04444.5 6 - 7 15593.03604.5 5 - 6 15593.02864.5 4 - 5 15593.02334.5 3 - 4 15593.01924.5 2 - 3 15593.01755.5 9 - 9 15592.22935.5 8 - 8 15592.22065.5 7 - 7 15592.2122Appendix GAssigned Rotational Lines of the Red System of CoO in cm -1325Appendix G: Line Frequencies of the Red Subbands of CoO^ 326J11ST = 7/2R-12" = 7/2,Q6127 AP5.5 16321.0346.5 16320.6527.5 16320.086 16313.146 16307.0188.5 16319.335 16311.580 16304.6389.5 16318.399 16309.830 16302.07110.5 16317.278 16307.888 16299.31811.5 16315.972 16305.768 16296.38412.5 16314.486 16303.465 16293.26013.5 16312.810 16289.95414.5 16310.950 16298.301 16286.46515.5 16308.902 16295.442 16282.79116.5 16306.672 16292.397 16278.93317.5 16304.261 16289.168 16274.88918.5 16301.662 16285.756 16270.66319.5 16298.880 16282.161 16266.25320.5 16295.914 16278.369 16261.65721.5 16292.762 16274.416 16256.87822.5 16289.426 16270.264 16251.91623.5 16285.893 16265.932 16246.77124.5 16281.464^16282.279 16261.386 16241.44224.5 16282.33825.5 16278.071^16278.369 16235.91725.5 16278.55626.5 16274.193^16274.332 16229.494^16230.31026.5 16274.889 16230.37027.5 16270.017^16270.055 16224.110^16224.40927.5 16224.59428.5 16265.659^16265.700 16218.244^16218.38528.5 16218.93329.5 16261.168^16261.208 16212.077^16212.11730.5 16256.621^16256.660 16205.731^16205.77331.5 16246.145^16246.222 16199.255^16199.29332.5 16248.374 16192.719^16192.75932.5 16242.927^16243.03733.5 16245.287 16186.330^16186.35833.5 16238.602^16238.711 16180.209^16180.330Appendix G: Line Frequencies of the Red Subbands of CoO^ 327(1' = 7/2 - Ii" = 7/2, 6127 A, Cont.J//^R^ Q^P34.5^16242.440 16180.50934.5 16233.518 16233.655^16175.058 16175.17135.5^16227.965 16175.43635.5 16168.741 16168.86136.5 16222.060 16222.138^16170.60336.5^16161.661 16161.81937.5^16216.061 16154.14438.5 16209.794^16146.262 16146.34539.5^16203.330 16138.28840.5 16196.641 16196.702 16130.04141.5 16189.770 16189.864^16121.60142.5 16182.740 16182.856 16112.933 16113.00643.5 16175.506 16175.665 16104.107 16104.19344.5^16095.095 16095.21145.5 16085.895 16086.054J"0' =R7/2 - 11" = 7/2, 6151 AQ^ P4.5 16258.7435.5 16258.4756.5 16258.0117.5 16257.350 16250.504 16244.4668.5 16256.498 16248.833 16241.9999.5 16246.988 16239.33810.5 16254.212 16244.949 16236.47711.5 16252.767 16242.699 16233.44512.5 16251.137 16240.263 16230.18613.5 16249.307 16237.638 16226.76614.5 16247.286 16234.795 16223.12015.5 16245.073 16231.776 16219.29216.5 16242.666 16228.562 16215.27117.5 16240.047 16225.161 16211.05818.5 16237.180^16237.274 16206.65019.5 16234.103^16234.288 16217.665^16217.748 16202.04020.5 16230.996^16231.109 16197.182^16197.27320.5 16231.53821.5 16227.723 16192.100^16192.287Appendix G: Line Frequencies of the Red Subbands of CoO^ 328jilIF = 7/2 - 0"R= 7/2, 6151 A, Cont.Q^P22.5 16224.184 16205.244 16187.000^16187.11222.5 16187.54923.5 16220.447 16200.689 16181.73324.5 16216.525 16195.964 16176.19925.5 16212.390^16212.453 16191.012 16170.47126.5 16208.103^16208.198 16164.55727.5 