UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

High resolution spectroscopy of niobium nitride and vanadium oxide Huang, Gejian 1988

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1988_A6_7 H82_3.pdf [ 8.32MB ]
Metadata
JSON: 831-1.0060385.json
JSON-LD: 831-1.0060385-ld.json
RDF/XML (Pretty): 831-1.0060385-rdf.xml
RDF/JSON: 831-1.0060385-rdf.json
Turtle: 831-1.0060385-turtle.txt
N-Triples: 831-1.0060385-rdf-ntriples.txt
Original Record: 831-1.0060385-source.json
Full Text
831-1.0060385-fulltext.txt
Citation
831-1.0060385.ris

Full Text

HIGH RESOLUTION SPECTROSCOPY OF NIOBIUM NITRIDE AND VANADIUM OXIDE By GEJIAN H U A N G B. Sc. (Chemistry) Zhongshan University (China), 1984 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F C H E M I S T R Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1988 © GEJIAN H U A N G , 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 A b s t r a c t T h i s thesis reports the spectroscopic studies of two gaseous molecules, n i o b i u m n i -t r i de ( N b N ) and v a n a d i u m ox ide ( V O ) . T h e 3 I1 — 3 A electronic t r an s i t i on of N b N was recorded by laser - induced f luorescence at D o p p l e r - l i m i t e d reso lut ion as wel l as at sub-Dopp l e r re so lu t ion . T w o almost i den t i c a l b ranch features are observed i n the 3 I Io — 3 A j sub-band because the A - d o u b l i n g in the 3 n o sub-state is large and a lmost J - i ndependent . T h e 3 I I i — 3 A ^ t r an s i t i on is sh i f ted 600 c m - 1 to the red of its cent ra l f i r s t -order po s i t i on as a result of very large second-order sp in -orb i t i n te rac t i on effects. T h e shift is be l ieved to be caused p r i n c i p a l l y by the coup l i ng of the 3 I I 1 component w i t h the 1 I I state f r o m the same e lectron conf i gu ra t ion , w i t h a smal ler c on t r i bu t i on f r o m coup l i ng of the 3 A 2 c ompo -nent of the g round state w i t h the l ow- l y i ng * A state. T h e 3 n and 1I1 states are unusua l i n t ha t the i r zero-order energies are ca lcu la ted to be w i t h i n 100 c m " 1 , based on the newly observed — 3 A 2 ( 1 , 0 ) t r an s i t i on ; this means that they form a very fine examp le of a " s u p e r - m u l t i p l e t " , where the sp in -orb i t effects w i t h i n and between the states of a pa r t i c -u la r e lect ron con f i gu ra t ion are larger than the i r separat ions. T h e sp in -orb i t in teract ions are so extens ive tha t the f ine s t ruc tu re can on ly be f i t ted us ing effective r o t a t i ona l and hyper f ine H a m i l t o n i a n s for the i n d i v i d u a l sub-states, as i n case (c) coup l ing . F r o m the de te rm ined hyper f ine constants h and Cl for the three 3H(v — 0) components , the F e r m i contact cons tant b was found t o be negat ive, wh i ch is consistent w i t h the conf igurat ion 7r6. R o t a t i o n a l analys i s gave the 3 I I and 3 A b o n d lengths as 1.6705 A and 1.6622 A, respect ive ly. T h e near - in f ra red e lectron ic sys tem of V O has been recorded i n emiss ion at Dopp le r -l i m i t e d re so lu t i on w i t h the 1-m F T spectrometer at K i t t Peak N a t i o n a l Observatory. n T h e s pec t r um in the 4000-14000 c m - 1 reg ion consists of numerous t ran s i t i on s w i t h most of t h e m extens ive ly ana l y zed . T w o i so lated sub-bands at 7200 c m - 1 have been assigned as the two spin component s of a 2 I I — 2 A t r an s i t i on and r o t a t i ona l l y ana lyzed. T h e r o t a t i ona l constant for the lower state is f ound to be larger t h a n t ha t for the a82 X4Y,~ g round state, i n d i c a t i n g t ha t the 2A s tate arises f r o m the electron conf i gurat ion o~28. T h e conf igurat ion ass ignment was conf i rmed by the der ived sp in-orb i t coup l i ng constant for the 2A state. S im i l a r reasoning app l i ed to the 2 I I uppe r state suggests tha t it may arise f r o m the con f i gu ra t ion a27r, though the p r e l im i na r y s tudy of the hyper f ine s t ructure argues against th i s ass ignment. m Table of Contents Abstract ii List of Tables vi List of Figures viii Acknowledgement ix 1 Electronic Transitions in Heteronuclear Diatomics 1 1.1 Introduction 1 1.2 The Molecular Hamiltonian 3 1.3 Angular Momenta and Hund's Coupling Cases 4 1.3.1 Angular Momenta 4 1.3.2 Hund's Coupling Cases 7 1.4 Effective Hamiltonian and Matrix Elements 12 1.4.1 Matrix Elements in Case (a )^ 12 1.4.2 A-Doubling in a 3 H State 15 1.5 Selection Rules and Transition Intensities 19 2 Laser Spectroscopy of N b N 24 2.1 Introduction 24 2.2 Experimental Details 25 2.2.1 Description of the source 25 2.2.2 Laser induced-fluorescence 26 iv 2.2.3 Intermodulated fluorescence 27 2.3 The Rotational Structure of the 3II - 3 A System 35 2.3.1 Description of the Doppler-limited spectra 35 2.3.2 Rotational constants 38 2.3.3 The perturbation of the 3JJi sub-state 41 2.4 Hyperfme Structure of the 3II State 45 2.4.1 Features of the sub-Doppler spectra 45 2.4.2 Non-linear least squares fit to the spectroscopic data 55 2.5 Discussion 59 3 Rotational Analysis of a 2II - 2 A System of V O 68 3.1 Background 68 3.2 2II - 2 A Transitions at 7200 cm" 1 69 3.3 Discussion 74 Bibliography 77 Appendix 80 v List of Tables 1.1 The simplified Hamiltonian matrix of the operator (Hrot + H L D ) in a case (a) basis 17 1.2 The Hamiltonian matrix after Wang transformation 18 1.3 Character Table of CTO„ Group 21 2.1 Rotational Constants of the 3 n and 3 A states of NbN° b 40 2.2 Common State Constants (in cm - 1) Reported by Other Workers 40 2.3 Molecular Constants of the 3 n statea 60 2.4 Correlation matrix for the molecular constants of the 3 n 2 ( f = 0) state. . 60 2.5 Expected Low-lying Electronic States of NbN 64 2.6 Comparison of the Hyperfine Constants of the 3 n and 3 $ ° States . . . . 64 3.1 Line Positions of the 2 n —2 A Transition (in cm - 1) 71 3.2 Rotational Constants for the 2 n - 2 A Transition at 7200 cm - 1 (in cm - 1 ) 75 3.3 Molecular Constants of the Quartet States of VO (in cm"1)' 75 vi List of Figures 1.1 Vector diagram of Hund's coupling case (a)[7] 8 1.2 Vector diagram of Hund's coupling case (b)[7] 11 1.3 Vector diagram of Hund's coupling case (c)[7] 13 2.1 Two level system 27 2.2 Gaussian profile of a resonant transition with the center frequency corre-sponding to the unshifted transition frequency 29 2.3 Gaussian Doppler-broadened velocity (vz) population (n) profile, showing two Bennet holes (solid lines) which converge at zero velocity (dotted line) [25] 30 2.4 Lamb dip formed at the center (w = w0) of the profile of intensity versus laser tuning frequency[25] 31 2.5 Schematic drawing of the intermodulated fluorescence experiment used in this laboratory. The discharge cube where the sample and the laser light are combined is shown in the top left corner 33 2.6 a. The formation of crossover resonance, b. Stick diagram of a spectrum with AF ^ A J transitions and the crossover resonances accompanying a AF = A J Q transition 34 2.7 Doppler-limited spectrum of the (0,0) and (1,1) bands of the 3 H 2 — 3 A 3 transition 36 2.8 Doppler-limited spectrum of the 3Ho — 3 A i sub-band 37 2.9 Resonance fluorescence induced by A r + (5145 A) laser pumping 43 vn 2.10 The Q heads of the three 3II - 3A(0,0) sub-bands of NbN 46 2.11 Reverse hyperfine patterns between the low J lines and the high J lines in the3II0(/) - 3 Aj sub-band 47 2.12 The begining of the Q head of the 3T72 - 3A 3(0,0) sub-band 49 2.13 Continuation of Fig. 2.12 50 2.14 The high J portion of the 3 n 2 - 3A 3(0,0) Q head 51 2.15 a) P(3) and b) P(4) lines of the 3 n 2 - 3 A3(0,0) sub-band, illustrating the A F = A J as well as the A F ^ A J transitions and the crossover resonances (labelled as •) 52 2.16 The Q head of the 3n 0(/) - 3 A a sub-band. Only A F = A J lines (labelled with the lower state F values) are observed 53 2.17 The hyperfine patterns of the first four P lines of the 3n0(e) — 3 Ai sub-band, showing that only those A F ^  AJ transitions which do not share the lower levels with any A F = A J transition are observed 54 2.18 a) P(2), b) P(3) and P(4) lines of the 3U.i - 3 A2(0,0) sub-band, showing the A F = A J and A F ^  A J transitions separated into two sections. . . 56 2.19 The relative energies of the molecular orbitals of NbN, formed from the linear combinations of the atomic orbitals of Nb and N 62 2.20 The energy levels and transitions of NbN observed so far 67 3.1 The two components of the 2II - 2 A (0,0) band of VO near 7200 cm"1. . 70 viii Acknowledgement I s incerely w i sh t o thank D r . A . J . Me re r for his adv ice , he lp fu l comments and constant encouragement th roughout th i s work. I a m gratefu l to D r . Y . A z u m a for his assistance w i t h the n i o b i u m n i t r i de pro ject . I wou ld l ike to t hank D r . J . B a r r y for her help i n the least squares fit to the N b N d a t a and Dr . C. C h a n for his techn i ca l assistance. T h a n k s also go to D r . A . A d a m and Dr . V . S rdanov for the i r useful and interest ing discussions. F i na l l y , I wish to thank m y parents for the i r support and encouragement. i x Chapter 1 Electronic Transitions in Heteronuclear Diatomics 1.1 Introduction Spect roscop ic s tudy starts w i t h the analys is of the energy levels of a quant i zed sys-tem. T h e energy levels of a molecu le are g iven by the eigenvalues of the t ime- independent Schrodinger equat i on = (1.1) where H is the t o t a l H a m i l t o n i a n operator , wh i ch may be w r i t t e n as[l]: H — H0 + Hrot + Hspin + Hhjs (1.2) Ho represents the nonre la t i v i s t i c H a m i l t o n i a n of a non - ro ta t i ng molecu le, i.e. the k i ne t i c and po ten t i a l energies of the electrons and the nuclei other t h a n the nuclear r o t a t i o na l energy, HTOi symbol izes the r o t a t i ona l mot ion of the nucle i , H s p i n inc ludes the magnet i c terms t ha t cause the electron sp in fine s t ructure and Hhfs conta ins a l l the electr ic and magnet i c te rms for the nuc lear sp in that cause the hyperf ine s t ructure, is the eigen-f unc t i on assoc iated w i t h a s ta t i onary state and the eigenvalue E is the energy of th is state. T h i s thesis is concerned w i t h the ro ta t i ona l and hyperf ine st ructures of i n d i v i d u a l v i b r on i c sub-states. T h e e lect ron ic and v i b r a t i ona l energy te rms i n Ho are therefore not needed exp l i c i t l y , and the i r effects are wr i t t en as v i b ron i c energy levels for the var ious sub-states. T h e r o t a t i ona l and hyperf ine Ham i l t on i an s w i l l be discussed i n Sect ion 1.2. 1 Chapter 1. Electronic Transitions in Heteronuclear Diatomics 2 For a mu l t i - e l e c t r on sys tem, equat i on (1.1) cannot be solved exact ly. In p rac t i ce , the e igenfunct ion is expanded as l inear comb ina t i on s of the members of a f in i te basis set * = £ arfi (1.3) T h i s then reduces the so lu t ion of equat ion (1.1) to f i nd i ng the roots of the fo l l ow ing secular de te rminant \Hl3 - E6i:\ = 0 (1.4) T h e quant i t ie s HlJ are the m a t r i x elements of the t o t a l H a m i l t o n i a n opera to r H, def ined i n D i r a c no t a t i on as: H{j =<1>i\H\il>j> (1.5) In p r i nc ip le , one can choose any complete basis set ip, to ca lcu la te the elements of the H a m i l t o n i a n m a t r i x . However, a wise choice of the basis set is the one wh ich leads to the m i n i m u m number and size for the off -d iagonal m a t r i x elements. In other words, a basis set is chosen in such a way that the d iagona l e lements of the H a m i l t o n i a n m a t r i x rough ly represent the energy levels of the system. H u n d [7] proposed five sets of angular m o m e n t u m eigenfunct ions as basis sets for the ca l cu l a t i on of the energy levels. T h e five basis sets correspond to five different s i tuat ions of angu la r m o m e n t u m coup l ing , and are now k n o w n as the Hund ' s coup l ing cases (a), (b) , (c), (d) a n d (e), wh ich w i l l be discussed i n Sect ion 1.3. W e bel ieve tha t H u n d ' s case (a) prov ides a good work ing a p p r o x i m a t i o n for the states of bo th N i o b i u m N i t r i d e , N b N , and V a n a d i u m Ox ide , V O , s tud ied in th is thesis. T h e m a t r i x elements of the r o t a t i ona l and hyper f ine H a m i l t o n i a n s ca l cu la ted i n a case (a) basis are given i n Sect ion 1.4. Sect ion 1.5 descr ibes the select ion rules for an elect ron ic t r an s i t i on and the l ine in tens i ty express ions wh i ch are needed when ass igning the s t ructures of e lectronic t rans i t ions . Chapter 1. Electronic Transitions in Heteronuclear Diatomics 3 1.2 The Molecular Hamiltonian A s descr ibed above, the v i b r on i c H a m i l t o n i a n Ho i n equat ion (1.2) is of l i t t l e interest here a n d w i l l be hand led in a phenomeno log i ca l manne r by i n c l u d i n g a T0 t e r m i n the d iagona l pos i t ions of the energy m a t r i x to represent the or ig ins of the v i b ron i c states. T h e r o t a t i o na l H a m i l t o n i a n for a d i a t o m i c mo lecu le is taken as: HTOt = BR2 - DR* (1.6) where R is the angu lar m o m e n t u m operator for the nuclear r o t a t i on , B is the r o ta t i ona l constant and D is the cent r i fuga l d i s t o r t i on constant represent ing the inf luence of r o t a t i on on the b o n d length. T h e constant B ( in un i t s of rec ip roca l wavelength) is def ined as: B{r) = h/(8ir2cpr2) (1.7) where c is the ve loc i ty of l ight , p is the reduced mass of the molecu le (wh i ch for a d i a tom ic mo lecu le is the p roduc t of the a t o m i c masses d i v i ded by their sum) and r is the bond length . T h e magnet i c hyperf ine H a m i l t o n i a n inc ludes the cont r ibut ions f r o m a l l the interac-t ions of the magnetic, moment of the nuclear sp in w i t h the other magnet i c moments i n the molecu le. In H u n d ' s coup l i ng case (a) (discussed in Sect ion 1.3), these interact ions con-sist of nuc lear sp in-overa l l r o t a t i o n , nuclear sp in-e lectron sp in, and nuclear sp in-e lectron o r b i t a l i n teract ions . T h e magnet i c i n te rac t i on of nuclear spin and nuclear ro ta t i on can be represented by the i r scalar p r oduc t [2]: Hi.j = Cii-J (1.8) where C, is the i n te rac t i on constant , J and J are the nuclear sp in a n d molecu la r r o ta t i on angu lar m o m e n t u m operators respectively. T h e i n te rac t i on of nuc lear sp in and e lectronic Chapter 1. Electronic Transitions in Heteronuclear Diatomics' 4 o r b i t a l m o t i o n is also expressed as the i r scalar product[3] [4]: HIL = al L (1.9) where a is the i n te rac t i on constant and L is the e lect ron ic o r b i t a l angu lar m o m e n t u m . T h e H a m i l t o n i a n for the nuc lear sp in-e lectron sp in i n te rac t i on is defined as[3]: HLS = bl- S + cIzSz (1.10) where b is the Fe rm i contact i n te rac t i on constant and c is the d ipo la r hyper f ine constant; Iz and Sz are the cartes ian z -component operators for the nuclear sp in J and the e lectron sp in S. T h e constant b represents the ove r l app ing of a nuc lear sp in wavefunct ion and an elect ron sp in wavefunct ion. In other words, it is the d i rect measurement of the e lectron dens i ty at the nucle i . S ince on ly s electrons have non-zero e lectron densit ies at a nucleus, electronic, states con ta in ing unpa i red s electrons w i l l have large and pos i t ive b values. O n the contrary, a negat ive b va lue indicates t ha t no unpa i red s electrons are present i n the e lect ron ic states. C o m b i n i n g equat ions (1.8), (1.9) and (1.10) gives the t o t a l magnet i c hyper f ine H a m i l -t on i an operator . Hhfs = al • L + {bi • S + cI:Sz) + CXI 3 (1.11) 1.3 Angular Momenta and Hund's Coupling Cases 1.3.1 Angular momenta T h e angu lar m o m e n t u m operators used i n the above Ham i l t on i an s correspond to the i n te rna l and the overa l l angular m o m e n t a present i n a molecule. In terna l angu lar m o m e n t a result f r o m the par t i c le mot ions w i t h i n t he mo lecu la r f rame, wh i l e the over-a l l angu la r m o m e n t u m corresponds to the molecu la r r o ta t i on . In general, a q u a n t u m Chapter 1. Electronic Transitions in Heteronuclear Diatomics 5 mechanical operator P is defined as an angular momentum operator if it obeys the com-mutation rules [5]: [Px, Py] = ihPz etc., in cyclic