16203.652^16203.786 16158.435^16158.49928.5 16199.064^16199.255 16152.153^16152.24729.5 16194.399^16194.686 16145.716^16145.84729.5 16192.35530.5 16189.794^16190.187 16139.138^16139.33130.5 16188.006^16188.07231.5 16183.217^16183.389 16132.486^16132.77231.5 16130.402^16130.44232.5 16178.094^16178.428 16125.901^16126.28932.5 16124.099^16124.17033.5 16172.828^16173.362 16117.327^16117.50334.5 16167.522 16110.223^16110.56135.5 16161.962^16162.289 16102.971^16103.50236.5 16095.69037.5 16088.159^16088.477jilIT = 7/2R- .0" = 7/2,Q6221 AP6.5 16074.033 16067.820 16062.4177.5 16073.583 16066.517 16060.3088.5 16072.955 16065.071 16058.0209.5 16072.152 16063.450 16055.55610.5 16071.189 16052.94111.5 16070.051 16050.14012.5 16068.763 16047.16713.5 16067.268^16067.306 16044.03614.5 16065.631^16065.827 16040.73015.5 16063.403^16063.829 16037.265^16037.30016.5 16061.573^16061.863 16033.603^16033.80217.5 16059.471^16059.736 16029.388^16029.815Appendix G: Line Frequencies of the Red Subbands of CoO^ 329...pf,fl' = 7 / 2 - (1" = 7/2,R^Q6221 A, Cont.P18.5 16057.170 16025.565 16025.85118.5 16057.420^16057.47419.5 16054.683^16055.013 16021.459 16021.72920.5 16052.014^16052.430 16017.16720.5 16017.413 16017.47221.5 16049.233^16049.699 16012.686 16013.03022.5 16046.254^16046.83623.5 16043.118^16043.84123.5 16042.14724.5 16039.823^16039.27724.5 16040.72225.5 16036.436^16036.29025.5 16037.49526.5 16034.144^16033.07227.5 16029.027^16029.64427.5 16030.68028.5 16025.136^16026.03728.5 16027.09929.5 16022.227^16021.09029.5 16023.40730.5 16018.227^16019.37830.5 16016.94131.5 16014.05732.5 16009.705IF =- 7 / 2 - ir = 7/2, 6388 AJ,, R Q^P8.5 15651.0929.5 15650.083 15633.86010.5 15648.951 15631.04411.5 15647.608 15628.07312.5 15646.096 15624.91813.5 15644.405 15621.59814.5 15642.522 15618.07715.5 15640.453 15614.37916.5 15638.219 15610.49817.5 15635.802 15606.436Appendix G: Line Frequencies of the Red Subbands of CoO^ 330J"(1' = 7/2 - (2" = 7/2,It^Q6388 A, Cont.P18.5 15633.233 15602.20219.5 15630.516 15597.80020.5 15627.690 15593.21921.5 15624.822 15588.51422.5 15618.756 15583.69423.5 15615.642 15578.83024.5 15612.155 15570.76725.5 15608.371 15565.66926.5 15604.328 15560.18427.5 15600.066 15554.40528.5 15595.602 15548.37229.5 15542.13130.5 15535.678jf/Si' = 5/ 2R- 12" = 5/2,Q6142 AP4.5 16275.2645.5 16279.669 16274.296.5 16279.345 16267.7527.5 16278.827 16271.807 16265.6008.5 16278.138 16263.2499.5 16277.139^16277.292 16260.73210.5 16276.249 16258.02011.5 16275.02012.5 16273.612 16252.12013.5 16272.02814.5 16270.264^16270.586 16245.47315.5 16268.313^16268.644 16241.86216.5 16266.532 16238.101^16238.41517.5 16263.863^16264.264 16234.46418.5 16261.386^16261.793 16230.001^16203.13018.5 16235.08119.5 16259.169^16232.684 16199.10220.5 16256.395^16230.088 16221.182^16194.88921.5 16253.464^16227.310 16216.978^16190.48622.5 16224.341 16185.90123.5 16221.182 16181.109Appendix G: Line Frequencies of the Red Subbands of CoO^ 331J11Si' = 5/2 - S2" = 5/2, 6142 A, Cont.R^Q P24.5 16217.840 16176.14125.5 16214.308 16170.98326.5 16210.582 16165.63727.5 16206.651 16160.09928.5 16202.561 16154.37829.5 16198.239 16148.46130.5 16193.771 16142.36231.5 16189.087 16136.08032.5 16184.216 16129.57933.5 16122.90134.5 16116.039jilSi' = 5/2R- fl" = 5/2,Q6152 AP2.5 16254.881 16251.8153.5 16255.180 16251.3684.5 16255.386 16250.671 16246.8565.5 16255.410 16249.845 16245.1306.5 16255.277 16248.864 16243.3087.5 16255.001 16247.734 16241.3238.5 16254.575 16246.454 16239.1889.5 16253.999 16236.90110.5 16253.271 16243.444 16234.46411.5 16252.400 16241.715 16231.87912.5 16251.376 16239.828 16229.14513.5 16250.115^16250.242 16226.26214.5 16248.802^16248.885 16223.22615.5 16247.286^16247.446 16219.976^16220.08116.5 16245.583^16245.825 16216.635^16216.73017.5 16243.747^16244.038 16213.101^16213.26218.5 16241.715^16242.070 16209.420^16209.64819.5 16239.533 16205.567^16205.85020.5 16201.538^16201.89821.5 16234.600^16235.260 16197.69222.5 16232.605 16192.970^16193.50523.5 16228.910 16188.407^16189.06324.5 16225.789 16183.641^16184.404Appendix G: Line Frequencies of the Red Subbands of CoO^ 332jil0' = 5/2 - (1" = 5/2,R^Q6152 A, Cont.P25.5 16222.412 16178.72226.5 16218.737 16173.58827.5 16214.615 16168.19828.5 16162.53029.5 16156.412jilST = 5/2 - il" = 5/2,R^Q6195 AP4.5 16142.9045.5 16142.7846.5 16142.489 16130.8707.5 16140.604^16142.009 16128.7108.5 16140.031^16141.388 16126.3929.5 16139.269^16140.604 16122.496^16123.90410.5 16138.375^16139.694 16119.924^16121.28811.5 16137.244^16138.609 16117.164^16118.49312.5 16135.894^16135.962 16114.219^16115.55813.5 16134.393^16134.488 16111.105^16112.46414.5 16132.685^16132.829 16107.755^16107.81515.5 16130.787^16130.993 16104.232^16104.33116.5 16128.710^16128.985 16100.521^16100.66617.5 16126.461^16126.802 16096.618^16096.82218.5 16124.462 16092.526^16092.80519.5 16121.962 16088.280^16088.62120.5 16119.314 16084.27521.5 16116.538 16079.76922.5 16113.597 16075.11323.5 16110.604 16070.32424.5 16107.222^16108.487 16065.40125.5 16103.778^16104.93726.5 16100.142^16101.232 16056.28227.5 16096.440 16050.73428.5 16045.023Appendix G: Line Frequencies of the Red Subbands of Co0^ 333jil(2' = 5/2R- 0" = 5/2,Q6305 AP2.5 15860.2543.5 15860.5454.5 15856.0095.5 15860.667 15850.4846.5 15860.491 15854.1237.5 15860.163 15852.944 15846.5878.5 15859.686 15851.639 15844.3999.5 15859.057 15850.212 15842.06310.5 15858.275 15839.57711.5 15857.347 15836.93412.5 15856.273 15834.14413.5 15855.057 15831.20714.5 15853.696 15828.12615.5 15852.204 15824.90116.5 15850.576 15821.53517.5 15848.820 15818.03318.5 15846.942 15814.40019.5 15844.941 15810.63720.5 15842.820 15806.75321.5 15840.582^15840.611 15802.74622.5 15838.246^15838.287 15798.63323.5 15835.790^15835.847 15794.386^15794.40924.5 15833.217^15833.302 15790.041^15790.08125.5 15830.492^15830.640 15785.587^15785.64626.5 15827.606^15827.809 15781.015^15781.10027.5 15776.292^15776.440dl' =5/2-11"=5/2,6373 1jil R Q P3.5 