order (1-12) where Px, Py, and P, are the cartesian components of the operator P in a fixed coordi-nate system. When referred to a molecule-fixed coordinate system, the internal angular momenta such as J, L and S obey commutation rules analogous to rule (1.12), but the overall angular momenta such as the angular momentum J obey an anomalous rule[6]: [Px, Py] = ~lhPZ (1.13) Angular momenta are important in quantum mechanics because P 2 and Pz commute with the Hamiltonian operator H. [P\ H] = 0 (1.14) This relation indicates that P2 and H possess simultaneous eigenfunctions. As a result, the eigenfunctions of angular momentum operators can be used as basis functions to construct the Hamiltonian matrix for the evaluation of the eigenvalues of the operator H. Since the Hamiltonian operator H can be expressed in terms of angular momentum operators, its matrix elements can be easily worked out by first considering the matrix elements of the angular momentum operators. For simplicity, the basis functions are defined in terms of quantum numbers relevant to the individual angular momenta rather than in terms of explicit forms for the wavefunctions. An angular momentum function \PMp > is simultaneously an eigenfunction of the operators P2 and Pz, with respective eigenvalues h2P(P+ 1) and hMp[5}. These relations can be expressed as follows: P'\PMP> = h2 P(P + 1)\PMP > (1.15) PZ\PMP> = hMp\PMP> (1.16) Chapter 1. Electronic Transitions in Heteronuclear Diatomics 6 P is the q u a n t u m number for the angu lar m o m e n t u m P , and takes integra l or ha l f - in tegra l values. Mp is the q u a n t u m number referr ing t o the p ro jec t i on of the angu lar m o m e n t u m P a long the z axis (as yet undef ined) and takes the (2P + 1) values: P, P — 1, . . ., —P. S ince the component operators Px, Py, and P . do not c ommute , the funct ions \PMp > are not e igenfunct ions of the operators Px and Py. T h e effect of the operators Px and Py on a f unc t i on \PMp > is most easi ly t rea ted in terms of the ladder operators P± for wh i ch P± \PMP > = (Px i i Py)\PMp > = sJP(P + 1) - MP(Mp ± l)h\PMP ± 1 > (1.17) Obv ious l y , the operators P+ and P_ have the effect of ra i s ing and lower ing the p ro jec t i on q u a n t u m number Mp by one un i t . W h e n referred to a molecule- f ixed coo rd ina te sys tem, equat ion (1.17) appl ies to the ladder operators of the in te rna l angular m o m e n t a direct ly. However, i t must be app l i ed w i t h c au t i on to the overal l angu lar m o m e n t u m ladder operators because the anomalous c o m m u t a t i o n ru le (Eq.(1.13)) leads to a sign reversal of equat ion (1.17) and produces a different re la t ion i n the molecule- f ixed axis sys tem: P ± \PMP >= sJP(P + 1) - MP(MP T l)h\PMP T 1 > (1.18) T h e sca lar p roduct s of angular m o m e n t u m operators used i n the express ion of the H a m i l t o n i a n opera to r (see Sect ion 1.2) are n o r m a l l y replaced by the fo l l ow ing express ion when eva lua t i ng m a t r i x elements in an angular m o m e n t u m basis. P1 P2 - PlzP2z + ( P 1 + P 2 - + A -P 2 + ) /2 (1.19) Th i s equa t i on is der ived by expand ing the angu la r m o m e n t a P j and P 2 as cartes ian component s and rep lac ing the x, y components w i t h the l adder operators. Chapter 1. Electronic Transitions in Heteronuclear Diatomics 7 T h e r e are var ious k inds of angu la r m o m e n t a that arise i n a molecu le. Because of the different scales of the angular m o m e n t u m interact ions , not a l l of the angu lar m o m e n t a are equa l l y impo r t an t . A s a consequence, one usua l l y on l y inc ludes cer ta in angu lar m o m e n t u m e igenfunct ions in the basis set for the ca l cu l a t i on of the H a m i l t o n i a n mat r i x . In the next sect ion on H u n d ' s coup l i ng cases, we w i l l cons ider the angu lar m o m e n t a present i n a mo lecu le and the i r c oup l i n g in teract ions , as wel l as the choices of different sets of angu lar m o m e n t u m basis funct ions . 1.3.2 Hund's coupling cases H u n d descr ibed the angu lar m o m e n t a present i n a d i a t om ic mo lecu le as " c o u p l e d " acco rd ing to the re lat ive strengths of the i r var ious interact ions . He d i s t ingu i shed five such types of coup l i ng , labe l led case (a) to case (e), and represented each by a pa r t i cu la r set of e igenfunct ions of the relevant angu lar momenta . H u n d ' s coup l i ng cases are l i m i t i n g s i tuat ions . A given state of a g iven molecu le is n o rma l l y descr ibed by one of the l i m i t i n g cases wh ich is chosen such that the m a t r i x representat ion t ha t arises is the most near ly d iagona l . O f course sometimes i t m a y be an advantage to use a basis where the ca lcu la t ion of the m a t r i x elements is easy bu t the m a t r i x i tself is far f r o m d iagona l . S ince the states of NbN and V O s tud ied in th i s thesis fo l low case (a) coupl ing, the present d iscuss ion m a i n l y focuses on H u n d ' s case (a). O t h e r l i m i t i n g cases are also br ief ly ment ioned . F i g u r e 1.1 is a vector d i a g r am representat ion of H u n d ' s coup l i ng case (a). B o t h the t o t a l e lect ron ic o r b i t a l angu la r m o m e n t u m L and the t o t a l e lect ron sp in angular m o m e n t u m S are s t rong ly coup led t o , and rap id l y precess about the internuc lear axis ( the z ax i s ) , w i t h constant p ro ject ions Lz and Sz. L and S can be cons idered as coup led to t he nuc lear r o t a t i ona l angu lar m o m e n t u m , R, to f o r m the t o t a l angu la r momen tum, Figure 1.1: Vector diagram of Hund's coupling case (a)[7]. Chapter 1. Electronic Transitions in Heteronuclear Diatomics 9 J, e x c l ud i n g the nuc lear sp in. J = S + L + R (1.20) T h e vectors S, L, R, J and the z component s Lz, Sz are represented by the respect ive q u a n t u m number s 5, L, R, J , A, and S i n F i g . 1.1. T h e p ro jec t i on of the angular m o m e n t u m J a long the molecu lar axis is represented by the q u a n t u m number Q: = A + S (1.21) B o t h A and R take i n teg ra l values 0, 1, 2, 3, ..., but the c o m m o n l y seen A values are less t h a n 4. Since 5 = 53,5,, where st = ± 1 / 2 for a single e lect ron, the quan tum numbe r 5 and consequent ly the q u a n t u m number J are i n tegra l or ha l f - in tegra l depend ing on whether there are even or odd number s of electrons present in the molecule. A l so the q u a n t u m number J cannot be smal le r t h a n the cor respond ing component q u a n t u m number fi, w h i c h impl ies tha t J has a m i n i m u m value of Q, in a pa r t i c u l a r sub-state. T h e q u a n t u m numbe r £ can take the 2 5 + 1 values S = 5, S - 1, - 5 (1.22) In the usua l spectroscopic nomenc la tu re , the sp in mu l t i p l i c i t i e s ( 25 + 1) are w r i t t en as left superscr ipts to the e lectron ic o r b i t a l angu lar m o m e n t u m symbols , wh i ch are labeled as S , II, A , . . . , cor respond ing to the |A| values of 0, 1, 2, 3, .... T h e basis funct ions i n H u n d ' s case (a) are def ined i n te rms of the above good q u a n t u m numbers , and are therefore w r i t t e n as \r)A; 5X1; J f i > where the s ymbo l r\ refers to the e lectron ic state. A case (a) coup l ing scheme is a good app rox ima t i on to a s i t ua t i on where the coupl ings of the angu la r m o m e n t a to the internuc lear axis are more impo r t an t t h a n the coupl ings between themselves. H u n d ' s coup l i ng case (b) differs f r o m case (a) in tha t the spin angu lar m o m e n t u m S is on l y weak l y coupled to the internuc lear axis so that the quan tum number S is no Chapter 1. Electronic Transitions in Heteronuclear Diatomics 10 longer denned. A basis f un c t i on for case (b) is des ignated \r],NASJ > where the t o t a l angu lar m o m e n t u m J is ob ta i ned as[7]: R + L = N; N + S = J (1.23) T h e vector d i a g r am for case (b) coup l i ng is given i n F i g . 1.2. Because S is not a good q u a n t u m number i n H u n d ' s case (b) , the q u a n t u m numbe r Q is also undef ined. Case (b) coup l i ng appl ies in S states where there is no o r b i t a l angu lar m o m e n t u m coup l i ng the e lect ron sp in to the internuc lear axis. E v e n i f the o r b i t a l angu la r m o m e n t u m is non-zero, the e lect ron spin tends to uncoup le f r o m the in ternuc lear axis w i t h increas ing r o t a t i on due to the increas ing sp in - ro ta t i on i n te rac t i on . Therefore , g iven large enough J values, any case (a) state w i l l uncoup le toward case (b) because of sp in -uncoup l i ng effects [6]. If the nuclear sp in J is i n c l uded in the coup l ing scheme, H u n d ' s case (a) and case (b) must be subd i v i ded because of the var ious coup l ing mechan i sms i n vo l v i n g I. T h e most c o m m o n l y used sub-categories of the H u n d ' s cases are a^ and b^ j where (3 or )3J denotes a s t rong coup l i ng i n te rac t i on between the nuclear sp in / and the r o t a t i on J. T h e t o t a l angu lar m o m e n t u m is therefore ob ta ined as: F = J + / (1.24) T h e basis funct ions for case a^ and bpj are des ignated [77A; S E ; JQ.IF > and \NASJIF > respectively[2][3], where the q u a n t u m number F takes t he values \J — 7|, \J — I -f 1 | , . . . , ( J + / ) . Figure 1.2: Vector diagram of Hund's coupling case (b)[7]. Chapter 1. Electronic Transitions in Heteronuclear Diatomics 12 Case (c) coup l i ng occurs i n molecules con ta i n i ng an a t o m suf f ic ient ly heavy tha t the sp in -o rb i t i n te rac t i on is so large tha t on ly the resu l tant angu lar m o m e n t u m Ja(Ja = L+S) is su i tab le to represent the e lectron m o t i o n . Ja is then coup led to the ro ta t i ona l angu la r m o m e n t u m R to f o r m the t o t a l angu lar m o m e n t u m J, as i l l u s t r a ted i n F i g . 1.3[7]. T h e basis funct ions for case (c) are therefore \nJa, JSIM > where f2 is the on ly wel l -def ined a x i a l component . Case (d) coup l i ng is se ldom found i n l ow - l y i n g e lectron ic states. It occurs i n molecules where an e lect ron has been p r omoted to a R y d b e r g o rb i t a l w i t h h igh p r i n c i p a l quan tum numbe r n. T h e large d i s tance between the e lectron and the nucle i results i n a weak coup l i ng of the e lectron m o t i o n to the in ternuc lear axis, but a s t rong coup l i ng to the ro-t a t i o n a l angu lar m o m e n t u m , R. Case (d) is equiva lent to case (b) bu t w i t h the difference t ha t L is uncoup led f r o m the in ternuc lear axis r a the r than 5. 1.4 Effective Hamiltonian and Matrix Elements 1.4.1 Matrix elements in case (a )^ S ince R = J — L — S i n case (a) coup l i ng (see E q . (1.20)'), the r o ta t i ona l H a m i l -t o n i a n BR' — D J? 4, in a f o r m appropr i a te for ca l cu la t ion in a case (a) basis, becomes: HTOt = B(J - L - S)2 - D(J - L - S ) 4 (1.25) Fo r s imp l i c i t y , we neglect the last t e r m in the above equat ion f o r . the t ime be ing and e x p a n d the B t e r m to Hrot = B{J3 + J> + 5 2 - 2 J • L - 2 J • S-+ 2L- S) (1.26) S ince L and its cartes ian components other t h a n Lz are not defined because of the non - spher i ca l s y m m e t r y of the molecu le, the x, y component s ' o f the operator L w i l l be Chapter 1. Electronic Transitions in Heteronuclear Diatomics 13 Figure 1.3: Vector diagram of Hund's coupling case (c)[7j. Chapter 1. Electronic Transitions in Heteronuclear Diatomics 14 o m i t t e d f r o m the subsequent calculat ion[8]. E q u a t i o n (1.26) therefore s impl i f ies to HTOt = B\J* + Ll + S* - 2 Jz Lz - 2 Jz Sz + 2 Lz Sz - ( j + 5_ + j _ S+)] (1.27) T h e ope ra to r — B(J+ S_ + J _ S + ) is k n o w n as the sp in -uncoup l i ng opera to r , because i t is respons ib le for the t r an s i t i on of H u n d ' s case (a) to case (b) when r o t a t i on is excited[9]. T h e d i agona l m a t r i x elements of the operator Hrot i n a case (a) basis are ca lcu la ted by a p p l y i n g equat ions (1.15) and (1.16) to equat ion (1.27) < S E ; LA; J O | Hroi | S E ; LA; J f i > = B [ J ( J + 1) - ft2 + 5 ( 5 + 1) - E 2 ] (1.28) T h e of f -d iagona l m a t r i x elements are ob ta i ned by app l y i n g equat ion (1.17) to the oper-ators S± and the anomalous equat ion (1.18) to the operators J± i n equa t i on (1.27). < 5 E ± 1; LA; J O ± 1 | Hrot \ 5 E ; LA; J O > = - B ^J{J + 1) - 0 ( 0 ± 1) x ^ 5 ( 5 + 1) - £ ( £ ± 1) (1.29) T h e m a t r i x elements for the D t e r m i n the r o t a t i ona l H a m i l t o n i a n are ca lcu la ted by m u l t i p l y i n g the constant. — D by the square of the coefficient m a t r i x of the B te rm. T h e der i va t ion of the m a t r i x e lements of the hyperf ine H a m i l t o n i a n operator , (a / • L + (bl • S + clz- Sz) + C{ i • J), i n case (a^) basis requires the app l i c a t i on of i r reduc ib le spher i ca l tensor techniques wh i ch we o m i t i n th is thesis. However, the results are given be low [10]. < JIFSlZM\Hhfs\ J / F O E M > = SlhR(J)/[2 J ( J + 1)] + ( l / 2 ) C , f l ( J ) (1.30) < JIFQHM\Hhf,\J - 1 , / F 0 E M > = -h(J2 - 0 2 ) 1 / 2 P ( J ) Q ( J ) / [ 2 J ( 4 J 2 - l ) 1 / 2 ] (1.31) Chapter 1. Electronic Transitions in Heteronuclear Diatomics 15 < JIFSlEM\Hhft\JIFSl ± 1, E ± 1, M > = b[(J + fi)(J ± Q + 1 ) ] 1 / 2 # ( J ) V ( 5 ) / [ 4 J ( J + 1)] (1.32) < JI FQY,M \Hhfs\ J - 1 , / F Q ± 1 , E ± 1 , M > = ^b[(J qp T n + 1 ) 1 / 2 P ( J ) Q ( J ) V ( 5 ) / [ 4 J ( 4 J 2 - 1 ) 1 / 2 ] (1.33) where the following abbreviations have been used: R(J) = F{F + 1) - J(J + 1) - / ( / + 1) (1.34) P(J) = [(F - I + J)(F + J + / + 1 ) ] 1 / 2 (1.35) Q(J) = [ ( J + / - F ) ( F - J + / + 1 ) ] 1 / 2 (1.36) V ( S ) = [ 5 ( 5 + 1) - E ( S ± 1 ) ] 1 / 2 (1.37) h = o A + (b + c ) E (1.38) Combining all the diagonal and off-diagonal matrix elements discussed above, one obtains the full Hamiltonian matrix in a case (a^ ) basis. Diagonalization of this matrix gives the hyperfine energy levels in terms of the total angular momentum quantum number F and the corresponding rotational quantum number J. For the rotational analysis, the hyperfine Hamiltonian is not needed; the energy levels can then be expressed in terms of the quantum number J. 1.4.2 A-Doubling in a 3n state In this section, we conduct a simplified calculation of the A-doubling in a 3n state, since A-doubling phenomena are observed in the present work on NbN. A-doubling results from a breakdown of the Born-Oppenheimer approximation which allows the separation of electronic motion and nuclear motion. It is the lifting of the ±A degeneracy that results when the onset of nuclear rotation prevents the orbital angular Chapter 1. Electronic Transitions in Heteronuclear Diatomics 16 m o m e n t u m f r o m be ing s t r i c t l y quant i zed a long the mo lecu la r z-axis. T h e m a i n oper-ators caus ing A - d o u b l i n g are the X -uncoup l i ng operator , — 2BJ • L, w h i c h arises f r o m the r o t a t i o n a l H a m i l t o n i a n operator(see equat ion (1.26)) and the sp in-orb i t i n te rac t i on operator , J2x aJi • s j l l ] . Thu s the effects of A - d o u b l i n g m a y be taken as ar i s ing f r o m the operator HLD = -2BJ • L + Y^oili • Si (1.39) X Because of the d i f f i cu l ty i n eva lua t ing the m a t r i x elements of the operator HID i n the general case, an "e f fec t i ve " H a m i l t o n i a n opera to r is n o r m a l l y used ins tead. T h e effective H a m i l t o n i a n for the A -doub l i n g i n a 3 n s tate has been der ived by B r o w n et al[ l2] f r o m the HID operator of equat ion (1.39) us ing second order p e r t u r b a t i o n theory. T h e result is g iven below: HLD,eff = (l/2)(o + p + q)(Sl + S2_) -(l/2)(p + 2q)(J+S+ + J _ 5 _ ) + (l/2)q(J2+ + j ? ) (1.40) E q u a t i o n (1-40) refers to a case (a) coup l i ng scheme. T h e effects of the ladder operators i n this equat i on are to be eva luated on the i m p l i c i t unde r s t and ing tha t they l i nk the A = 1 and A = — 1 components of the n s tate only. Therefore, i n a case (a) basis, the fo l l ow ing m a t r i x elements of the operator Hj,D,ejf a r e ob ta i ned . < + 1, £ ± 2, J, n\HLD \ ± l , S J f i > -l-(o + p + q)[{S(S + 1) - £ ( £ ± 1 ) } { S ( S + 1) - ( E ± 1 ) ( E ± 2 ) } ] 1 / 2 (1.41) < + 1 , E ± 1, J, 0 =F 1\HLD | ± 1, E J O > = -l-(p + 2q)[{S(S + 1) - E(E ± 1 ) } { J ( J + 1) - 0 ( 0 =F l ) } ] 1 ' 2 (1.42) < + 