15689.4274.5 15689.427 15681.2405.5 15689.251 15683.892 15679.4016.5 15688.897 15682.764 15677.3407.5 15688.347 15681.346 15675.1868.5 15687.620^15687.721 15679.771 15672.8069.5 15686.708^15686.932 15670.25110.5 15685.352^15685.620 15667.505^15667.611Appendix G: Line Frequencies of the Red Subbands of CoO^ 334jil(1' = 5/2 - 0" = 5/2,R^Q6373 A, Cont.P11.5 15684.095^15684.341 15664.570^15664.81012.5 15682.618^15682.881 15661.225^15661.49313.5 15680.945^15681.240 15657.951^15658.17614.5 15679.102^15679.401 15654.471^15654.73915.5 15677.072^15677.388 15650.799^15651.09216.5 15674.821^15675.186 15646.955^15647.25117.5 15672.398^15672.806 15642.888^15643.22618.5 15669.771^15670.251 15638.658^15639.02219.5 15667.011^15667.505 15634.231^15634.64320.5 15664.052^15664.571 15629.613^15630.05021.5 15660.921^15661.446 15624.822^15625.30722.5 15657.610^15658.176 15619.855^15620.36923.5 15654.164^15654.652 15614.712^15615.24424.5 15650.590^15650.971 15609.406^15609.93725.5 15646.955^15647.132 15603.961^15604.44526.5 15643.103^15643.283 15598.395^15598.77727.5 15638.891^15639.659 15592.708^15592.92128.5 15634.517 15586.897^15587.07629.5 15629.969 15580.66630.5 15625.259 15574.31631.5 15620.410 15567.76032.5 15615.417 15561.07333.5 15610.338 15554.22034.5 15605.193 15547.247J1112' = 5/2 - 0" = 5/2,R^Q6418 AP8.5 15580.4609.5 15579.50010.5 15578.329 15560.35011.5 15577.001 15557.37712.5 15575.500 15554.22013.5 15573.801 15550.86214.5 15571.962 15547.32915.5 15569.961 15543.64016.5 15567.776 15539.79417.5 15565.493 15535.79718.5 15531.627Appendix G: Line Frequencies of the Red Subbands of CoO^ 335J„(1' =R5/2 - 11" = 7/2, 6294 AQ P3.54.55.5 15886.60915882.40015881.37115880.38515876.9186.5 15880.107^15880.192 15874.8497.5 15885.742 15878.786 15872.610 15872.7088.5 15884.977 15877.073^15877.223 15870.180 15870.2819.5 15884.044 15875.258^15875.464 15867.564 15867.71510.5 15873.230^15873.534 15864.748 15864.95311.5 15881.649 15871.019^15871.419 15861.736 15862.02012.5 15880.192 15868.601^15869.147 15858.512 15858.91213.5 15878.580 15865.950^15866.691 15855.62214.5 15876.814 15863.080^15864.073 15851.44715.5 15874.869 15859.989^15861.305 15847.569 15848.56416.5 15872.785 15858.367 15844.78217.5 15870.579 15855.289 15840.84918.5 15868.245 15852.07519.5 15848.74420.5 15863.224 15845.28421.5 15841.72722.5 15838.08623.5 15834.36524.5 15830.58725.5 15826.7390' = 9/ 2 - fl" = 7/2, 6186 Ajil R Q P8.5 16161.2269.5 16159.991 16151.73310.5 16158.550 16149.47911.5 16156.925 16147.04412.5 16155.051 16144.40113.5 16152.970 16141.53714.5 16150.687 16138.47715.5 16148.182 16135.15316.5 16145.484 16131.67817.5 16142.570 16127.971 16114.21918.5 16139.442 16124.033 16109.466= 7/2, 6186 A, Cont.Q P16104.55016099.43816094.08616088.52516082.72616076.786 16119.92516115.55916111.01116106.24216101.26216096.07616090.66516085.05516079.218a" = 7/2, 6295 AQ^P15877.51815876.23515874.77015873.12015869.282^15867.101^15856.77015864.758 15853.60015862.238^15850.26015846.72215856.700^15842.98915853.690 