1 , E , J, 0 =F 2\HLD\ ± 1, E J O > = ^q\{J(J + 1) - 0 ( 0 =f 1)}{J(J + 1) - ( 0 T l ) ( f i T 2 ) } ] 1 / 2 (1.43) Chapter 1. Electronic Transitions in Heteronuclear Diatomics 17 Tab le 1.1: T h e s imp l i f i ed H a m i l t o n i a n m a t r i x of the opera to r (HTOt + Hm) i n a case (a) basis. |A = 1 > A = -1. > |J2 = 2 > |fi = 1 > |n = o > \n = o > |fi = 1 > \n = 2 > E2 \qy/x2 - 2x Ei i (p+2<7) \qx E0 (o + p+q) \(p + 2q) \qVx2 - 2x \qy/x2 — 2x (o + p+q) E0 \qx |(p + 2 9) E, \qVx2 - 2x E2 x — J(J + 1). T h e r o t a t i ona l H a m i l t o n i a n Hrot has been taken as B J 2 — D J 4 so that En = Tn + BJ(J + 1) - DJ2(J + l ) 2 In order to find out the effects of the above m a t r i x elements on the energy levels of a 3 n s tate, let us consider the H a m i l t o n i a n m a t r i x of the operator (HROT + HLD) M A case (a) basis. Fo r the sake of s imp l i c i t y , the r o t a t i o na l H a m i l t o n i a n HTOT w i l l be taken as BJ2 — DJ4 so tha t i t on ly gives d iagona l elements. T h e m a t r i x is g iven i n Tab le 1.1. T h e two sets of i den t i ca l d iagonal -e lements [E0, E\ and E2) represent the ± A-degeneracies of the 3 n 0 , 3Hi and 3 n 2 sp in sub-states before the A - d o u b l i n g is considered. T h i s A-degeneracy is l i f ted by the ac t i on of the off -d iagonal elements of the A -doub l i n g operator . T h e mechan i sm can be under s tood by app l y i n g a W a n g t r an s f o rmat i on to the r o t a t i o na l m a t r i x of Tab le 1.1. P e r f o r m i n g a W a n g t r an s f o rmat i on is equivalent to t a k i n g n o r m a l i z e d sums and differences of the i n i t i a l basis funct ions as the new basis set. W i t h the new basis funct ions , the H a m i l t o n i a n m a t r i x factor izes i n to two 3 x 3 sub-matr ices as shown i n Tab le 1.2. To first order, the d iagona l elements i n these two sub-matr ices can be taken as the Chapter 1. Electronic Transitions in Heteronuclear Diatomics 18 Tab le 1.2: T h e H a m i l t o n i a n m a t r i x after W a n g t r an s fo rmat i on . |A = 1 > + |A = - 1 > |A = 1 > - |A = - 1 > |ft = 2 > \n = I > \n = o > in = o > |ft = 1 > |fi = 2 > E2 \q\/ x2 — 2x Ei + \qx i(p + 2«) \qVx2 - 2x \(P + 2q) E0-o* -\(p + 2q) - ^ ( P + 2 9 ) Ei ~ \qx -\q\/x2 - 2x o* = (o -f P + q) energies of the A -component s of the three 3n sp in sub-states. For a t r ip le t state the A - componen t s co r respond ing to the basis funct ions ^ ( | A = 1 > + |A = —1 >) are def ined as the / components , wh i l e the other A -component s w i t h basis funct ions ^ ( | A = 1 > — |A = — 1 >) are l abe l l ed as the e components[13]. T h e energy separat ion between the e and / components of a sp in sub-state is the magn i tude of the A -doub l i ng . F r o m the d iagona l elements of the m a t r i x in Tab le 1.2, the A -doub l i ng s i n the 3 n 0 and 3n! component s can a p p r o x i m a t e d as: 3 n 0 : Tj - Te » 2 ( o + p + 9) (1.44) ^ :T5 - Te % qJ{J + 1) (1.45) T h e fact that, the e and / levels rema in degenerate in first order for the 3n2 state on ly i nd ica tes t h a t the A - d o u b l i n g is ext remely sma l l . Fo r the 3 n 0 sub-state of the N b N molecu le s tud ied i n th i s thesis, the separat ion of the two A - component s is about 6.61 c m - 1 wh i ch gives the A - d o u b l i n g pa rameter (o + p + q) as 3.3 c m - 1 . T h e A - d o u b l i n g of the 3Hi sub-state is cons iderab ly smal ler t h a n that of Chapter 1. Electronic Transitions in Heteronuclear Diatomics 19 3n0. In fact , A - d o u b l i n g is not observed for the low J levels of t he N b N 3 I I 1 sub-state. It does, however, become pa r t i a l l y resolved at J % 20 at sub -Dopp le r reso lut ion and is then found to increase w i t h J(J + 1), ju s t as equat i on (1.45) ind icates. For the N b N 3n 2 sub-state, the A - d o u b l i n g is too sma l l to be observed. A - d o u b l i n g not on ly affects the r o t a t i ona l energies of the two A - component s of a 3 I I sub-state, i t also affects the hyperf ine structures of the two A -component s . A s shown by Frosch and Foley[3], the effect of the A - doub l i n g on the hyperf ine s t ruc tu re of a 3 I I state can be represented by the operator HHFILD) = \d(e-**S+I+ + e 2 ^ S _ / _ ) (1.46) where d is a pa ramete r for the state of interest. T h e operators e±2l<^ act on the funct ions |A = =pl > t o give mu l t i p l e s of |A = ± 1 >, i.e. < A = ± l | e ± 2 , * | A = qpl > = - 1 (1.47) Thu s the m a t r i x e lements of the operator Hhf(LD) m a c a s e (a/3) basis |JIFQ.HM > are eva luated on the i m p l i c i t unde r s tand ing that they l i nk the A = 1 a n d A = — 1 components only. I nc lud ing th is A - d o u b l i n g operator into the hyperf ine H a m i l t o n i a n operator in equa t i on (1.11) results i n a sma l l change i n the m a t r i x element of equat ion (1.32) < JI FQ.'LM \Hhjs\ JIFQ. ± 1, E ± 1, M > = (b±d)[(J + Q)(J ± Q + 1 ) ] 1 / 2 R{J)V(S)/[4J{J + 1)] (1.48) T h e upper and lower signs i n equat i on (1.48) refer to the e and / levels respectively. A t sub -Dopp le r re so lu t ion , b o t h b and d are determinab le hyperf ine constants. 1.5 Selection Rules and Transition Intensities T h e e lec t romagnet i c f ie ld of a l ight wave can induce e lectr ic d ipo le t rans i t ions i n a molecu le between the r o t a t i ona l energy levels of different v ib ron i c states. T h e t ran s i t i on Chapter 1. Electronic Transitions in Heteronuclear Diatomics 20 p r obab i l i t y is p r o p o r t i o n a l to the square of the t r an s i t i o n momen t wh i ch is def ined as[7]: Rnm = JKvVmdT (1.49) tyn and tym are the eigenfunct ions of the upper and the lower states. A is the e lect r ic d ipo le momen t opera to r for a molecu le and is def ined as: N A = £ e, f , (1.50) i In D i r a c no t a t i on , equat ion (1.49) becomes Rnm = < * n | A I * m > (1.51) T h e B o r n - O p p e n h e i m e r app rox ima t i on al lows a wave f unc t i on \T/ to be separated i n t o a p roduc t of e lect ron ic , v i b r a t i o na l , r o t a t i ona l and nuc lear - sp in wave funct ions . T h e nuc lear - sp in wavefunct ion is o m i t t e d here for s imp l i c i t y . Thu s , equat ion (1.51) can be rewr i t ten as: Rnm — < Cnvn | A I ^m V m > (1-52) where e, v and r are the e lectron ic , v i b r a t i o n a l and ro ta t i ona l factors wh ich are expressed in terms of the e lect ron ic quantum/numbers (A , S, S ) , the v i b r a t i o n a l q u a n t u m numbe r v and the r o t a t i ona l q u a n t u m numbers J and 0 respect ively. T h e d ipo le moment p is defined i n the molecu le - f i xed axis system. O n l y its p r o jec t i on onto the space-f ixed Z axis is important, to the b e a m of photons defined i n the space-f ixed axis sy s tem[ l4 ] . Az = £ § z 9 h (1-53) g=:x,y,z In the above equat ion $zg is the d i rec t i on cosine wh i ch gives the o r i en ta t i on re l a t i on of the two axis systems. S ub s t i t u t i n g equat ion (1.53) in to equat ion (1.52) gives R n m = < e n vn rn I £ $Zg > (1.54) Chapter 1. Electronic Transitions in Heteronuclear Diatomics 21 Tab l e 1.3: C h a r a c t e r Tab le of the C^ G r o u p n E 2C* • ooo~v Ai = E + 1 1 1 z x2 + t/2, z 2 A2 = S " 1 1 -1 Rz E l = H 2 2 cos $ 0 ( z . y ) ; {Rx,Ry) ( x z , y z ) ( x 2 - j / 2 , x y ) E2 = A 2 2 cos 2$ • 0 E 3 = $ 2 2 cos 3 $ • 0 For an electronic transition, equation (1.54) factorizes to[14] < en | /i° | e m >< vn | u m >< rn \ § Z g \rm > (1.55) 9 where p\° is the electr ic d ipo le momen t operator for the molecu le at e q u i l i b r i u m . A n e lect ron ic t r an s i t i on can occur on ly if the t r an s i t i on moment Rnm is non-zero. Th i s c ond i t i on subsequent ly gives rise to the r o t a t i ona l , v i b r a t i o n a l and e lect ron ic selection rules in an e lect ron ic t rans i t i on . A l t h o u g h there are no st r ic t se lect ion rules regard ing the change of v i b r a t i o n a l q u a n t u m numbers i n an e lectron ic t r an s i t i on , the intens i ty of a v i b r on i c t r an s i t i on depends on t he in tegra l < vn \ vm >, known as the F r anck -Condon over lap in tegra l . T h e selection rules for the electronic pa r t can be easi ly der ived f r o m the i n teg ra l < en | p°g | em > by the app l i c a t i on of g roup theory. T h e s y m m e t r y propert ies of the electronic, e igenfunct ions and the Ca r t e s i an compo-nents of the e lect r i c d ipo le moment can be descr ibed by the representat ions of the C^ group, to wh i ch a heteronuclear d i a t o m i c belongs. Inspect ion of the C^y character tab le (Tab le 1.3) reveals tha t the i r reduc ib le representat ions are labe l led i n the same way as the e lec t ron i c states, n ame l y they are labe l led as: S, II, A , ..., wh i ch t e l l us immed ia te l y the s y m m e t r y species of the electronic, states. T h e components of the electr ic d ipo le Chapter 1. Electronic Transitions in Heteronuclear Diatomics 22 moment behave l ike t rans l a t i ons a long the mo lecu la r axes, and t r an s fo rm as and Tl(x,y)- Fo r equa t i on (1.55) to be non-zero, the d i rect p r oduc t of the s ymmet r y species of the upper state, the lower s tate and the d ipo le moment component must conta in the t o t a l l y s y m m e t r i c species £ + . E x a m i n a t i o n of the character tab le y ie lds the fo l l ow ing select ion rules for an e lect r ic d i po l e t r an s i t i on i n the case (a) app rox ima t i on : 1. A A = 0, ± 1 2. A E = 0 3. AS = 0 T h e second and the t h i r d rules occu r because the d ipo le moment operator does not depend on the sp in coord inates [15]. C o m b i n i n g ru le one and ru le two gives: A f i = 0, ± 1 T h e de r i va t i on of the r o t a t i o n a l selection rules f r om the integra l < rn | | rm > w i l l not be discussed in th is thesis (Readers can refer to Ref [ l6 ] ) . However, the results are g iven be low. • R o t a t i o n a l selection rules i n a case (a) basis: A J = - 1 , 0, +1 < except A J = + 1 , - 1 , on ly , when A A = 0 and A' = A" = 0 T h e r ov i b ron i c t rans i t ions cor respond ing to the three A J values ( — 1, 0, 4-1) are des-i gna ted P, Q and R branches respectively. T h e re la t i ve intens it ies of the three branches can be useful i n the analys is of e lectron ic bands. T h e re lat ive in tens i ty of a ro ta t i ona l l ine depends on the popu l a t i on of the i n i t i a l s tate and the l ine s t rength, Sj, associated w i t h the t r an s i t i on . Sj accounts for the Chapter 1. Electronic Transitions in Heteronuclear Diatomics 23 rotational dependence of the transition moment. For transitions starting from the same energy level, the relative intensities of the rotational lines depend solely on the transition line strengths. The formulae for the line strengths of a symmetric top molecule were first derived by Honl and London [17]. The formulae for the R, Q and P lines in a A A = — 1 emission transition are given below:[7] where J' and 0' are the quantum numbers for the upper state. Because hyperfine interactions split a rotational level of quantum number J into sev-eral hyperfine energy levels with different F quantum numbers, each rotational transition in fact consists of a number of hyperfine transitions. The selection rules for the hyper-fine transitions are identical with those for the rotational transitions with the additional selection rule: A P = 0, ± 1. The relative intensities of the hyperfine transition lines within a rotational line have a very simple pattern for large J values: the intensity is proportional to the total angular momentum quantum number F. ( j 1 - o')(j' - l - o ' ) AJ' (J' - Q')(J' + 1 + Q')(2J' + 1) AJ'(J' + 1) (J' + 1 + Q')(J' + 2 + 0') A(J> + 1) (1.56) (1.57) (1.58) Chapter 2 Laser Spectroscopy of NbN 2.1 Introduction T h e op t i c a l s pec t rum of N i o b i u m N i t r i d e , N b N , was first r epo r ted by D u n n and R a o in 1969[18]. W i t h a theoret i ca l reso lv ing power of about 500,000, t he emiss ion s p e c t r u m of a 3 $ _ 3 A s y s tem i n the red region was found to exh ib i t large hyper f ine sp l i t t ings in some of the low J R l ines, i n d i c a t i n g a large Fe rm i contact in teract ion. T h e hyperf ine constants o,6, and c of b o t h the upper and lower states were later ca lcu la ted by Femen ia s et al[ l9] f r o m more intense spect ra taken w i t h a g ra t i ng spetrograph. Femen ia s ' s result suggested tha t the 3 $ state makes a non-neg l ig ib le c o n t r i b u t i o n to the hyper f ine s t ructure of the 3<|> _ 3 A t ran s i t i on . Recen t l y Femen ias and Dunn[20] have pub l i s hed a f u l l r o t a t i ona l analys i s of the 3<J> — 3 A s y s tem f r o m the g ra t i ng spectra. In 1979, an op t i ca l emiss ion s tudy revealed four new systems, one of w h i c h was assigned as two of the three sp in-orb i t component s of a 3EI — 3 A system; these two were the 3 I Io — 3 A i and the 3H2 — 3 A 3 , whose (0,0) sub-bands were observed at 17900 c m - 1 and 17415 c m - 1 , respectively[21]. A few years la ter , a Ru s s i an group pe r fo rmed a laser spectroscopy s tudy of the 3 $ — 3 A sy s tem i n wh i ch they proposed a set of r o t a t i ona l , cent r i fuga l d i s to r t ion and sp in -orb i t coup l i ng constants (B, D and A), and an energy level scheme for the sy s tem [22]. However, some of the i r results are erroneous, most n o t a b l y a suggestion that the order ing of the sp in-o rb i t man i fo ld s for the two electron ic states is i nver ted , and an ass ignment of the 17057 c m - 1 and the 17415 c m - 1 sub-bands as sp in -orb i t satell ites of the 3 $ — 3 A system. A 24 Chapter 2. Laser Spectroscopy of NbN 25 complete investigation of the red-orange region of the NbN spectrum has been carried out during the past two years at this laboratory at UBC. Extensive laser spectroscopy studies have led to some significant, results. These include the reassignment of the 17057 cm - 1 sub-band as the 3n1 — 3A 2(0,0) transition and the completion of full rotational and hyperfine analyses of the 3 $, 3 A and 3I1 electronic states. The 3II1 — 3A 2(0,0) sub-band has been rotationally analyzed by Lyne[23] and the 3II1 sub-state was found to be severly perturbed by an unseen electronic state lying at higher energy. Similar perturbations also affect the hyperfine structure of the 3 A ground state. As described by Barry in her Ph.D. thesis, the middle component of the 3 A state is shifted down by about 40 cm - 1 and unequal perturbations on the other two components of the 3 A state result in distorted hyperfine structure which cannot be handled by the conventional Hamiltonian using the hyperfine constants a, b, and c. Eventually the hyperfine structure of the 3 A state was interpreted with three h constants (h — a A + (b -f c)E) and two different off-diagonal b constants[24]. In the light of Barry's result, a similar method was used to analyze the hyperfine structure of the 3n state described later in this chapter. The rotational analysis of the 3n — 3 A transition, including the two sub-bands reported by Ranieri[2l], will also be given. 