15839.19815850.500^15835.17915847.121 15830.98615843.560^15826.61715839.881 15822.08515836.010^15817.37015831.955 15812.51115827.720^15807.46815823.363 15802.22515818.760^15796.88015813.957 15791.26615785.484Appendix G: Line Frequencies of the Red Subbands of CoO^ 336J"(1' = 9/2R- 11"19.5 16136.08020.5 16132.51021.5 16128.72622.5 16124.76023.524.5 16116.14425.5 16111.54026.5 16106.71627.5(1 ' = 9/2 -jil R6.5 15883.7617.5 15883.2728.5 15882.6209.5 15881.79010.5 15880.78911.5 15879.61812.5 15878.26813.5 15876.76314.5 15875.07015.5 15873.20116.5 15871.18717.5 15869.00118.5 15866.62219.5 15864.08420.5 15861.37121.5 15858.50322.5 15855.44823.5 15852.21524.5 15848.84425.5 15845.25826.5 15841.43927.528.5Appendix G: Line Frequencies of the Red Subbands of CoO^ 337IT = 9/ 2 - 12" = 7/2, 6313 Ajil^R^ Q^P7.5 15838.100^8.5^15837.541 15829.5929.5 15835.986 15836.822 15827.409 15828.02610.5 15834.875 15835.928 15825.480 15826.31011.5 15833.659 15834.875 15823.370 15824.42812.5 15832.304 15832.369 15821.174 15822.376 15810.88112.5^15833.64713.5 15830.870 15831.235 15818.791 15818.870 15807.66013.5 15820.13714.5 15829.284 15830.641 15816.362 15817.750 15804.28815.5 15827.569 15828.869 15813.779 15815.121 15800.85816.5 15825.759 15826.920 15811.081 15812.358 15797.27717.5 15823.798 15824.783 15808.254 15809.41318.5 15821.697 15822.490 15805.297 15806.28519.5 15819.430 15820.008 15802.199 15800.97720.5 15817.014 15817.370 15798.936 15799.51121.5^15814.436^15795.512 15795.86222.5 15811.574 15791.94123.5^15808.471^15788.07224.5 15783.98212' = 9/2 - fl" = 7/2, 6314 AJII^R^ Q^P^8.5^15836.4769.5 15835.467^15826.97010.5^15834.250 15824.96011.5 15832.869^15822.751^15813.45012.5^15831.313 15820.372 15810.24013.5 15829.592^15817.808^15806.87314.5^15827.690 15815.083 15803.29715.5 15825.253 15825.555^15812.152^15799.57016.5 15823.091 15823.300 15808.745 15809.050^15795.64717.5^15820.880^15805.600 15805.810 15791.250 15791.55018.5 15818.245 15802.376^15787.076 15787.30019.5 15815.322 15815.481^15798.746 15782.87619.5 15815.616 15815.688Appendix G: Line Frequencies of the Red Subbands of Co0^ 338ft12' = 9/2 - f2" = 7/2, 6314 A, Cont.R^ Q P20.5 15812.511^15812.643 15794.818^15794.980 15778.23320.5 15794.113^15794.18921.5 15809.264^15809.413 15791.014^15791.13222.5 15806.060 15786.759^15786.91823.5 15802.596 15782.57024.5 15778.105jil(2' = 7/2 - II" = 5/2, 6154 AR^ Q^P13.5 16242.04314.5 16227.45815.5 16237.639 16224.40916.5 16235.204 16221.07417.5 16232.556^16232.684 16217.62118.5 16229.675^16229.851 16213.949^16214.10319.5 16226.599 16210.080^16210.26620.5 16223.227 16206.00321.5 16219.809 16201.64022.5 16216.080 16197.20023.5 16192.49124.5 16208.006 16187.55025.5 16203.652 16182.40326.5 16199.026 16177.06927.5 16194.208 16171.42028.5 16189.174 16165.59129.5 16183.911 16159.57130.5 16178.276^16178.420 16153.31031.5 16172.507^16172.679 16146.671^16146.81932.5 16166.570^16166.726 16139.914^16140.08433.5 16160.599 