2.2 Experimental Details 2.2.1 Description of the source Gaseous NbN molecules were produced in a flow system by passing a mixture of gaseous niobium pentachloride (NbCl5), nitrogen and argon through a 2450-MHz mi-crowave discharge at low pressure. Gaseous NbCl 5 was obtained by subliming NbCl5 powder at about 100 °C, and the vapour was entrained in a stream of Ar gas containing a small amount of N 2 gas (Ar/N 2 % 10/1). The mixture was pumped rapidly through a Chapter 2. Laser Spectroscopy of NbN 26 mic rowave d ischarge cavity. A lavender-co lored " f l a m e " of N b N chemi luminescence was observed a few cent imeters downs t r eam f r om the discharge tube. 2.2.2 Laser induced-fluorescence T h e l ong flame downs t r eam f r o m the microwave discharge was p u m p e d across a cube-shaped m e t a l fluorescence cel l f i t t ed w i t h qua r t z w indows. A laser b e a m was passed th rough the flame pe rpend i cu la r l y , and the resu l t ing fluorescence observed at r ight angles to the b e a m a n d to the s t r eam of molecules. T h e laser source was a tunab le dye laser (Coherent R a d i a t i o n M o d e l CR599-21 ) p u m p e d by an argon i on laser (Coherent R a d i a t i o n M o d e l Innova-20) operated on the 514.5 n m l ine at 3 watts power .The t unab i l i t y of the dye laser arises f r o m the b road banded na tu re of the fluorescence of an organic dye so lut ion. B y p u m p i n g dye so lut ions of R h o d a m i n e 6 G and R h o d a m i n e 110, it was poss ib le to ob ta i n laser r ad i a t i on in the 16000-18500 c m - 1 region for the observat ion of N b N spectra. A nar row- l ine laser beam, w i t h an output power of about 100 m W and l i n e w i d t h of a few M H z , was obta ined after the b road b a n d dye fluorescence was narrowed by pass ing t h rough a birefr ingent f i l ter and two etalons ins ide the laser cavity[25]. For c a l i b r a t i on purposes, a sma l l po r t i on of the output beam, p icked off by a b e a m sp l i t ter , was sent to an iod ine abso rpt ion or emiss ion cel l ; another f ract ion was sent to a T r ope l con foca l F ab r y -Pe ro t interferometer (w i th a free spect ra l range of 299 M H z ) to p rov ide a l adde r of f requency markers. T h e laser b e a m w i t h the m a j o r i t y of the o u t p u t power was passed pe rpend i cu l a r l y through the s t ream of sample molecules. T h e laser - induced fluorescence of N b N was detected th rough a sharp-cut red filter us ing a pho-t o m u l t i p l i e r t u b e powered by a h igh voltage (300~500 V ) power supply. To increase the s ignal-to-noise r a t i o , phase-sensit ive detect ion was employed, w i t h a P r i n ce ton A p p l i e d Research ( P A R ) mode l 128A l ock - i n ampl i f ier select ing the m o d u l a t e d sample s ignal w i t h Chapter 2. Laser Spectroscopy of NbN 27 w0 = ( E 2 - E 1 ) / h E1 F i gu re 2.1: T w o level s y s tem a na r row-band e lect r i ca l f i l ter. T h e s ignal was m o d u l a t e d by chopp ing the laser beam w i t h a mechan i ca l chopper. T h e sample s pec t r um, the I 2 s p e c t r u m and the frequency markers p rov i ded by the Fab r y -Pe ro t inter ferometer were recorded s imu l taneous ly so that the sample spec t ra cou ld be referred to the I 2 s pec t r um for abso lute f requency ca l i b ra t i on . Laser exc i t a t i on spec t ra of N b N were taken over large regions between 17000 and 18000 c m - 1 . T h e reso lut ion of the spect ra was l i m i t e d by the Dopp le r effect, so that most of the r o t a t i ona l s t ructure in the spect ra was resolved, bu t the hyper f ine sp l i t t ings r ema ined unresolved. For the s tudy of the hyperf ine s t ruc tu re of N b N , some sub -Dopp le r spec t ra were recorded by the I n te rmodu la ted F luorescence techn ique ( IMF) [26] , a h igh reso lut ion techn ique capable of e l im i na t i n g the Dopp le r b roaden ing so t ha t the hyperf ine sp l i t t i n g cou ld be resolved in most of the r o t a t i ona l l ines. In the next sect ion, we consider the bas ic theory and the expe r imenta l ar rangement of th i s technique. 2.2.3 Intermodulated fluorescence A Dopp le r - b roaded spectra l fine is the sum of a great number of m u c h narrower component s cor respond ing to molecules w i t h different t he rma l ve loc i ty v. Cons ider a two level s y s tem i l l u s t r a ted i n F i g . 2.1 where w0 is the t r an s i t i on f requency for a fixed molecule. Mo lecu le s w i t h t h e r m a l ve loc i ty v in teract w i t h a coherent l ight wave of wave vector k at f requency (w0 + k • v) ra ther t han f requency w0 because of the Dopp l e r shift[26]. T h i s effect consequently produces a Gaus s i an l ine-shape cor respond ing to the Chapter 2. Laser Spectroscopy of NbN 28 Gau s s i a n ve loc i t y d i s t r i b u t i o n of the gas-phase molecules. T h e centre of the prof i le (see F i g . 2.2) corresponds to the t r an s i t i o n f requency of molecules w i t h zero ve loc i ty re lat ive to the p ropaga t i n g l i ght wave, and therefore corresponds to the a c tua l t r an s i t i on f requency w0. Because of the h i gh laser power used, the ve loc i ty d i s t r i bu t i on of molecules i n the lower s tate is pe r t u r bed : spec i f ica l ly the popu l a t i on of molecules w i t h one pa r t i cu l a r ve loc i t y is dep leted. T h i s dep let ion of the p o p u l a t i o n is descr ibed as " b u r n i n g a Bennet ho le " i n the lower-state ve loc i ty d i s t r i bu t i on . W h e n two beams f r o m the same laser pass t h r ough the sample molecules i n oppos i te d i rect ions , they b u r n two s ymmet r i c holes about the centre of the Dopp l e r prof i le ( F i g . 2.3) as a result of the oppos i te Dopp le r shifts of the molecules tha t are exc i ted by the laser. In th is s i t ua t i on , two groups of molecules are exc i ted and the t o t a l f luorescence in tens i ty is the s u m of the cont r ibu t ions f r o m each group. A s the laser f requency is tuned towards the a c tua l t r an s i t i o n f requency w0, t he two holes move towards each other; meanwh i l e the t o t a l f luorescence intens i ty increases because of the increas ing p o p u l a t i o n of exc i ted molecules i n accordance w i t h the Gau s s i an ve loc i ty d i s t r i bu t i on . A t the centre of the Dopp l e r prof i le, the two holes co inc ide and the two laser beams in teract w i t h on ly one group of molecules (see F i g . 2.3). A s a resu l t , fewer molecules are exc i ted and the t o t a l fluorescence decreases and forms a d i p at the centre of the Dopp l e r prof i le of the t r an s i t i on . T h i s decrease is known as a " L a m b D i p " ( F i g . 2.4) and has been used i n laser spectroscopy to ob ta i n Doppler - f ree t r a n s i t i o n signals. In pract i ce , L a m b d ip signals are not detected f r o m the change of the t o t a l fluorescence intens ity. Instead they are i so lated w i t h a sens it ive m o d u l a t i o n m e t h o d , the Inter m o d u l a t e d F luorescence technique. In an I M F exper iment , the two laser beams are chopped at frequences / j and f2 so tha t the laser - induced fluorescence s ignals are m o d u l a t e d . W h e n the laser f requency is not set at the t r an s i t i on frequency, the two beams exc i te two different groups of molecules to p roduce fluorescence signals m o d u l a t e d at fi and f2 respect ively. A t the centre of the Dopp le r prof i le, the two laser Chapter 2. Laser Spectroscopy of NbN 29 Figure 2;2: Gaussian profile of a resonant transition with the center frequency corre-sponding to the unshifted transition frequency. Figure 2.3: Gaussian Doppler-broadened velocity (vz) population (n) profile, showing two Bennet-holes (solid lines) which converge at zero velocity (dotted line)[25]. Chapter 2. Laser Spectroscopy of NbN Figure 2.4: Lamb dip formed at the center (w laser tuning frequency [25]. = wQ) of the profile of intensity Chapter 2. Laser Spectroscopy of NbN 32 beams in te rac t w i t h t he same molecules. A s shown by So rem and Schawlow[27], the resu l t i ng fluorescence signals are not on ly m o d u l a t e d at fi and f2 bu t also at (fi + f2) and (fi — f2). If the fluorescence is detected w i t h a l ock - i n ampl i f ie r , t u n e d to the sum frequency (fi -f / 2 ) , on ly the s ignal co r respond ing to the L a m b d ip at the centre of the Dopp l e r prof i le is detected wh i le the rest of the fluorescence is suppressed. T h e l ine w i d t h of the s igna l ob ta ined is dete rmined by the w i d t h of the L a m b d ip wh i ch is m u c h smal ler t han the Dopp l e r w i d t h . A s a result , the hyperf ine t rans i t i ons w i t h i n a r o t a t i o na l l ine are n o r m a l l y resolved unless the hyper f ine s p l i t t i n g is ex t reme ly sma l l . A compute r i zed I n te rmodu l a ted F luorescence exper iment [28] has been pe r fo rmed i n th i s l a b o r a t o r y to o b t a i n sub -Dopp le r spect ra of the 3 I I — 3 A system of N b N . T h e exper-imen ta l a r rangement is i l l u s t r a ted i n F i g u r e 2.5, where the reference s igna l is p rov ided by a He -Ne laser chopped at f requency (fi -f f2). T h e reso lu t ion of th i s exper iment is l i m i t e d by pressure effects in the m i c romave discharge source to about 15 M H z , wh i ch is cons iderab ly smal ler t h a n most of the hyperf ine sp l i t t ings in N b N . W e encountered l i t t l e p r o b l e m in record ing most of the spect ra at sub -Dopp le r reso lut ion except for the pe r t u rbed 3 D i — 3 A 2 sub -band where the t ran s i t i on in tens i ty was so low tha t the s ignal-to-noise r a t i o was insuff ic ient. In most of the sub -Dopp le r spect ra of N b N , cross-over resonances were observed in the low J r o t a t i o na l lines. These appear hal f -way between two hyperf ine t rans i t ions shar ing the same lower energy level whose Dopp l e r profi les over lap. A cross-over resonance is a false s igna l wh i ch occurs because a L a m b d ip s ignal is detected when two sets of s y m m e t r i c Bennet holes cor respond ing to two close t rans i t i ons co inc ide. T h e f o rma t i on of a cross-over resonance is descr ibed i n F i g . 2.6. Chapter 2. Laser Spectroscopy of NbN 33 Pump Discharge in flow system spectrum Fabry--^ Perot PDP-11/23 Micro-computer 13-pen chart recorder £«=iJ y PMT calibration hterpolation markers Figure 2.5: Schematic drawing of the inter modulated fluorescence experiment used in this laboratory. The discharge cube where the sample and the laser light are combined is shown in the top left corner. b) A.F=AJ=0 AF= X X I 1 AJ>I 1 1 1 1 Figure 2.6: a) The formation of crossover resonances (Fi + A 2 and F 2 + A \ ) as the result of allowed AF = AJ transitions (Ai and A 2 ) occurring within the same Doppler-broadened velocity profile as forbidden AF ^ A J transitions (Fj and F 2 ) . The diagram shows the laser scanning toward the non-Doppler-shifted A F = A J transition (occurring at A \ + A 2 ) and beyond toward higher frequency to the A F = A J + 1 transition (Fi + F 2 ) . If the F's and A ' s are exchanged, the first central Lamp dip is the A F = A J — 1 transition, b) Stick diagram of the spectrum of the four forbidden transitions that can accompany a A F = A J = 0 Q transition (X denotes a crossover). With an R line, the A F = 0 and A F = — 1 transitions and the associated crossovers occur to the red of the A F = A J + 1 transition, while with a P line the forbidden transitions lie to the blue to the A F = A J — 1 transition. Chapter 2. Laser Spectroscopy of NbN 35 2.3 The Rotational Structure of the 3n - 3 A System 2.3.1 Description of the Doppler-limited spectra T h e b road band s pec t r um of gaseous N b N i n the region 17000-18000 c m - 1 shows numerous t rans i t i ons . T h e most p rominent are the 3ITo — 3 A i (0,0) t r an s i t i on at 17905 c m " 1 and the 3 I I 2 - 3 A 3 (0, 0) t rans i t ion at 17415 c m 1 . O the r re la t i ve ly weak t rans i t ions i n c l ude the 3n2 - 3 A 3 ( 1 , 1 ) sub-band at 17366 c m " 1 and the ^ - 3 A 2 ( 0 , 0 ) sub-band at 17057 c m - 1 . T h e r o t a t i ona l s t ruc tu re of the 3 I 1 1 — 3 A 2 ( 0 , 0 ) sub-band has been ana l y zed by Lyne[23]. Its hyperf ine s t ruc tu re w i l l be discussed in th i s thesis. T h e D o p p l e r - l i m i t e d spec t ra of the 17366 c m - 1 , 17415 c m - 1 and 17905 c m - 1 sub-bands are i l l u s t r a ted in F i g . 2.7 and F i g . 2.8, where each is seen to consist of red-degraded R, Q and P branches. U n l i k e the other two sub-bands, the 3 I l o — 3 A i (0,0) t r an s i t i on shows two sets of ( R ,Q ,P ) branches wh i ch f o r m two a lmost i den t i ca l sub-bands w i t h Q heads at 17902 c m - 1 and 17908 c m - 1 . T h e i den t i ca l s t ructures of these two sub-bands result because there is a J - i ndependen t A - d o u b l i n g i n the 3 I Io uppe r state, as descr ibed i n C h a p t e r 1. T h e A -doub l i n g in the other 3 I I sub-states is cons iderab ly smal ler and is usua l l y not observed at D o p p l e r - l i m i t e d reso lut ion except, at the highest J values. For convenience, the 17902 c m - 1 and 17908 c m - 1 sub-bands w i l l be descr ibed i nd i v i dua l l y and referred to as 3n o(e) — 3 A j an d 3n0(/) - 3 A ! respect ive ly. 1 A l l the sub-bands i l l u s t r a t ed in F i g . 2.7 a n d F i g . 2.8 show re lat ive in tens i ty pat te rns character i s t i c of a A A = — 1 t r an s i t i on : a st rong Q b ranch , a modera te l y s t rong P b ranch a n d a relat ively weak R b ranch . In most of the branches, the r o t a t i ona l l ines are wel l separated and can be fo l lowed to r o t a t i ona l q u a n t u m numbers of about J = 50 T h e on ly unreso lved ro ta t i ona l s t ruc tu re i n the spect ra appears i n the dense Q head areas where the sma l l r o ta t i ona l 1The e and / assignments were made very recently in this laboratory, based on the newly observed 3IIo — 3S]" transition. 17410.63 c m - l 17445.48 c m - 1 I 1.11. ln 2- 3A 3(o,o) J U U U U ^ M I H I H I I I * 40 35 30 25 20 15 10 5 3 QllillllilUl p20^ 25 130 135 3 5 17406.45 c m - 1 I i 1 7 3 6 , 6 mjJyiiJL^^ 1 6.88 c m - l IT I PL f1 PI 1  1 40 I 45 1 1 I I I 35 30 25 20 , . I I 1 I I I I . 1 I | K 15 10 5 3 17323.55 c m . I .5 i i i w i ^ ^ i„ LLi, i ,I„J... L i -1 n 20125 130 I 35 I 140 I 45 1 I l i „ • 3 1 (1 l l l l M i l l 1 1 n 2-*A 3 ( i , i ) I I 1 I 1 I I 1 1 3 5 10 15 20 25 30 Figure 2.7: Doppler-limited spectrum of the (0,0) and (1,1) bands of the 3 n 3 — 3 A 3 transition. 17939.24 c m - 1 I JuULlJL^ 17910.02 cm «-J j ,1 1..JL, L> H|l|i[i|i! liiiliiiiiiuigi nn II in II l| <|l|<l um i i i | R , 35l 1 1 1 l3i 2 5 2 0 1 5 • , 11 1 1 , , 1 0 1 40 35 30 2 5 2 0 Nb atomic lines 15 ^17908.14 c m - 1 <^ t^mmiiiriifiaiinniniiii^ iiHiiiittniiuiiiiiiPini i nir m ~ • 20 25 30 35 40 20 25 |30 I 35 40 45 p>Ui 1 3 5 35 40 1_L 20  1  I   1' I1 I1 1' I' 10 1 5 2 0 1 0 15 1 0 17878.60 c m " 1 I i 2 5 1 1 I 2 0 Figure 2.8: Doppler-limited spectrum of the 3TJ 0 - 3 Aj sub-band. Chapter 2. Laser Spectroscopy of NbN 38 l i ne separat ion compared to the large hyper f ine l ine w i d t h results i n the over lapp ing of successive r o t a t i ona l l ines. S ince the R branches and the P branches are s t i l l we l l resolved i n the low J regions, they can be t raced down to the i r first l ines. T h e J values of these first l ines give the J values of the lowest r o t a t i ona l levels in the two comb in i n g states, and therefore the fi-values. T h i s m e t h o d of de te rm in ing the fi va lue cannot be appl ied to the 3 n 0 — 3 A i (0, 0) sub -band because the b lend ing of the r o t a t i ona l l ines i n the two Q heads prevents the first r o t a t i ona l l ines f r o m be ing observed. A l t e r n a t i v e l y fi values can be ca l cu l a ted f r o m the H o n l - L o n d o n factors (see equat ions (1.56) to (1.58)) by compar i ng the re lat ive intens it ies of two r o t a t i ona l l ines w i t h a c o m m o n energy level. T h e r o ta t i ona l q u a n t u m number J is assigned to a r o t a t i ona l l ine by means of g round state comb ina t i on differences. T h e pos i t ions of the assigned r o t a t i ona l l ines of the 3L1 — 3 A s y s tem are l i sted i n A P P E N D I X I. O n l y the re la t i ve ly nar row l ines, wh ich are more accurate ly measured, were i n c l uded in the least squares fit to der ive the r o ta t i ona l constants of the 3 I I and 3 A states. 2.3.2 Rotational constants R o t a t i o n a l constants are determined f r o m the r o t a t i ona l b ranch s t ruc tu re of an elec-t ron i c t r an s i t i on by the m e t h o d of least squares. Since the sp in fine s t ructure is not the concern of the present p rob l em, the r o t a t i ona l energy level express ion for the i n d i v i d u a l sub-states can be taken as: E(J) = T0 + BV[J(J + 1) - fi2] - DV[J(J + 1) - fi2]2 (2.1) T h i s equa t i on is der ived from the effective H a m i l t o n i a n Bv J2 — Dv J 4 w i t h Bv and Dv be i ng the effective r o t a t i o n a l constants for v i b r a t i ona l level v. It essential ly represents the d iagona l m a t r i x elements of the operator H0 + Hrot w i t h a l l the ro tat iona l - independent terms i n c l uded i n the e lect ron ic or ig in T 0 . Chapter 2. Laser Spectroscopy of NbN 39 The actual procedure for the least squares sets the line frequencies equal to the dif-ferences between the upper state and lower state energies according to the equation: B' D' B" D" (2.2) or A £ 2 i J[(Jl + i) -J?(J{ + I) 2 -J['(j[' + i) AE2 — 1 J2{J2 + 1) -JZ(j% + i) AEn 1 Wn+l) -Wn + i) 2 W n + 1)2 AE = B C Equation (2.3) can be rearranged to C = ( B T • B ) _ 1 B T AE (2.3) (2.4) In equation (2.3) the vector contains the measured transition frequencies. On cal-culating the matrix ( B T • B ) _ 1 • B T with the known quantum numbers, it is possible to determine the vector of molecular constants C . The results appear in Table 2.1. For comparison, some constants reported by other workers for these states are given in Ta-ble 2.2. The symbol v0 is normally used to describe the origin of an electronic sub-band. It is actually the position of the hypothetical Q(0) rotational line. In pure case (a) coupling the rotational constants Btjj^ for the three components of a triplet state are related to the 'true' rotational constant Bv by the equation 2 Bv E , Beff,S = Bv(l + AA 0 (2.5) where the constant A is the spin-orbit coupling constant, representing the magnitude of Chapter 2. Laser Spectroscopy of NbN 40 Table 2.1: Rotational Constants of the 3II and 3 A states of NbN a Subband B: <LLL i o7 / r 3n 0(e) - 3A a(0,0) 17902.281(3) 0.49647(9) 3 n 0 ( / ) - 3A!(0,0) 17908.891(3) 0.49657(9) Average(e & /) 3 n a - 3A2(O,O) 3 n 2 - 3A3(O,O) 3 n 2 0.49652 17057.474 0.49530 17415.443(12) 0.49619(9) 5.08(42) 0.50013(9) 4.58(42) 5.13(45) 0.50014(9) 4.68(45) 5.10 0.50013 4.63 4.98 0.50173 4.69 5.06(24) 0.50253(9) 4.72(27) 5A3(1,1) 17366.030(12) 0.49221(12) 4.13(60) 0.49988(12) 4.48(60) a All values in cm l , with error limits of three standard deviation. b The standard deviation a of the least squares fit is about 0.0017 cm - 1 . c Values are from Ref[23]. Table 2.2: Common State Constants (in cm 1) Reported by Other Workers Subband 3 n, 3 n 3 $ 3 $ 3 3 $ 4 3A!(0,0) a 3A 3(0,0) a ^(0,0)* 3A 2(0,0) h 3 A ^ n 3(0,0)b 17915 17422 16144.648 16542.980 16860.319 ^11 0.4964 0.4964 0.49532 0.49570 0.49642 1 0 7 v. ff 107£>;. 0.5002 0.5026 4.95 0.50015 4.71 4.88 0.50160 4.59 4.84 0.50263 4.54 a Values Teported in Ref[21] b Values reported in Ref[20]. Chapter 2. Laser Spectroscopy of NbN 41 the s p l i t t i n g of the sp in component s i n an e lect ron ic state. In the present case equa-t i o n (2.5) does not app l y to the m i d d l e components of the 3 I I and 3 A states because b o t h sub-states are p e r t u r b e d to different degrees. F o r t una te l y it can s t i l l be app l ied to the outer two sp in components as they appear not to be pe r tu rbed . T h u s the ' t r u e ' r o t a t i o n a l constant Bv can be a p p r o x i m a t e d as the s imple average of the B e / / , E constants of the two outer sub-states: BV=0(3U) = 0.49635 c m - 1 (2.6) BV=0(3A) = 0.50133 c m - 1 (2.7) S u b s t i t u t i n g the Bv=o values i n to equat i on (1.7), we ob t a i n the e q u i l i b r i u m b o n d lengths as: r 0 ( 3 l T ) = 1.6705 A (2.8) r 0 ( 3 A) = 1.6622 A (2.9) T h e sma l l difference between the upper and lower state b o n d lengths suggests a re lat ive ly sma l l change in the chemica l b o n d i n g as a result of the e lectron ic t r an s i t i on . 2.3.3 The perturbation of the 3I11 sub-state A s ment i oned before, the m i d d l e component of the 3 I I state, 3 I I 1 , was found to be severely p e r t u r b e d by an e lect ron ic state at h igher energy; to be exact , i t is pushed down to 17457 c m - 1 f r o m its expected po s i t i on at 18106 c m - 1 , ( that is m i d w a y between the two outer components of the 3 I I s tate). T o under s tand the e lectron ic i n te rac t i on w i t h i n an e lect ron conf i gurat ion or between e lect ron conf igurat ions, i t is essent ia l to f ind the p e r t u r b i n g state. If add i t i ona l t rans i t i ons are later f ound wh i ch invo lve the new states of N b N near 5000 c m - 1 there m a y also be fu r ther i n f o rmat i on ava i lable about the pe r tu rb i ng Chapter 2. Laser Spectroscopy of NbN 42 state. In th i s sect ion, we report a p re l im i na r y s tudy of the new ly f ound 1 I I e lectronic s tate w h i c h we bel ieve per tu rbs the 3 I I i sub-state. T h e 1 I I s tate was discovered when N b N molecules were exc i ted by an A r ion laser ope ra t i ng at the 5145 A l ine. State-selected f luorescence was detected at numerous l oca -t ions, the most in teres t ing one being resonance f luorescence near 5137 A. It is i l l u s t r a ted i n F i g . 2.9. A c c o r d i n g to the selection rules, the resonance fluorescence consists of an R l ine, a Q l ine and a P l ine shar ing the same upper energy level. Th i s means that the lower levels of the r o t a t i ona l lines mus t have q u a n t u m numbers J — 1, J , a n d J + 1 w i t h energies app rox ima ted as BJ(J — 1), BJ(J + 1) and B(J 4- l ) ( J + 2) respect ively. Thu s f r o m the measured separat ions between the three r o t a t i o n a l l ines the q u a n t u m number J is ca l cu la ted to be 16. Since a l l three lines o r i g inate f r o m the same upper level, the H o n l - L o n d o n fo rmu lae (Eqns (1.56) to (1.58)) can be app l ied . F r o m the re lat ive in ten-sities of the R l ine and the P l ine, th i s t r an s i t i on is f ound to be an Q' — 1 —> Q" — 2 t r an s i t i on . T h e lower state of th is t r an s i t i on is p re sumed to be the 3 A g round state. Thu s i t seems qu i te l i ke ly that the observed t ran s i t i on is a — 3 A 2 t r an s i t i on . O t h e r ev idence ind icates tha t th is is a (1,0) v i b r a t i o n a l band . If the above ass ignment is correct, we expect tha t the (0,0) band of the 1Hi — 3 A 2 t r an s i t i on should lie near 18470 c m - 1 , about 1000 c m - 1 below the (1,0) t rans i t i on . Inspect ion of the N b N b roadband spec t rum i n th i s region shows a weak band sy s tem at 18454 c m - 1 . Hence it seems resonable to bel ieve t h a t the 1ni — 3 A 2 ( 0 , 0 ) t r an s i t i on lies near 18454 c m - 1 . F r o m the pos i t i on of the 1n i(v = 0) e lectron ic state, i t seems fa i r l y cer ta in t ha t th i s is the s tate wh i ch causes the p e r t u r b a t i o n of the 3 H j state. S ince the *n s tate arises f r om the same e lect ron conf igurat ion as the 3n state, the sp in -orb i t operator J2i at'r*t w i l l have of f -d iagonal m a t r i x elements of the t ype < 3n! | Y,i a i l i ' * i 11TIi >[6]. Consequent ly , one can set up a H a m i l t o n i a n m a t r i x Chapter 2. Laser Spectroscopy of NbN 43 Figure 2.9: Resonance fluorescence induced by A r + (5145 A) laser pumping. Chapter 2. Laser Spectroscopy of NbN 44 us ing t he basis funct ions \3Ili > and j 1 J I a >, as fol lows: #11 #12 #21 #22 where # n = T^U,), H22 = T^Tl^ and H12 = H2l =< 3nx | £t o-iU • «; I >• T h e d i agona l m a t r i x elements Hu and H22 are the depe r tu rbed energies of the 3 I I i and ' I i i states. It is assumed tha t the deper tu rbed energy of the 3 I I i state is at the m i d -po int of the sp in -orb i t man i f o l d for the 3 I I state and tha t the pe r t u rba t i on shifts of the 3 I I 1 and 1 I I i states are equa l and oppos i te; i t is then poss ib le to ca lcu late the zero-order level pos i t ions of 3 I I i and 1 I i i as 18106 c m - 1 and 18200 c m - 1 , respect ively. Since the eigenvalues A i and A 2 of the above m a t r i x ( that is, the ac tua l energies of the 3 I I 1 and 1 I I i states observed) are known , one can ca lcu la te the off -d iagonal element # J 2 f r o m the A's and H's by the fo l l ow ing equat ion. #12 = ^\A A i -V ) 2 - ( # n - # 2 2 ) 2 (2.10) #12 is f ound to be 695 c m - 1 . Based on the s ingle-electron conf igurat ion mode l for the e lect ron conf igurat ion Sir of NbN[6] , one can show tha t the off-d iagonal element < 3 I I i | J2, o-i''i • &i 11 TIi > shou ld have the same magn i tude as the sp in-orb i t constant AA for the 3 $ state, wh i ch was measured to be 803 c m - 1 [24]. S ince the measured H\2 (695 c m - 1 ) is in reasonable agreement.with the ca lcu la ted H^2 (803 c m - 1 ) , i t seems l ike ly t ha t the s ingle-electron conf igurat ion m o d e l is app l i cab le i n the molecule of N b N . Thu s the conc lu s ion is t h a t the Q' = 1 upper state of the 5137 A band must be the v = 1 level of the Sir1!! s tate w h i c h is responsible for the a s ymmet r y i n the sp in -orb i t s p l i t t i n g of the 8TT 3n state. W h a t is pa r t i c u l a r l y interest ing is tha t the zero-order pos i t ions of the xn and 3 I I i v = 0 levels ( that is, before the effects of sp in-orb i t i n te rac t ion are cons idered) l ie w i t h i n 100 c m - 1 . T h i s is a remarkab ly sma l l i n te rva l , whose size reflects how very sma l l the effect of the n i t rogen l i gand is on the meta l o rb i ta l s . Chapter 2. Laser Spectroscopy of NbN 45 2.4 Hyperfine Structure of the 3II State 2.4.1 Features of the sub-Doppler spectra In order to s tudy the hyper f ine s t ruc tu re of the 3 I I state, we recorded sub -Dopp le r spec t ra of the 3 I I - 3 A ( 0 , 0 ) t r an s i t i on and the 3n2 - 3 A 3 ( 1 , 1 ) sub -band w i t h the techn ique of I n te rmodu la ted F luorescence. O u r sub -Dopp le r spec t ra cover the region of the dense Q branches and the low J P and R lines; in these la t te r t he hyperf ine s t ruc tu re is we l l resolved and al lows precise measurement of the l ine pos i t ions . A l s o the hyper f ine s p l i t t i n g i n these l ines is ex t reme ly large and contains an enormous amount of i n f o rma t i on on the hyperf ine interact ions. T h e Q head regions of the sub -Dopp le r spec t ra of the three 3 I I - 3 A ( 0 , 0 ) sub-bands are shown in F i g . 2.10. T h e 3n2 - 3 A 3 ( 0 , 0 ) sub-band shows cons iderab ly larger hyperf ine s p l i t t i n g t han the other two sub-bands, w i t h a l l the hyperf ine components of a ro ta t i ona l t r an s i t i on c lear ly resolved. For the 3 I L j ( / ) — 3 A;i and the 3Tli — 3 A 2 sub-bands, the hyper f ine sp l i t t i ng decreases so r a p i d l y w i t h increas ing J that the hyperf ine s t ructure is on ly resolved to about J = 10. In the 3n0(/) — 3 A j (0,0) sub-band, the r o t a t i ona l l ine w i d t h passes t h rough a m i n i m u m at J % 14, and then widens w i t h increas ing r o ta t i on . A f t e r J — 20, the r o t a t i ona l l ines are resolved again but w i t h a reversed hyperf ine pa t te rn as is i l l u s t r a ted i n F i g . 2.11. However, for the 3 D i — 3 A 2 (0,0) sub-band, the hyper f ine s t ructure remains unresolved w i t h increas ing r o ta t i on . Theo re t i ca l l y , the " s p i k i n g " at Q(14) in the 3n0 — 3 A ] sub-band results f r o m sp in -u n c o u p l i n g effects i n the 3 A state, wh i ch is a ' regu la r ' state (i.e. w i t h the sp in-orb i t coup l i ng pa rameter A pos i t ive) . A s descr ibed ear l ier, w i t h increas ing r o t a t i o n , the elec-t r o n sp in g radua l l y uncouples f r o m the internuc lear ax is , and thus the coup l i ng changes f r o m case (a) t o case (b). It is k n o w n tha t i n a pure case (a) t r ip le t s tate, the hyper -fine sp l i t t ings i n the outer two components decrease w i t h increas ing r o ta t i on . It is also Q(14) % f - s A i s n 2 - SA 8 Figure 2.10: The Q heads of the three 3II - 3 A ( 0 , 0 ) sub-bands of NbN. Chapter 2. Laser Spectroscopy of NbN 47 Q(8) Q(9) Q(10) Q(24) Figure 2.11: Reverse hyperfine patterns between the low J lines and the high J lines in the 3 n 0 / — 3 A ! sub-band. Chapter 2. Laser Spectroscopy of NbN 48 k n o w n t ha t a pure case (b) t r ip le t state has inverted hyper f ine sp l i t t ings for the outer two components compared to those for a regular case (a) t r i p l e t state. Thus , a t ran s i t i on f r o m case (a) to case (b) w i l l cause the hyperf ine sp l i t t ings i n the outer two components of a regu lar t r i p l e t state to go th rough a m i n i m u m w i d t h . S ince the hyperf ine sp l i t t i ng is m u c h more p ronounced i n the 3 A state than i n the 3I1 state, the chang ing of the hyperf ine p a t t e r n i n the 3 A state results i n the " s p i k i n g " observed i n the 3 n o — 3 A i sub-band. " S p i k i n g " is also seen i n the 3 n 2 — 3 A 3 sub -band, a l though i t occurs at a m u c h higher J v a l u e ( ~ 3 8 ) . S ince the hyper f ine s p l i t t i n g i n the m i d d l e component of a t r ip le t state is m u c h smal ler t h a n those in the other two components and less sensit ive to r o ta t i on , the increase of r o t a t i on does not affect the hyper f ine pa t te rn i n the 3 n i — 3 A 2 sub-band. T h e ass ignment of the hyper f ine F q u a n t u m numbers is easy for the c lear ly resolved and we l l separated r o t a t i o na l l ines because the hyperf ine s t ruc tu re fol lows the Lande i n te rva l - t ype pa t te rn : the hyper f ine components open out at the higher F s ide, and the h igher F l ines have greater intensity. However th is p rocedure is not easi ly app l i cab le to the spec t ra in the densely over lapped Q head regions. In fact , the hyperf ine t rans i t ions i n those regions were later assigned by m a t c h i n g the observed t r an s i t i on frequencies w i t h those ca l cu la ted . T h e assignment, of the c rowded Q head of the 3 n 2 - 3 A 3 (0, 0) sub-band is i l l u s t r a t ed i n F i g . 2.12 to F i g . 2.14. Some A F ^ AJ l ines are observed in this region, wh i l e more such t rans i t i ons are c lear ly seen i n the low J P l ines i l l u s t r a ted in F i g . 2.15. Supr i s ing ly , a lmost no A F ^ AJ t rans i t ions are observed i n the 3 n o ( /) — 3 A ; i (0,0) Q head shown in F i g . 2.16, and those few tha t appear in the low J P l ines have no cross-over resonances (see F i g . 2.17). W h a t this ind icates is that the hyper f ine s p l i t t i n g i n the upper s tate is so sma l l tha t the A F / A J t rans i t ions l ie at exac t l y the same pos i t ions as the A F = A J t rans i t ions shar ing the same lower levels. Because of the poor s ignal to noise r a t i o i n the 3 n a — 3 A 2 (0,0) sub-band, the hyperf ine s t ruc tu re of the Q b ranch at low J is on l y p a r t i a l l y resolved. However some low J P lines are c lear ly resolved and show some Figure 2.12: The begining of the Q head of the 3 n 2 - 3A3(0,0) sub-band. All lines are assigned with the lower state F quantum numbers. The AF = A J and A F ^ AJ transitions starting from the same F level are connected with a thick line. — 17415.1287 cm" 1 17414,4394 cm" 1 I Q(9) Q(8) Q(7)|0 Q(io)Pf P I \ 5.5 4.5 7.5 L 10.5 y lQ(ii) 1 Q(i2)l 111 14.5 7.5 J—L 13.5 13.5 8.5 12.5 15.5 Q(6)l| Hllll Q(5)iL Q(4)_ Q(3) 9.5 I— 12.5 11.5 J 10.5 9.5 1 8.5 —u 7.5 Figure 2.13: Continuation of Fig. 2.12. Cn O 17414.5366 cm I - l 17414.0350 cm - 1 I Q(12) 7 . 5 1 2 . 5 1 6 . 5 Q(13)_ 1 1 h Q(14) Q(15) 1 0 . 5 1 5 . 5 1 9 . 5 9.5 1 4 . 5 1 8 . 5 1 3 . 5 1 7 . 5 Figure 2.14: The high J portion of the 3 n 2 - 3A3(0,0) Q head. Chapter 2. Laser Spectroscopy of NbN 52 a) P(3) Figure 2.15: a) P(3) and b) P(4) lines of the 3 n 2 - 3 A3(0,0) sub-band, illustrating the A F = A J as well as the A F ^ AJ transitions and the crossover resonances (labelled as •). 17908.6340 cm-1 17909.0345 cm"1 I Figure 2.16: The Q head of the 3n 0(/) - 3 A X sub-band. Only A F = A J lines (labelled with the lower state F values) are observed. w Chapter 2. Laser Spectroscopy of NbN 54 a) P(l) 9/2-11/2 d) P(4) 15/2-17/2 Figure 2.17: The hyperfine patterns of the first four P lines of the 3Ilo(e)— 3 A i sub-band, showing that only those A P ^ A J transitions which do not share the lower level with any A P = A J transition are observed. Chapter 2. Laser Spectroscopy of NbN 55 in teres t ing hyper f ine pat te rns : for the P lines w i t h J " = 2 to J " = 5 the A P = A J and the A P 7^  A J component s are separated in to two sections w i t h the A P = A J t rans i t ions on the low frequency side; the o rde r i ng of the A P = A J hyperf ine component s of the P (2 ) l ine is such tha t the h igh F component, lies at the h igh f requency side, but the order for the higher J P lines is ju s t the oppos i te. F i g . 