16132.988^16133.13634.5 16154.378 16126.02135.5 16118.783Appendix G: Line Frequencies of the Red Subbands of CoO^ 339(1 ' = 7 / 2 - 12" = 5/2, 6370 AR Q P6.5 15698.0547.5 15697.5648.5 15697.037 15689.0369.5 15696.326 15687.44310.5 15695.467 15685.73611.5 15694.435 15683.892 15674.18112.5 15693.196 15681.838 15671.35713.5 15691.783 15679.622 15668.29214.5 15690.187 15677.205 15665.03815.5 15688.393 15674.598 15661.62216.5 15686.425 15671.817 15658.01617.5 15684.277 15668.855 15654.24518.5 15681.923 15665.686 15650.26519.5 15679.401 15662.339 15646.09720.5 15676.649 15658.793 15641.73921.5 15673.726 15655.062 15637.22022.5 15670.611 15651.140 15632.46123.5 15667.323 15647.032 15627.53324.5 15663.827 15642.719 15622.41825.5 15660.141 15638.219 15617.11426.5 15656.262 15633.536 15611.61827.5 15652.191 15628.660 15605.93028.5 15647.929 15623.582 15600.06529.5 15643.469 15618.320 15593.97830.5 15638.820 15612.873 15587.72931.5 15633.951 15607.222 15581.27332.5 15628.935 15601.374 15574.63333.5 15623.708 15595.334 15567.76034.5 15618.270 15589.115 15560.753
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High resolution spectroscopy of some gaseous transition metal containing diatomic molecules Huang, Gejian 1993
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Title | High resolution spectroscopy of some gaseous transition metal containing diatomic molecules |
Creator |
Huang, Gejian |
Date Issued | 1993 |
Description | This thesis reports studies of the electronic spectra of some gaseous diatomic molecules containing transition metals. The (0,0) bands of the C³Π - X³Δ, e¹Π - X³Δ and f¹Φ - a¹Δ electronic transitions of niobium nitride (NbN) have been recorded at sub-Doppler resolution by intermodulated laser-induced fluorescence. The C³Π and e¹Π states, which belong to the same electron configuration (4dδ)¹(4dπ)¹, interact strongly with each other by a second-order spin-orbit/Fermi contact mechanism; the structure of the resulting e¹Π/ C³Π complex has been treated by a full Hamiltonian matrix analysis, which resulted in a deperturbed set of spin and hyperfine parameters as well as interaction parameters. From the well resolved low J lines of the f¹Φ - a¹Δ (0, 0) band, the hyperfine parameters "a" were measured accurately for the ¹Δ and ¹Φ states; they were found to have been modified by the spin-orbit/Fermi contact cross terms < a¹Δ | Ĥ | X³Δ > and < f¹Φ | Ĥ | B³Φ > respectively. Wavelength-resolved fluorescence studies of the emission spectra from the laser-excited e¹Π and f¹Φ states located a new ¹Σ⁺ and a new ¹Γ electronic state at 5863 and 9919 cm⁻¹, respectively. The low energy of the ¹Σ⁺ state suggests that it arises from the electron configuration (5sσ)²; while the ¹Γ state can only come from the configuration (4dδ)². The vibrational sequence bands of the B - X and C - X systems, along with some weak sub-bands in the region 18200 - 18500 cm⁻¹, have been rotationally analyzed from the laser