2.18 shows the ass igned hyperf ine s t ruc tu re of the first three P lines of the 3 I 1 1 — 3 A 2 sub-band. 2.4.2 Non-linear least squares fit to the spectroscopic data In p r i n c i p l e , hyper f ine in teract ions in a case (a) coup l i ng scheme can be descr ibed by the hyper f ine H a m i l t o n i a n discussed in Sect ion 1.4. T h e hyper f ine constants a, b, c, and Cj can be dete rmined by a least squares fit of the sub -Dopp le r spectroscop ic d a t a of an e lect ron ic band sy s tem to the energy expressions der ived f r o m the hyperf ine H a m i l t o n i a n m a t r i x in a case (a^) basis. A n i n i t i a l a t tempt by D r . B a r r y to ob ta i n such a set of mo lecu la r constants for the 3 A state, us ing the d a t a ob ta i ned f r om the 3 $ — 3 A system, d id not succeed. T h e d i f f i cu l ty i n her i n i t i a l least squares fit was caused by unequa l sp in-orbit pe r tu rba t i on s of the hyper f ine st ructures of the three sp in components of the 3 A state. T h i s abno rma l s i tua t ion was later hand led by rep lac ing the convent iona l hyperf ine constants w i t h three different h constants, h+, h0 and (corresponding to the three 3 A component s ) and us ing two different b constants in the hyper f ine H a m i l t o n i a n matr ix[24]. In the l ight of B a r r y ' s result , and cons ider ing the severe pe r t u r ba t i o n observed i n the m i d d l e component of the 3 n state, we dec ided to treat the hyperf ine s t ructures of the three 3 n component s i nd i v i dua l l y , and descr ibed each of t h e m w i t h two hyperf ine constants: h and C t . Hence the basis funct ions i n th is pa r t i c u l a r case (a^) coup l i ng can be taken as |T?A; 5 E ; JQ.IF > where J and F are the on ly va r y i ng q u a n t u m numbers . T h e to t a l H a m i l t o n i a n operator consists of the r o t a t i ona l H a m i l t o n i a n operator and the hyperf ine H a m i l t o n i a n operator ; therefore i ts m a t r i x elements are sums of the r o t a t i o na l m a t r i x Chapter 2. Laser Spectroscopy of NbN 56 Figure 2.18: a) P(2), b) P(3) and P(4) lines of the 3 I 1 , - 3 A 2(0,0) subband, showing the A F = A J and A F ^ A J transitions separated into two sections. Chapter 2. Laser Spectroscopy of NbN 57 elements discussed ear l ier and the m a t r i x e lements of the hyper f ine H a m i l t o n i a n g iven by equat ions (1.30) and (1.31). Fo r a given F q u a n t u m number , J runs f r o m \F — I\ to F + I w i t h the nuc lear spin / equa l to 4.5 for the N b nucleus. T h e t o t a l H a m i l t o n i a n m a t r i x is therefore e i ther a 10 x 10 m a t r i x or a (2F -f 1) x (2F -f 1) m a t r i x depend ing on whether F is larger t h a n / or not. T h e presence of the of f -d iagonal m a t r i x element g iven by equa t i on (1.31) results i n a non- l inear energy express ion i n terms of the mo lecu la r constants after the d i agona l i za t i on of the H a m i l t o n i a n mat r i x . Consequent l y a non- l inear least squares fit is requ i red to der ive the mo lecu la r constants f r o m the spectroscopic data . U n l i k e a l inear least squares f i t , a non- l inear fit is per fo rmed by i te ra t i ve l y i m p r o v i n g an es t imated set of constants u n t i l a successful f it is achieved. In the fo l l ow ing discuss ion of the least squares fit pe r fo rmed for the four 3 n sub-states, the energy levels of the 3 n state were ca l cu la ted by add ing the l ine pos i t ions of the 3 n — 3 A t r an s i t i on l i s ted i n A P P E N D I X II to the energy levels of the 3 A s tate wh ich have been ob ta i ned by B a r r y f r o m the stronger a n d less pe r t u rbed 3 $ — 3 A t r an s i t i on . S t a r t i n g w i t h a set of es t imated molecu la r constants (T, B, D, h, C t ) represented by a vector X , t he t o t a l H a m i l t o n i a n m a t r i x H can be expressed as: H - B X (2.11) where B is a coefficient m a t r i x of the same order as the H m a t r i x . T h e eigenvalues (or the energy levels) E of the H m a t r i x are ob ta ined by d iagona l i za t ion w i t h the eigenvectors U. U T H U = E (2.12) In p rac t i ce , th i s is done by a compute r d i agona l i za ton rout ine w h i c h gives bo th the eigenvalues E and eigenvectors U. T h e ca l cu la ted energy levels are then subt rac ted f r o m the observed energy levels, i n th i s case the 3 n energy levels, to p roduce the vector of res iduals AE. N o w AE is impo r t an t because i t can be used to ca lcu la te the correct ions Chapter 2. Laser Spectroscopy of NbN 58 to the e s t ima ted mo lecu la r constants, AX, f r o m the H e l l m a n n - F e y n m a n theo rem, wh i ch states: = J * * ( ^ ) * ^ T (2.13) ax W h e n referred to a pa r t i cu l a r pa rameter X m , the H e l l m a n n - F e y n m a n t heo rem becomes: I r = l u T i ( ^ - ) i u l « = D - (2-14) oXm dXm T h e H e l l m a n n - F e y n m a n theo rem der ivat ives D i m f o r m the der ivat ives m a t r i x D AE = D AX (2.15) B y a s imp le rearrangement, A X can be expressed as: AX = ( D T D T1 D T AE (2.16) S ince the vector of res iduals AE is known , and the der ivat ive m a t r i x D is easi ly ca l cu -l a ted , the AX vector, con ta i n i ng the correct ions to the e s t ima ted parameter s , can be ca l cu la ted . AX is then added to the i n i t i a l constants to prov ide the improved constants for the next i te ra t ion . T h e same procedure is per formed i te ra t i ve l y un t i l the residuals are reduced to a level comparab le to the expe r imenta l prec is ion. T h e goodness of the least squares fit is measured by the s tandard dev i a t i on , wh ich is ca l cu l a ted as: (AE)T{AE) . { . 1 K ' 2.17 \ (n — m) where n is the number of independent measurements, m the number of unknowns to be de te rm ined , a n d (n — m) the number of degrees of f reedom. T h e uncerta int ies i n the measured parameters are g iven by the square roots of the d iagona l elements of the var iance-covar iance m a t r i x 0 def ined as: 0 = a 2 ( D T D ) " 1 (2.18) Chapter 2. Laser Spectroscopy of NbN 59 T h e factor ( D T D ) _ 1 is a quan t i t y re lated to the s t ruc tu re of the fitting m o d e l and has a large in f luence on how wel l the constants are dete rmined. T h e re lat ions between the constants are descr ibed by the co r re la t i on m a t r i x C w i t h elements c{j = eni^l&ii • en (2.i9) T h e off-diagonal e lements of the C m a t r i x represent the degree of dependence of the mo lecu la r constants on each other. A n element w i t h a value close to un i t y ind icates tha t the constant i and constant j cannot be dete rmined independent ly . A least squares fit of the ava i lab le spectroscopic d a t a of the 3II — 3 A system per-fo rmed accord ing to the above procedure produces five sets of we l l -de te rmined mo lecu la r constants for the five 3II sub-states: 3 n 0 ( e ) , 3 n 0 (/), 3 n a , 3 n 2 (u = 0) and 3Tl2(v = 1). T h e constants are s ummar i zed i n Tab le 2.3 and an examp le cor re la t ion m a t r i x is g iven in Tab le 2.4. T h e s tanda rd dev iat ions of these fits are about 15 M H z wh i ch is close to the expe r imen ta l prec i s ion. T h e hyperf ine constants h for the two A-components of the 3 n 0 state were i n i t i a l l y floated i n the least squares fit. T h e y tu rned out not to be determinab le , w i t h the i r s tandard errors even bigger t han themselves. Th i s is bel ieved to be caused by the fact that the h constant is m u l t i p l i e d by a factor fi = 0 in the d iagona l element of the H a m i l t o n i a n m a t r i x for the 3 n 0 state; thus to first order, h is not determinab le . Since the 3Tli sub-state was fitted w i t h a l i m i t e d amount of da ta , because the hyperf ine s t ructure of the 3 n ! — 3 A 2 sub -band is mos t l y unresolved, the hyperf ine constants were determined w i t h lower accuracy t h a n i n the other sub-states. 2.5 Discussion Mo lecu l a r constants are i m p o r t a n t i n spectroscopy because they conta in va luable i n f o rma t i on on the i n te rna l and ex te rna l structures of molecules, w h i c h one needs i n order Chapter 2. Laser Spectroscopy of NbN Tab le 2.3: M o l e c u l a r Cons tant s of the 3 I I s t a t e 2 60 Sub-state To Beff h 105C, 10 4a 3 n 0 ( e ) ( t ; = °) 17902.2773(1) 0.496550(1) 5.056(3) -6.95(11) 5.6 3 n 0 ( / ) ( t ' = 0) 17908.8881(1) 0.496628(1) 5.031(3) -8.29(11) 5.0 3 n 0 ( ^ e . ) 17905.5837 0.496589 5.044 -7.62 3 I L > = 0) 17459.2260 0.495233(12) 4.98fc 0.0501(4) 5.4(20) 6.9 3n2(v = o) 18308.4033(1) 0.496174(1) 4.806(12) 0.0171(1) 7.06(18) 5.0 3U2{v = 1) 17363.5060(2) 0.49221c 4.13c 0.0191(3) 7.15(30) 5.3 3A3(v = 1) 0.0C 0.499960(2) 5.14(4) 0.1115(3) -13.88 c 5.3 A l l values i n c m - 1 , T h e numbers in parentheses are three t imes the s tandard errors of the constants. 4 T h e D constant is f ixed because on ly low J l ines are i nc luded in the least squares f i t . c Cons tant s are fixed because of h igh cor re la t ion between the cor respond ing constants for the 3 l l 2 ( i > = 1) and 3A3(v = 1 ) states. Tab l e 2.4: Co r re l a t i on m a t r i x for the mo lecu la r constants of the 3H2(v = 0) state. T0 DVieff h To 1.0000 Bv.efJ -0.7106 1.0000 Dv,eff -0.5703 0.9582 1.0000 h 0.3285 -0.2070 -0.1588 1.0000 -0.0789 0.1365 0.1333 -0.4142 1.0000 Chapter 2. Laser Spectroscopy of NbN 61 to unde r s t and the i r deta i led s t ructures. Take the constants B and h as an example. T h e ro t a t i o na l constant B is a d i rect measurement of the b o n d length of a d i a t om i c molecu le, wh i l e the hyper f ine constant h is an ind i rec t measurement of the F e r m i contact i n te rac t i on between the electrons and the nucle i in an e lect ron ic state. Therefore extens ive studies of the mo lecu la r constants of the N b N molecu le are ind i spensab le for us when we w i sh to in terpret the propert ies of th is molecule. B u t before we examine the molecu lar constants of the 3 I I s tate of N b N , it is i m p o r t a n t to discuss the e lectron conf igurat ions and the re lated e lect ron ic states i n the N b N molecule. P u t in s imp le terms, e lect ron conf igurat ions descr ibe how the electrons of a molecu le occupy the ava i lab le mo lecu la r orb i ta l s . For d i a t om i c molecules, mo lecu la r o r b i t a l func-t ions are cons t ruc ted f r om the l inear comb inat ions of the a t om i c orb i ta l s of the two atoms. Acco rd i ng l y , a qua l i t a t i ve m o d e l of the valence mo lecu la r orb i ta l s of N b N is proposed [29] and is i l l u s t r a ted i n F i g . 2.19. In th is f igure a , 7r and 8 denote the magn i tudes (A = 0, 1, and 2) of the angu lar m o m e n t u m components of the i n d i v i d u a l electrons a long the inter-nuc lear axis. It is emphas i zed tha t orb i ta l s have no ac tua l rea l i t y in molecules: on ly the e lect ron ic states have phys i ca l s ignif icance, but orb i ta l s are a useful app rox imat i on . Ba sed on the " b u i l d i n g - u p " p r i nc ip l e , six of the eight valence electrons of the N b N molecu le w i l l f i l l the molecu la r orb i ta l s w i t h the lowest energy in F i g . 2.19 to give the e lect ron con f i gu ra t ion 2pa22pnA. T h e rema in i ng two electrons w i l l e i ther bo th occupy the 5 s a o r b i t a l w i t h an t i -pa ra l l e l spins or go independent ly , w i t h pa ra l l e l spins, i n to the bsc and 4d8 o rb i ta l s . T h e re la t i ve size of the energy separat ion (E^s ~ E5sa) and the e lectron repu l s ion energy ga ined when b o t h electrons are conta ined i n the 5so~ o r b i t a l w i l l decide if the e lect ron conf i gurat ion for the g round state is 2pa22pn45sa2 or 2pc22p-KAbsa14d81. O t h e r conf igurat ions arise when an electron is exc i ted to a pa r t i a l l y filled or empty mo lecu la r o r b i t a l . E l ec t ron i c states are classif ied by the t o t a l e lectron sp in and o rb i t a l angu lar m o m e n t a of the molecu le , wh i ch are der ived f r o m the sp in and o rb i t a l angular Chapter 2. Laser Spectroscopy of NbN 62 Nb orbitals NbN orbitals N orbitals 4da Figure 2.19: The relative energies of the molecular orbitals of NbN, formed from the linear combinations of the atomic orbitals of Nb and N. Chapter 2. Laser Spectroscopy of NbN 63 m o m e n t a of the i n d i v i d u a l electrons accord ing to the a d d i t i o n p r inc ip le s for angular m o m e n t u m . Thus , g iven the poss ib le e lectron conf igurat ions of the N b N molecule, we can deduce the cor respond ing l ow- l y i ng e lectron ic states, w h i c h are shown i n Tab le 2.5. T h e 3 A state wh i ch has been extens ive ly s tud ied by D u n n , Femen ias a n d recently by Barry[l8][19][24] is be l ieved to arise f r o m the 2pcr2 2pirA bso1 Ad8l conf igurat ion. It is ev ident t ha t an unpa i red s e lectron is present because the Fe rm i contact in te ract ion in th is 3 A electron ic state is f ound to be large and pos i t i ve [24]. T h e e lect ron conf igurat ion for the 3LT s tate can be shown to be 2pcr2 2p~nA Adn1 AdS1, wh i ch is the same conf igurat ion as tha t for the 3<i> s tate invo lved in the 3 $ - 3 A transition[24],[20]. C o m p a r i n g the h constants for the 3 I I and the 34> states (Tab le 2.6), one finds a lmost i den t i ca l h+ constants for the 3n2 and the 3 $ 4 components . Fo l l ow ing the def in i t ion of equat i on (1.38), the measured h constants of the 3 I I sub-states are expanded as: 3nj : h0 = a A = 0.5001 c m - 1 ( £ = 0) 3 I I 2 : h+ = Q A + ( H C ) = 0.0171 c m - 1 ( E = 1) Sub t r a c t i n g ho f r o m the h+ gives a negat ive (6 -f c) va lue, wh ich is consistent w i t h the conf i gurat ion TTS where the Fe rm i contact i n te rac t i on wou ld be negat ive because of sp in po l a r i z a t i on of the electrons in the 7r and 8 o rb i ta l s . A l t e rna t i ve l y , one can ca lcu late the app rox ima te b constants f r o m the measured C, constants accord ing to second order p e r t u r b a t i o n theory. Th i s is done as fol lows. Con s i de r i n g the fact t ha t the pe r tu rba t i on s i t ua t i on i n the 3I1 state is s imi la r to tha t i n the 3 A state, we app ly B a r r y ' s H a m i l t o n i a n m o d e l for the 3 A state to the 3n state. Th i s means that, the fu l l H a m i l t o n i a n m a t r i x w i l l i n c lude the fo l l ow ing two off -d iagonal elements wh i ch do not exist i n the H a m i l t o n i a n m a t r i x for a sp in sub-state as discussed earl ier. < 0 = 0, E = - 1 , Jl F\Htotal\n = 1, E = 0, J IF > = Chapter 2. Laser Spectroscopy of NbN Table 2.5: Expected Low-lying Electronic States of NbN 64 Electron Configurations Electronic States 5s<r2 J E + *A, 3 A r 5s<T14dir1 SsaHdcj1 4d82 1 E + , 3 E - , Jr AdvHdS1 Jn, 3nr, 3$r UoHdS* *A, 3 A r a - Subscript r refers to regular (opposed to inverted) multiplet structure. Table 2.6: Comparison of the Hyperfine Constants of the 3 n and 3 $ a States Units: c m - 1 3 n 3$ /i+(E = +l) 0.0171 0.0168 /i o(E = 0) 0.0501 0.0411 A_(E = -1) 0.0633 a - Values are from Ref[24]. Chapter 2. Laser Spectroscopy of NbN 65 V ^ T T ) [ ^ f ^ - B] (2.20) < fi = 1, E = 0, J I F | Htotai I ft = 2, S = l , J / F > = ^ ^ - • l i x ) - " (2-21) where i ? ( J ) = + 1) - J ( J + 1) - / ( / + 1)] and the A -doub l i n g i n 3 n 0 is i n c l uded i n equat i on (2.20) (see Sect ion 1.4.2). To second order, the effects of these two m a t r i x elements can be expressed as correct ions to the d iagona l m a t r i x elements w i t h magn i tude: A£(2) = 1 < 0 I Htotai 1 0' > (2.22) Sub s t i t u t i n g equat ion (2.20) in to equat ion (2.22), we ob ta i n the cor rect ion to the first d iagona l m a t r i x element ($7 = 0, £ = — 1) as: AE(2) = ZJyJ + v n 4j(j+i) D \ E(Q=O) - F(n'=i) { h ^ l ) r ~ ±QB R ( J ) + 2B2 J ( J + 1) S ince 6j and ci are n o r m a l l y very sma l l constants, (6j zbd)2 results in a negl ig ib le first t e r m i n the numerator . T h e t h i r d t e r m is inc luded i n the r o t a t i ona l t e r m B J(J + 1 ) . T h a t means that the measured r o t a t i o na l constant B is ac tua l l y an effective constant ra ther t han the t rue r o t a t i ona l constant. T h e second t e r m , — •=—(bl±d^B R(J), corresponds to the | C, R(J) t e r m i nc l uded i n the present least squares fit. Thu s the measured constants C t ( 3 r i o ) equa l t he quan t i t y ~ E(f-o)-<E^1-1) • s a m e a rgument can be app l i ed to the Cx constant of the 3 n 2 sub-state. A s a result, we have the fo l l ow ing re lat ions: c'^ f» = -^V-Epn,,— 0 - 8 2 9 - 1 0 - 4 < 2' 2 5' c - ( 3 n = » = - ^ n j - g ( » n , ) = 0 - 7 0 8 x l 0 ' < < 2 2 6 ) Chapter 2. Laser Spectroscopy of NbN 66 In the above de r i va t i on , we d i d not cons ider the effect of the p e r t u r b a t i o n of the 3 I I 1 component . Fo r an app rox ima te ca l cu l a t i on , the depe r tu rbed energy of the 3 O i sub-state, ( that is about 18106 c m - 1 ) is used i n the above equat ions; and the fo l l ow ing constants are ob ta i ned . &i = -0 .0154 c m - 1 d = 0 . 