excitation spectra of NbN. Two of the weak sub-bands were assigned as vibrational bands of the transition (5sσ)¹(4dπ)¹ Y³Π₂ - X³Δ₃, from preliminary analysis of the hyperfine structure; the other weak sub-bands were found to originate from the X³Δ state and to have charge transfer upper states of Π symmetry with B constants of about 0.45 cm⁻¹. The B⁴Π - X⁴Σ⁻ (1,0) band of VO has been investigated at sub-Doppler resolution by intracavity laser-induced fluorescence spectroscopy and at Doppler-limited resolution by emission spectroscopy. Values of the principal hyperfine parameters "a" and (b + c) for the B⁴Π state have been deduced from the resolved hyperfine structures of the low J lines of the B⁴Π [5/2 subscript] - X and B⁴Π [3/2 subscript] - X sub-bands: a = 0.00964 cm⁻¹ and (b + c) = -0.01295 cm⁻¹. Estimates of the dipolar parameter c show that it is small compared to b, so that the Fermi contact parameter b is sizeable and negative; this allows the electron configuration of the B⁴Π state to be assigned as (3dδ)²(3dπ)¹. Rotational analysis of the emission spectra of the B⁴Π - X⁴Σ⁻ (1,0) band proved that the B⁴Π [1/2 subscript] (v = 1) substate is perturbed by an orbitally non-degenerate Σ state. The perturbing state has a fairly large rotational constant, B = 0.5405 cm⁻¹, and very large hyperfine structure; this suggests that it is the ²Σ⁺ state from the electron configuration (3dδ)²(4dσ)¹. A portion of the red system of CoO, from 6100 to 6450 Å, has been studied by intracavity laser-induced fluorescence spectroscopy. Hyperfine analysis of two ⁴Δ [7/2 subscript] - X⁴Δ [7/2 subscript] sub-bands and one ⁴Δ [5/2 subscript] - X⁴Δ [5/2 subscript] sub-band, with origins at 6338 Å, 6411 Å and 6436 Å, gave the principal hyperfine parameters "a" and (b + c) for the upper and lower states. From the derived Fermi contact parameters b, it was possible to assign the X⁴Δ state to the configuration (4sσ)²(3dδ)³(3dπ)², and the two excited ⁴Δ states to (4sσ)¹(3dδ)³(3dπ)²(3dσ)¹ and (O,2pπ)³(4sσ)²(3dδ)³(3dπ)³. The spin structure of the X⁴Δ ground state of CoO was measured directly by wavelength-resolved fluorescence with the observation of spin satellites from the excited ⁴Δ [7/2 subscript] and ⁴Δ [5/2 subscript ] levels. Values of the first- and second-order spin-orbit coupling constants A and λ for X⁴Δ have been deduced from the sub-state origins: A = 166.2 cm⁻¹ and λ = 7.32 cm⁻¹. Rotational analysis of 16 other sub-bands of the red system of CoO revealed a very complex upper state energy level structure with extensive global and local perturbations. Vibrational levels belonging to various ⁴Δ, ⁴Φ and ⁴Π excited states have been partially identified. |
Extent | 17295484 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-07-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0061821 |
URI | http://hdl.handle.net/2429/1211 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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