0 0 1 1 c m " 1 b3 = - 0 .0143 c m " 1 T h e negat ive b constants once again i nd i ca te the sp in po l a r i z a t i on effect i n the 3 I I s tate resu l t ing f r o m the 7T1 (51 e lectron conf igurat ion. T h e hyper f ine s t ruc tu re of the 3 I I 2 — 3 A 3 ( l , l ) sub-band was also ana lyzed in th i s thesis, g i v i ng hyper f ine constants h and C j for the upper and lower sub-states. These con-stants are qu i te close to the cor respond ing constants for the v = 0 sub-states, i n d i c a t i n g that v i b r a t i o n a l e x c i t a t i on has on ly a very sma l l inf luence on the hyperf ine interact ions . T h e r o t a t i ona l constants B and D have been discussed i n Sect ion 2.3.2, wh i l e the t e r m values To give the pos i t ions of the three 3 I I sub-states re lat ive to the lowest energy level of the 3 A state, wh ich is bel ieved to be the g round state. T h e energy level d i a g r am i l l u s t r a ted i n F i g . 2.20 shows the re lat ive pos i t ions of the e lectronic states of NbN detected so far and the observed t rans i t ions. T h e 3 E ~ and 1 A states are two electronic states tha t have been recent ly d iscovered by the H i g h Reso lu t i on Spectroscopy G r o u p at U B C . T h e y were observed i n the two new t rans i t ions 3 n — 3 E ~ and 3Hi — 1 A , whose pos i t ions enable us to ca lcu la te the sp in -orb i t coup l ing constants for the 3 A and 3 $ states wh i ch had been i nco r rec t l y e s t imated by other workers [20] [22]. For the i m m e d i a t e fu tu re , a r o t a t i ona l and hyperf ine analys is of the new ly found elec-t ron i c states, J n , 3 E ~ and J A , must be made. Fu tu re tasks w i l l also i nc lude d i scover ing other l ow- l y i ng e lect ron ic states such as CT21E+, 6 2 1 F and J E + and Sir1^ Chapter 2. Laser Spectroscopy of NbN .2 8 n f o± (18850) 18306.7 17902.281 8.891 17751.6 17457 16943 16144.648 VISIBLE REGION BANDS INFRA-RED SYSTEMS B o oo oo oo 8 oo o 1 V = l 3 v = 0 CO o 1—( CO © CO 8 o oo OJ CM I O CD OO CD .2 IA o CO CO o C M CO OO T f • • t— o> © CN co CN oo o CO CN CN 5604.4 5197 5111 1033.739 891.3 400 0 Figure 2.20: The energy levels and transitions of NbN observed so far. Chapter 3 Rotational Analysis of a 2 I T — 2 A System of V O 3.1 Background T h e f irst e lectron ic band sy s tem of V O , the C4H~ — X4T,~ sy s tem, was discovered by M a h a n t i in 1935 [30], and la ter observed in stel lar spec t ra by K u i p e r in 1947 [31]. S ince then , four quar tet systems of V O have been found and ana l yzed [32-36]. B u t no doub let systems were observed expe r imenta l l y u n t i l the emiss ion s pec t r um of V O in the in f ra - red region was invest igated at th is l abo ra to ry at U B C . In th is C h a p t e r , we report the r o ta t i ona l analys i s of one of the doublet systems, the 2 I I — 2 A ( 0 , 0 ) sys tem, at 7200 c m - 1 . T h e e lect ron ic t rans i t ions of gasous V O i n the region 4000-14000 c m - 1 were recorded i n emiss ion us ing the 1-m Four ier t r an s f o rm spect rometer bu i l t by Dr . J . W . B r a u l t for the M c M a t h So lar Telescope at K i t t Peak N a t i o n a l Observatory, Tucson , Ar izona[37]. T h e V O emiss ion s pec t rum obta ined is a lmost t o ta l l y dom ina ted by two intense t rans i t ions , A4U - X4Y,~ at 9500 c m - 1 and B4U - X4T,~ at 12700 c m - 1 . Besides these two systems and the i r assoc iated sequence s t ruc tu re , two band systems at 11900 c m - 1 and 9700 c m - 1 are ass igned as the D4A — A'4$ t r an s i t i on and the D4A — J4 4II t r an s i t i on respectively. S ince one of the D4 A — J 4 4 I I sub-bands is v i r t u a l l y ent i re ly b lended w i t h stronger l ines of the J4 4II — X 4 S ~ ( 0 , 0 ) band at 9550 c m - 1 , the quartet na tu re of the D — A system was not obvious. Nea r 9550 c m - 1 , there is i n fact another e lect ron ic t r an s i t i on over lapp ing w i t h the A—X and the D — A t rans i t ions . Th i s band system is assigned as a 2 A — 2 $ t rans i t i on . 68 Chapter 3. Rotational Analysis of a 2H — 2 A System of VO 69 T h e 2 I I — 2 A ( 0 , 0 ) t r an s i t i o n at 7200 c m - 1 , w h i c h w i l l be discussed i n th i s C h a p t e r , is compa ra t i v e l y weak, bu t f o r t una te l y does not over lap w i t h other b and systems. Its Av = 1 sequence lies near 8100 c m - 1 and w i l l not be discussed here. 3.2 2LT - 2 A Transitions at 7200 c m - 1 T w o i so lated sub-bands w i t h R heads at 7157 and 7265 c m - 1 are assigned as the component s of the (0,0) b a n d of the 2II — 2 A t ran s i t i on . T h e y are i l l u s t r a ted i n F i g . 3.1. T h e 2 I l 3 / 2 — 2 A 5 / 2 sub -band (7157 c m - 1 ) is decept ive ly s imp le i n appearance, w i t h clear R, Q and P branches and a sma l l uppe r state A - d o u b l i n g at the highest J values. However, s ix branches rather t han three are observed i n the other sub -band, 2 n i ( / 2 — ^ 3 / 2 , because of cons iderab ly larger A - d o u b l i n g in the upper state. S ince the A - d o u b l i n g is l inear i n J + 1/2, the upper state must have Q = l/2[38]. T h e fact t h a t the P b ranch is stronger t h a n the R b ranch i n b o t h sub-bands conf i rms tha t th is is a A A = —1 t r an s i t i on . T h e J q u a n t u m number s were assigned to the r o t a t i ona l l ines by means of comb ina t i on differences. T h e ass ignments of the r o t a t i ona l l ines and the comb ina t i on differences for b o t h the upper and the lower states are given i n Tab le 3.1. A l inear least squares f i t of the upper state comb ina t i on differences to the case (a) f o r m u l a Erot{J,o) = ?b + BN J(J + 1) - Dn J2(J + l ) 2 (3.1) produces r o t a t i o na l constants BQ and DQ for the two upper sub-states. However, the lower s tate is c lear ly p e r t u r b e d , because its comb ina t i on differences do not fit equa-t i on (3.1). Instead, the energy levels of the lower state are pushed down by @ amount tha t increases w i t h J and wh i ch cannot be hand led by add i ng the t e r m + HQ J3(J + l ) 3 t o equa t i on (3.1). In other words, the h igher the range of J values i n c l uded i n the da ta set, the more the least-squares res iduals show systemet ic t rends. T h i s t ype of behav iour is characte r i s t i c of a s t rong i n te rac t i on w i t h a nearby unseen level , where the unseen level Figure 3.1: The two components of the 2II - 2A(0,0) band of V O near 7200 c m - 1 . Upper tracing: 2 I I 1 / 2 - 2 A 3 / 2 Lower tracing: 2 n 3 / 2 — 2 A 6 / 2 . The spacing of the Q branch of 2U3/2 — 2 A 5 / 2 starts to close up near J = 35.5 and small A-doublings can be seen. Chapter 3. Rotational Analysis of a 2T1 — 2 A System of VO Tab le 3.1: L i n e Pos i t ions of the 2 I I — 2 A T r an s i t i on ( in c m - 1 ) 71 2 n 1 / 2 — 2A3/2 J - l / 2 R Q P A a P ' ( J ) A a P " ( 4 7255.634* 7255. 572* 5 7255.318 7255. 182 6. 656 7.052 6 7254.951 7254. 758 7248. ,278* 7248. 119 7. ,723 8.210 7 7263.434 7254.484 7254. 252* 7246. .741 7246. 548* 8. .821 9.366 8 7263.886* 7253.952* 7253. 695 7245. ,127 7244. ,877 9. .863 10.476 9 7264.282 7263.934 7253.353 7253. 068 7243. ,478 7243. 216* 10. .890 11.571 10 7264.632 7252.692 7252. 367 7241. ,783 7241. 496 11. .924 12.664 11 7264.908 7251.961 7251. ,618 7240. ,032 7239. ,699 12. .951 13.768 12 7265.149 7251.150 7238. .199* 7237. ,844 14, .000 14.849 13 7265.354 7264.908 7250.313 7249. ,901 7236, .301 7235. .914* 15, .029 15.965 14 7265.465* 7265.003 7249.388 7248. .953 7234, .345 7233. .938 16. .071 17.076 15 7265.510 7248.388 7247. ,943 7232. .303 7231. ,885 17 .098 18.157 16 7265.546* 7247.352 7246. .862 7230, .233 7229. .784* 18. .144 19.273 17 7265.398 7264.875 7246.239 7245. .717 7228, .071 7227. .596 19 .160 20.364 18 7265.262 7264.707 7245.053 7244. .513 7225. .353 20 .193 21.463 19 7265.073 7264.463 7243.811 7243 .216 7223 .599 7223. .041 21 .254 22.564 20 7264.790 7264.166 7242.502 7241. .905 7221, .245 7220. .654 22 .265 23.662 21 7264.463 7263.788 7241.132 7240 .481 7218, .843 7218. .240 23 .305 24.746 22 7264.045 7263.366 7239.699 7239 .033 7216 .372 7215, .749 24 .337 25.852 23 7263.587 7262.861 7238.199 7237, .504 7213 .845 7213, .183 25 .376 26.952 24 7263.053 7262.310 7236.637 7235 .914 7211 .244* 7210, .554 26 .389 28.046 25 7262.445 7261.686 7234.984 7234 .254 7208, .589 7207. .870 27, .438 29.135 26 7261.796 7260.979 7233.308* 7232 .535 7205 .857 7205. .110 28 .482 30.245 27 7261.067 7260.232 7231.557 7230 .760 7203, .070 7202. .283 29, .497 31.334 28 7260.285 7259.432* 7229.740 7228 .903 7200 .219 7199. .430 30 .534 32.429 29 7259.432* 7258.529* 7227.854 7226 .998 7197 .308 7196. .476 31 .563 33.527 30 7258.529* 7257.592 7225.903 7225 .021 7194 .330 7193. .468 32 .581 34.611 31 7257.527 7256.598 7223.896 7222 .968 7191 .292 7190. .409 33 .621 35.702 32 7256.487 7255.519 7221.822 7220 .880 7188 .190 7187. .269 34 .649 36.802 33 7255.373 7254.382 7219.684 7218 .712 7185 .020 7184. .078 35 .676 37.885 34 7254.209 7253.182 7217.488 7216 .489 7181.796 7180. .829 36 .717 38.994 35 7252.974 7251.925* 7215.229 7214 .193 7178.499 7177 .489 37 .734 40.078 Chapter 3. Rotational Analysis of a 2H — 2A System of VO 72 36 7251.676 7250.601 7212.896 7211. 839 7175.146 7174.120 38.763 41.160 37 7250.313* 7249.204 7210.515* 7209. 428 7171.740 7170.676 39.794 42.261 38 7248.885 7247.737 7208.053 7206. ,955 7168.251 7167.169 40.805 43.342 39 7205.541 7204, .393 7164.728 7163.595 41.845 44.423 40 7202.972 7201, .806 7161.101 7159.987 41 7200.328 7199 .147 42 7197.631 7196, .423 43 7194.868 7193 .633 44 7192.054 7190 .783 45 7189.169 7187 .869 46 7186.215 7184 .900 47 7183.222 7181 .876 48 7180.164 2 n 3 / 2 - 2A5/2 J - 1/2 R Q P AlF"(J) 5 7140.927 6.779 6 7146.217* 7139.438 7.835 8.301 7 7145.751 7137.916 8.847 9.392 8 7145.206 7136.359 9.898 10.481 9 7155.541* 7144.623 7134.725 10.938 11.597 10 7155.952 7143.964 7133.026 11.979 12.690 11 7156.280 7143.253 7131.274* 13.027 13.803 12 7156.519 7142.477 7129.450 14.071 14.912 13 7156.733 7141.636 7127.565 15.102 16.005 14 7156.881 7140.733 7125.631 16.149 17.109 15 7156.956 7139.773 7123.624 17.176 18.209 16 7157.008* 7138.740 7121.564 18.231 19.311 17 7156.926* 7137.660 7119.429 19.261 20.415 18 7156.808 7136.506 7117.245 20.307 21.509 19 7156.628 7135.304 7114.997 21.335 22.610 20 7156.413 7134.029 7112.694 22.379 23.703 21 7156.125 7132.705 7110.326 23.425 24.813 22 7155.778 7131.317 7107.892 24.437 25.907 23 7155.365 7129.847 7105.410 25.490 26.983 24 7154.894 7128.354 7102.864 26.532 28.098 25 7154.342 7126.788 7100.256 27.568 29.189 Chapter 3. Rotational Analysis of a 2II — 2 A System of VO 73 26 7153.776 7125. 167 7097. .599 28.605 30.287 27 7153.121 7123. .485 7094. .880 29.635 31.381 28 7152.419 7121. .739 7092. .104 30.679 32.471 29 7119. .947 7089. .268 31.714 33.569 30 7150.846 7118. 092 7086, .378 32.748 34.651 31 7149.964 7116. ,189 7083, .441 33.785 35.744 32 7149.047* 7114, .230 7080 .445 34.822 36.827 33 7148.059 7112. ,225 7077, .403 35.836 37.901 34 7147.051 7110. .160 7074, .324 36.885 38.975 35 7108. .070 7071 .185 37.908 40.069 36 7105.923 7105.896 7068.001 38.947 41.138 37 7103.732 7103.705 7064.785 7064.758 39.975 42.200 38 7101.514 7101.473 7061.545 7061.492 41.008 43.260 39 7099.255 7099.228 7058.253 7058.213 42.025 44.315 40 7096.962 7096.941 7054.947 7054.906 43.059 45.352 41 7094.687 7094.629 7051.626 7051.572 44.090 46.372 42 7092.396 7092.356 7048.306 7048.265 43 7090.113 7090.075 44 7087.870 7087.823 45 7085.674 7085.624 46 7083.572 7083.504 * = blended Chapter 3. Rotational Analysis of a 2H 2 A System of VO 74 lies at h igher energy bu t has a smal ler B value so tha t the two levels approach each other w i t h increas ing J . 3.3 Discussion R o t a t i o n a l constants for th is 2 I I — 2 A t r ans i t i on are g iven in Tab l e 3.2; the mo lecu la r constants of the quar tet states ana l yzed so far are g iven in Tab le 3.3 for compar i son. T h e 2 A lower state is pa r t i cu l a r l y in teres t ing because it has a s l i ght ly bigger B va lue (and consequent ly a shorter b o n d length) t h a n the g round state, wh i ch i m m e d i a t e l y tel ls us tha t i t comes f r o m the e lectron conf igurat ion tr2^1. T h e reason is that in T i O the o molecu la r o r b i t a l is known to be ma rg i na l l y more s t rong ly bond i n g t h a n the 8 o r b i t a l , so t ha t a s suming the same appl ies to V O , as wou ld seem reasonable, the o~282A state shou ld have a shorter bond length t han the g round state, a82 X4Y,~. The re can be no doubt about th is conc lus ion since i t is conf i rmed by the sp in -orb i t coup l ing . T h e equat ion BQ = B(l + 2BYZJAK) (see equat ion (2.5)) must be app l i ed w i t h c au t i on to the 2 A s tate because i t is known to be pe r tu rbed , and therefore the effective B constant, Befjtf, may be s l ight ly con tamina ted . O n the other hand , the 2I1 state at 7200 c m - 1 is u n p e r t u r b e d , and also the difference between its two Bejj^s is larger so tha t E q . (2.5) gives a more accurate es t imate of Ah. A p p l y i n g E q . (2.5) to the 2I1 state we find ylA ( 2 n) ~ 207 c m - 1 , and add i ng the separat ion of the two 2 I1 — 2 A sub-bands we obta ined AA(2A) ~ 316 c m - 1 . [Direct app l i ca t i on of equat ion (2.5) to the 2 A state gives A A ( 2 A ) ~ 400 c m - 1 . ] N o w the CT2.52A s tate has an unpa i r ed e lectron i n the same 8 mo lecu la r o r b i t a l as the 080* DAA state ana lyzed previously[39]. Therefore, as suming no con f i gu ra t ion i n te rac t ion , we expect where the factor of three comes f r o m the rat io of eigenvalues of Sz for the highest £ , 4 A ( a 2 6 2 A ) = 3 x AA{a8a' D4A) (3.2) Chapter 3. Rotational Analysis of a 2U — 2 A System of VO 75 T a b l e 3.2: R o t a t i o n a l Cons tan t s for the 2I1 — 2 A Tran s i t i on at 7200 c m 1 ( in c m : ) T£ Be] 1 0 7 / J n A - doub l i n g r f 2 n 3/2 a + 7147.735(7) 0.52118(14) 6.94(121) (5.2 ± 1.8) x 1 0 ~ 7 1/2 7256.427(5) 0.51857(7) 6.25(29) 0.02850(7) 2 A 5/2 a 0.55239(15) 9.16(201)b 3/2 0 0.55087(7) 6.84 (34) c E r r o r l im i t s are three s t anda rd dev iat ions , in un i t s of the last s igni f icant f igure. a Re l a t i ve to 2 A 3 / o , v = 0. h\0uH = - 1 5 . 8 ± 1 2 . 2 . c 10uH = - 1 . 9 ± 0.5. d Coeff ic ient of ( J + l / 2 ) 2 " . Tab l e 3.3: M o l e c u l a r Cons tant s of the Qua r te t States of V O (in c m 1 ) a E l e c t r on i c States T0 B A A E l e c t r o n Conf i gu ra t ions A'4$ 7254.951 0.52231 170.79 Asa^dS^dir1 D 4 A 19148.08 0.48704 95.66 4sa13d813dai C 4 £ ~ 19420.103 0.49379 3dS23da1 B4U 12605.57 0.5124 63 3d823dw1 AAIi 9498.878 0.51693 35.19 4scrl3d8*3d^ A ' 4 E - 0 0.54638 Asal3d82 a - A l l values are f r o m Ref[39]. Chapter 3. Rotational Analysis of a 2TI — 2 A System of VO 76 sp in components[40]. T h e e s t imated value, AA(2A) ~ 316 c m - 1 , is i n very sat i s factory agreement w i t h the value 3 x AA(D4A) =287.0 c m - 1 der ived f r o m the observed sp in-o rb i t spl i tt ings[39]. It is tempting to app ly s im i l a r arguments to determine the e lect ron conf igurat ion of the 2 I I state, bu t the ev idence is less conclus ive. W h a t we can say def in i te ly is t ha t on ly the conf igurat ions c T 2 7 r a and 081s give l ow- l y i ng 2 I I states that cou ld emi t to the a282A s tate in one-e lectron j umps , so tha t the 2LT state p resumab ly has to come f r o m one of these conf igurat ions . N o w the c 2 7 r 2 I I state w i l l have a sp in-orb i t coup l i ng in terva l AA that is three t imes that for the O~2TT B4U s tate (by the same reasoning as for E q . (3.2)), i n other words ~ 1 8 9 c m - 1 [35]. Th i s is suff ic ient ly close to the rough value 207 c m - 1 f ound for the 2 I I s tate of the 7200 c m - 1 s y s tem tha t the ass ignment of CT27T for i t might seem c lear -cut . O n the other hand , the hyper f ine s t ructure poss ib ly argues against this. W e do not expect large hyper f ine effects i n the a28 2A state because there is no unpa i red e lect ron in the a mo lecu la r o r b i t a l wh ich (being der ived f r o m the 4s(V) a tomic o rb i t a l ) is k n o w n to p roduce a large Fe rm i contact pa rameter [41]. E x p e r i m e n t a l l y we find a def in i te b roaden ing of the low J l ines of the 2 n 3 / 2 — 2 A 5 / 2 ( 0 , 0 ) sub-band at 7157 c m - 1 . A s s u m i n g t ha t the terms a I-L and cIz-Sz i n the hyperf ine H a m i l t o n i a n (see E q . (1.9) and E q . (1.10)) are negl ig ib le (which cou ld i n fact be a weak l ink in the a rgument ) , this impl ies that an unpa i r ed a e lectron may be present, in the 2 n state. A t present, we def in i te ly favor the sp in -orb i t evidence but must reserve final judgement on the conf igurat ions u n t i l the hyper f ine s t ruc tu re has been ana lyzed i n deta i l f r om sub-Dopp le r exper iments . Bibliography [1] A . S-C. Cheung , R. C. Hansen and A . J . Merer , J. Mol. Spectrosco., 91, 165-208 (1982). [2] A . S-C. C h e u n g and A . J . Merer , Mol. Phys., 46, 111-128 (1982). [3] R. A . Frosch and H. M . Foley, Phys. Rev., 88, 1337 (1952). [4] J . M . B r o w n , I. K o p p , C. M a l m b e r g and B. R y d h , Phys. Scr., 17, 55 (1978). [5] A . R. E d m o n d s , Angular Momentum, in Quantum Mechanics, P r i n c e t o n Un i ve r s i t y Press , P r i n c e t o n (1974). [6] H . Le febv re -B r i on and R. W . F i e l d , Perturbation in The Spectra of Diatomic Molecules, A c a d e m i c Press, New York , 1972. [7] G . Herzbe rg , Spectra of Diatomic Molecules, E d . 2, V a n No s t r and Co . Inc., N e w Yo r k , 1950. [8] J . T . Hougen, The Calculation of Rotational Energy Levels and Line Intensities in Diatomic Molecules, N a t i o n a l B u r e a u of Standerds M o n o g r a p h 115, 1970. [9] E. H i l l and J . H. V a n V l e ck , Phys. Rev., 32, 250 (1928). [10] A . J . Mere r , U n p u b l i s h e d Notes. [11] J . M . B r o w n , A . S-C. Cheung and A . J . Merer , J. Mol. Spectrosco., 124, 1987. [12] J . M . B r o w n and A . J . Merer , J. Mol. Spectrosco., 74,488-494 (1979). 77 Bibliography 78 [13] J . M . B r o w n et a l . , J. Mol. Spectrosco., 55, 500-503 (1975). [14] A . J. Me re r , Notes from Chem.420 Course (Unpub l i s hed ) , U B C , 1986. [15] J . I. S te in fe ld , Molecules and Radiation: An Introduction to Modern Molecular Spec-troscopy, T h e M I T Press, Mass., 1978. [16] G . He r zbe rg , Infrared, and Raman Spectra of Polyatomic Molecules, E d . 2, V a n Nos-t r a n d Co . Inc., New York , 1950. [17] H. H o n l and F. L o n d o n , Z. Physik, 33, 803 (1925). [18] T . M . D u n n and K. M . Rao , Nature, 222, 266 (1969). [19] J . -L . Femenias , C. A t h e n o u r and T . M . D u n n , J. Chem. Phys., 63, 286 (1975). [20] J . -L . Femenias , T . M . D u n n and G . Cheva l , S u b m i t t e d to J. Mol. Spectrosco., A c -cepted. [21] N. L. R a n i e r i , Ph.D. dissertation, U. of M i c h i g a n , Diss. Ab s t . Int. B., 40, 772 (1979). [22] E. A . P a z y u k , E. N. M o s k v i t i n a and Y u . Y a . K u z y a k o v , Spectrosc. Letter, 19, 627 (1986). [23] M . L y n e , M.Sc. Thesis, U .B .C . (1987). [24] J . Ba r r y , Ph.D. Dissertation, U .B .C . (1988). [25] W . D e m t r o d e r , Laser Spectroscopy, ( Spr inger -Ver lag , New Y o r k , 1982), Ch.6. [26] W . D e m t r o d e r , i b i d , Ch.3. [27] M . S. So rem and A. L. Schawlow, Saturation Spectroscopy in Molecular Iodine by Intermodulated Fluorescence, O p t . C o m m u n . 5, 148 (1972). Bibliography 79 [28] J . Ba r r y , i b i d , Ch.II. [29] O. A p p l e b l a d and A . Lagerqu i s t , Phys. 5cr., 10, 307 (1974). [30] P. C. M a h a n t i , Proc. Phys. Soc, 47, 433 (1935). [31] G . P. K u i p e r , W . W i l s o n and R. J . C a s h m a n , Ap. J., 106, 243-250 (1947). [32] K -P . H u b e r and G . Herzberg , Constants of Diatomic Molecules, V a n No s t r and -R e i n h o l d , N e w Yo rk , 1979. [33] A . S-C. C h e u n g , R. C. Hansen, and A . J . Mere r , J. Mol. Spectrosc, 91, 165-208, (1982). [34] D. R i cha rd s and R. F. B a r r o w , Nature ( L ondon ) , 217, 842 (1968). [35] L. Veseth, Phys. Scr., 12, 125-128 (1975). [36] A . S-C. Cheung , A . W . Tay lo r , and A . J . Merer , J. Mol. Spectrosc, 92, 391-409, (1982). [37] A . S-C. C h e u n g , PhD. Dissertation, U.B.C. (1981). [38] J . B r o m n and A . J . Merer , J. Mol. Spectrosc, 74, 488-494 (1979). [39] A . J . M e r e r et a l , J. Mol. Spectrosc, 125, 465-503 (1987). [40] R. S. M u l l i k e n , Phys. Rev., 33, 730-747 (1929). [41] P. H. K a s a i , J. Chem. Phys., 49, 4979-4984 (1968). A P P E N D I X Assigned lines of the 3II — 3 A transition 81 APPENDIX I. Transitions of the 3II - 3 A System of NbN (Doppler-limited resolution) 3JT0(e) - 3A> (o,o) J R Q P 4 17898.2378 5 17897.2071 6 17909.0708 17896.1687 7 17910.0186 17895.1264 8 17910.9489 17894.0769 9 17911.8798 17901.9518 17893.0145 10 17912.8022 17901.8787 17891.9500 11 17913.7109 17901.7976 17890.8783 12 17914.6136 17901.7103 17889.7996 13 17915.5090 17901.6145 17888.7118 14 17901.5115 17887.6177 15 17917.2801 17901.4005 17886.5158 16 17918.1535 17901.2838 17885.4057 17 17919.0201 17901.1586 17884.2861 18 17919.8774 17901.0253 17883.1584 19 17920.7277 17900.8856 20 17921.5708 17900.7350 21 17922.4062 17900.5820 22 17923.2309 17900.4193 17878.5963 23 17924.0489 17900.2469 17877.4352 24 17924.8620 17900.0712 17876.2660 25 17925.6553? 17899.8841 26 17926.4639 17899.6882 17873.9084 27 17927.2460 17899.4890 17872.7194 28 17928.0244 17899.2776 17871.5230 29 17928.7977 17899.0630 17870.3160 30 17929.5598 17898.8387 17869.1038 31 17930.3138 17898.6182? 17867.8896 32 17931.0612 17898.3656 17866.6582 33 17931.7994 17898.1178 17865.4233 34 17932.5286 17897.8607 17864.1835 35 17933.2482 17897.5961 17862.9338 36 17933.9267? 17897.3231 17861.7131? 37 17934.7048? 17897.0435 17860.4103 38 17935.3595 17896.7570 17859.1365 39 17936.0462 17857.8517 40 17856.5633 82 I, Continued. 3 i T 0 ( / ) - 3 A (o,o) J R Q P 5 17914.7448 6 17915.6917 17902.7828 8 17917.5697 17900.6890 9 17918.4984 17908.5692 17899.6306 10 17919.4196 17908.4981 17898.5678 11 17920.3337 17908.4194 17897.4959 12 17921.2387 17908.3335 17896.4180 13 17922.1376 17908.2400 17895.3321 14 17923.0282 17908.1395 17894.2406 15 17923.9107 17908.0320 17893.1383 16 17924.7895 17907.9161 17892.0314 17 17925.6553 17907.7936 17890.9202 18 17926.5201 17907.6635 17889.7996 19 17927.3701 17907.5271 17888.6726 20 17928.2170 17907.3824 18887.5374 21 17929.0558 17907.2296 17886.3937 22 17929.8862 17907.0711 17885.2438 23 17930.7099 17906.9047 17884.0836 24 17931.5241 17906.7301 25 17932.3330 17906.5476 26 17933.1322 17906.3588 27 17933.9267 17906.1620 17879.3889 28 17934.7048 17905.9560 17878.1933 29 17935.4824 17905.7468 17876.9934 30 17936.2495 17905.5219 17875.7848 31 17937.0079 17905.2961 32 17937.7594 17905.0583 17873.3534 33 17938.5060 17872.1197 34 17939.2420 17904.5682 17870.8789 35 17939.9672 17904.3093 17869.6365 36 17940.6837 17904.0497 17868.3851 38 17903.4848 17865.8635 39 17903.1927 17864.5870 41 17862.0155 83 I, Continued. SJT 3 - *A3 (o, o) J R Q P 20 17392.9455 21 17391.6844 22 17390.3991 23 17435.7182 17389.1295 24 17436.3996 17387.8313 25 17437.0712 17386.5243 26 17437.7260 17410.9716 17385.2025 27 17438.3683 17410.6250 17383.8702 28 17438.9962 17410.2633 17382.5248 29 17439.6123 17409.8970 17381.1679 30 17440.2142 17409.5122 17379.7972 31 17440.8035 17409.1153 17378.4112 32 17441.3779 17408.7048 17377.0130 33 17441.9403 17408.2812 17375.6062 34 17442.4880 17407.8432 17374.1808 35 17443.0227 17407.3933 17372.7446 36 17443.5439 17406.9259 17371.2979 37 17444.0517 17406.4534 17369.8388 38 17444.5448 17405.9654 17368.3642 39 17445.0248 17405.4613 17366.8799 40 17445.4915 17404.9438 17365.3772 41 17445.9450 17404.4147 17363.8695 42 17446.3812 17403.8733 17362.3428 43 17446.8061 17403.3148 17360.7966 44 17447.2177 17402.7429 17359.2517 45 17447.6122 17402.1592 17357.6893 46 17447.9989 17401.5638 17356.1116 47 17448.3636 17400.9533 17354.5251 48 17448.7165 49 17449.0500 50 17449.3879 84 APPENDIX I, Continued. 3JTa - » 4 3 ( i , i ) J R Q P 20 17343.0694 21 17341.8413 22 17340.5238 23 17385.4110 17339.1886 24 17386.0302 17361.4452 17337.8415 25 17386.5780 17361.0650 17336.4763 26 17387.2120 17360.6731 17335.0994 27 17387.7783 17360.2565 17333.7078 28 17388.3339 17359.8351 17332.2979 29 17388.8737 17359.3873 17330.8776 30 17389.3970 17358.9298 17329.4418 31 17389.9063 17358.4604 17327.9912 32 17390.4297 17357.9756 17326.5238 33 17390.8772 17357.4747 17325.0413 34 17391.3412 17356.9576 17323.5476 35 17391.7878 17356.4271 17322.0409 36 17392.2230 17355.8830 17320.5168 37 17392.6413 17355.3254 17318.9798 38 17393.0412 17354.7500 17317.4280 39 17393.4272 17354.1603 17315.8616 40 17353.5554 17314.2822 41 17352.9512 17312.6867 42 17352.3020 43 17351.6526 44 17350.9838 45 17350.3038 46 17349.6145 47 17348.9148 85 A P P E N D I X II. Transitions of the SJT - 3 A System of N b N (sub-Doppler resolution) »270(e) - 3 A ( 0 , 0 ) R Q P J" p» 1 3.5 rR 1 5.5 rR 2 2.5 rR 2 3.5 rR 2 4.5 rR 2 5.5 rR 2 6.5 rR 3 5.5 rR 3 6.5 rR 3 7.5 rR 4 0.5 rR 4 1.5 rR 4 2.5 rR 4 3.5 rR 4 4.5 rR 4 5.5 rR 4 6.5 rR 4 7.5 rR 5 0.5 rR 5 1.5 rR 5 2.5 rR 5 3.5 rR 5 4.5 rR 5 5.5 rR 5 6.5 rR 5 8.5 rR 5 9.5 rR 6 1.5 rR 6 2.5 rR 6 3.5 rR 6 4.5 rR 6 5.5 rR 6 6.5 rR 6 7.5 rR 6 8.5 rR 6 9.5 rR freq.(cm 1) 17904.0959 17904.4059 17905.1299 17905.1614 17905.2051 17905.2618 17905.3330 17906.2047 17906.2381 17906.2787 17907.1114 17907.1157 17907.1219 17907.1315 17907.1435 17907.1589 17907.1777 17907.1995 17908.0810 17908.0834 17908.0879 17908.0933 17908.1011 17908.1100 17908.1220 17908.1521 17908.1708 17909.0422 17909.0447 17909.0490 17909.0540 17909.0608 17909.0687 17909.0776 17909.0878 17909.0998 J " 1 4.5 qQ 1 5.5 qQ 2 2.5 qQ 2 3.5 qQ 2 4.5 qQ 2 5.5 qQ 2 6.5 qQ 3 3.5 qQ 3 4.5 qQ 3 5.5 qQ 3 6.5 qQ 3 7.5 qQ 4 4.5 qQ 4 5.5 qQ 4 6.5 qQ 4 7.5 qQ 4 8.5 qQ 5 5.5 qQ 5 6.5 qQ 5 7.5 qQ 5 8.5 qQ 5 9.5 qQ 6 5.5 qQ 6 6.5 qQ 6 7.5 qQ 6 8.5 qQ 6 9.5 qQ 6 10.5 qQ 7 2.5 qQ 7 4.5 qQ 7 5.5 qQ 7 6.5 qQ 7 7.5 qQ 7 8.5 qQ 7 9.5 qQ 7 10.5 qQ freq.(cm 1 ) 17902.2463 17902.4191 17902.1451 17902.1727 17902.2200 17902.2776 17902.3542 17902.1788 17902.1998 17902.2266 17902.2602 17902.3010 17902.1727 17902.1885 17902.2072 17902.2289 17902.2556 17902.1475 17902.1593 17902.1727 17902.1885 17902.2072 17902.1019 17902.1094 17902.1188 17902.1293 17902.1416 17902.1554 17902.0411 17902.0475 17902.0521 17902.0573 17902.0634 17902.0706 17902.0788 17902.0877 J " pn 1 3.5 rP 1 4.5 qP 1 5.5 PP 2 2.5 rP 2 3.5 qP 2 4.5 PP 2 5.5 pP 2 6.5 pP 3 1.5 rP 3 2.5 qP 3 3.5 PP 3 4.5 PP 3 5.5 PP 3 6.5 PP 3 7.5 pP 4 0.5 rP 4 1.5 qP 4 2.5 PP 4 3.5 PP 4 4.5 PP 4 5.5 pP 4 6.5 PP 4 7.5 pP 4 8.5 PP 5 0.5 qP 5 1.5 PP 5 2.5 pP 5 3.5 PP 5 4.5 PP 5 5.5 PP 5 6.5 pP 5 7.5 PP 5 8.5 pP 5 9.5 PP 6 4.5 PP 6 5.5 pP freq. (cm 1) 17901.1178 17901.2591 17901.4261 17900.1646 17900.1968 17900.2388 17900.2981 17900.3698 17899.1788 17899.1892 17899.2057 17899.2275 17899.2545 17899.2877 17899.3283 17898.1755 17898.1792 17898.1857 17898.1948 17898.2073 17898.2230 17898.2416 17898.2649 17898.2909 17897.1581 17897.1614 17897.1650 17897.1711 17897.1783 17897.1886 17897.1994 17897.2139 17897.2300 17897.2488 17896.1442 17896.1505 86 APPENDIX H, Continued. 3iTo(e) - (o, o) R Q P J" F" 6 10.5 rR freq.(cm 1) 17909.1127 J" pn 7 11.5 qQ 8 3.5 qQ 8 4.5 qQ 8 5.5 qQ 8 6.5 qQ 8 7.5 qQ 8 8.5 qQ 8 9.5 qQ 8 10.5 qQ 8 11.5 qQ 8 12.5 qQ 9 5.5 qQ 9 6.5 qQ 9 7.5 qQ 9 8.5 qQ 9 9.5 qQ 9 10.5 qQ 9 11.5 qQ 9 12.5 qQ 9 13.5 qQ 10 6.5 qQ 10 7.5 qQ 10 8.5 qQ 10 9.5 qQ 10 10.5 qQ 10 11.5 qQ 10 12.5 qQ 10 13.5 qQ 10 14.5 qQ 11 10.5 qQ 11 11.5 qQ 11 12.5 qQ 11 13.5 qQ 11 14.5 qQ 11 15.5 qQ 17 17.5 qQ 17 18.5 qQ freq. (cm 1) 17902.0974 17901.9884 17901.9911 17901.9941 17901.9977 17902.0020 17902.0066 17902.0123 17902.0186 17902.0255 17902.0330 17901.9346 17901.9372 17901.9351 17901.9384 17901.9426 17901.9466 17901.9516 17901.9567 17901.9626 17901.8653 17901.8674 17901.8699 17901.8720 17901.8752 17901.8784 17901.8815 17901.8856 17901.8901 17901.7913 17901.7940 17901.7971 17901.7999 17901.8028 17901.8060 17901.1583 17901.1568 J" jp" 6 6.5 pP 6 7.5 pP 6 8.5 PP 6 9.5 PP 6 10.5 pP 7 2.5 pP 7 3.5 pP 7 4.5 PP 7 5.5 pP 7 6.5 pP 7 7.5 pP 7 8.5 pP 7 9.5 PP 7 10.5 PP 7 11.5 PP freq.(cm :) 17896.1579 17896.1667 17896.1775 17896.1901 17896.2037 17895.0962 17895.0992 17895.1025 17895.1066 17895.1118 17895.1180 17895.1250 17895.1335 17895.1431 17895.1535 APPENDIX H, Continued. 3iT0(e) - *A x ( 0 , 0 ) R Q J " F" freq.(cm 1 ) 17 19.5 qQ 17901.1550 17 20.5 qQ 17901.1532 17 21.5 qQ 17901.1514 18 13.5 qQ 17901.0350 18 14.5 qQ 17901.0332 18 15.5 qQ 17901.0317 18 16.5 qQ 17901.0295 18 17.5 qQ 17901.0273 18 18.5 qQ 17901.0254 18 19.5 qQ 17901.0232 18 20.5 qQ 17901.0210 18 21.5 qQ 17901.0188 18 22.5 qQ 17901.0162 19 14.5 qQ 17900.8971 19 15.5 qQ 17900.8948 19 16.5 qQ 17900.8925 19 17.5 qQ 17900.8901 19 18.5 qQ 17900.8877 19 19.5 qQ 17900.8846 19 20.5 qQ 17900.8821 19 21.5 qQ 17900.8796 19 22.5 qQ 17900.8769 19 23.5 qQ 17900.8741 20 15.5 qQ 17900.7513 20 16.5 qQ 17900.7490 20 17.5 qQ 17900.7463 20 18.5 qQ 17900.7433 20 19.5 qQ 17900.7401 20 20.5 qQ 17900.7370 20 21.5 qQ 17900.7339 20 22.5 qQ 17900.7308 20 23.5 qQ 17900.7269 20 24.5 qQ 17900.7238 21 16.5 qQ 17900.5978 21 17.5 qQ 17900.5948 21 18.5 qQ 17900.5917 21 19.5 qQ 17900.5886 APPENDIX H, Continued. 3JTc(e) - SAX (o, o) R Q J" pn freq.(cm 1) 21 20.5 qQ 17900.5851 21 21.5 qQ 17900.5817 21 22.5 qQ 17900.5779 21 23.5 qQ 17900.5745 21 24.5 qQ 17900.5705 21 25.5 qQ 17900.5666 22 17.5 qQ 17900.4370 22 18.5 qQ 17900.4338 22 19.5 qQ 17900.4302 22 20.5 qQ 17900.4263 22 21.5 qQ 17900.4223 22 22.5 qQ 17900.4186 22 23.5 qQ 17900.4143 22 24.5 qQ 17900.4103 22 25.5 qQ 17900.4060 22 26.5 qQ 17900.4018 23 18.5 qQ 17900.2680 23 19.5 qQ 17900.2643 23 20.5 qQ 17900.2606 23 21.5 qQ 17900.2565 23 22.5 qQ 17900.2521 23 23.5 qQ 17900.2476 23 24.5 qQ 17900.2432 23 25.5 qQ 17900.2388 23 26.5 qQ 17900.2340 2 J 27.5 qQ 17900.2292 24 19.5 qQ 17900.0912 24 20.5 qQ 17900.0870 24 21.5 qQ 17900.0826 24 22.5 qQ 17900.0785 24 23.5 qQ 17900.0738 24 24.5 qQ 17900.0691 24 25.5 qQ 17900.0639 24 26.5 qQ 17900.0591 24 27.5 qQ 17900.0540 24 28.5 qQ 17900.0485 25 20.5 qQ 17899.9073 A P P E N D I X LT, C o n t i n u e d . 3 2T 0(e) - *AX ( o , o ) R Q J" f req. (cm *) 25 21.5 q Q 17899.9030 25 22.5 q Q 17899.8983 25 23.5 q Q 17899.8932 25 24.5 q Q 17899.8881 25 25.5 q Q 17899.8831 25 26.5 q Q 17899.8779 25 27.5 q Q 17899.8723 25 28.5 q Q 17899.8668 25 29.5 q Q 17899.8611 26 21.5 q Q 17899.7157 26 22.5 q Q 17899.7108 26 23.5 q Q 17899.7057 26 24.5 q Q 17899.7002 26 25.5 q Q 17899.6951 26 26.5 q Q 17899.6893 26 27.5 q Q 17899.6837 26 28.5 q Q 17899.6776 26 29.5 q Q 17899.6721 26 30.5 q Q 17899.6659 27 22.5 q Q 17899.5163 27 23.5 q Q 17899.5111 27 24.5 q Q 17899.5054 27 25.5 q Q 17899.5002 27 26.5 q Q 17899.4943 27 27.5 q Q 17899.4882 27 28.5 q Q 17899.4819 27 29.5 q Q 17899.4758 27 30.5 q Q 17899.4696 27 31.5 q Q 17899.4628 28 23.5 q Q 17899.3093 28 24.5 q Q 17899.3033 28 25.5 q Q 17899.2977 28 26.5 q Q 17899.2917 28 27.5 q Q 17899.2851 28 28.5 q Q 17899.2788 2 8 . 29.5 q Q 17899.2724 28 30.5 q Q 17899.2659 APPENDIX H, C o n t i n u e d . 3 iT 0 (e) R J" 28 31.5 28 32.5 29 24.5 29 25.5 29 26.5 29 27.5 29 28.5 29 29.5 29 30.5 29 31.5 29 32.5 29 33.5 30 25.5 30 26.5 30 27.5 30 28.5 30 29.5 30 30.5 30 31.5 30 32.5 30 33.5 30 34.5 31 26.5 31 27.5 31 28.5 31 29.5 31 30.5 31 31.5 31 32.5 31 33.5 31 34.5 31 35.5 32 27.5 32 28.5 32 29.5 32 30.5 32 31.5 3 A ( o , o ) Q freq.(cm l ) qQ 17899.2593 qQ 17899.2522 qQ 17899.0941 qQ 17899.0880 qQ 17899.0822 qQ 17899.0758 qQ 17899.0692 qQ 17899.0627 qQ 17899.0557 qQ 17899.0485 qQ 17899.0413 qQ 17899.0337 qQ 17898.8712 qQ 17898.8651 qQ 17898.8586 qQ 17898.8518 qQ 17898.8446 qQ 17898.8378 qQ 17898.8307 qQ 17898.8231 qQ 17898.8156 qQ 17898.8077 qQ 17898.6402 qQ 17898.6341 qQ 17898.6273 qQ 17898.6196 qQ 17898.6124 qQ 17898.6053 qQ 17898.5977 qQ 17898.5893 qQ 17898.5814 qQ 17898.5733 qQ 17898.4022 qQ 17898.3950 qQ 17898.3879 qQ 17898.3807 qQ 17898.3729 A P P E N D I X I I , C o n t i n u e d . 3 U 0 ( e ) - 3 A ( o , o ) R Q J" F" freq.(cm ' ) 32 32.5 q Q 17898.3647 32 33.5 q Q 17898.3568 32 34.5 q Q 17898.3487 32 35.5 q Q 17898.3403 32 36.5 q Q 17898.3320 33 28.5 q Q 17898.1552 33 29.5 q Q 17898.1476 33 30.5 q Q 17898.1402 33 31.5 q Q 17898.1326 33 32.5 q Q 17898.1248 33 33.5 q Q 17898.1164 33 34.5 q Q 17898.1078 33 35.5 q Q 17898.0995 33 36.5 q Q 17898.0906 33 37.5 q Q 17898.0816 34 29.5 q Q 17897.8998 34 30.5 q Q 17897.8924 34 31.5 q Q 17897.8846 34 32.5 q Q 17897.8764 34 33.5 q Q 17897.8679 34 34.5 q Q 17897.8597 34 35.5 q Q 17897.8507 34 36.5 q Q 17897.8417 34 37.5 q Q 17897.8325 34 38.5 q Q 17897.8232 35 30.5 q Q 17897.6367 35 31.5 q Q 17897.6283 35 32.5 q Q 17897.6206 35 33.5 q Q 17897.6120 35 34.5 q Q 17897.6032 35 35.5 q Q 17897.5948 35 36.5 q Q 17897.5854 35 37.5 q Q 17897.5759 35 38.5 q Q 17897.5669 35 39.5 q Q 17897.5570 36 31.5 q Q 17897.3659 36 32.5 q Q 17897.3574 APPENDIX H, Continued. 32T0(e) - 3 A (o, o) R Q J" freq.(cm *) 36 33.5 qQ 17897.3492 36 34.5 qQ 17897.3401 36 35.5 qQ 17897.3313 36 36.5 qQ 17897.3218 36 37.5 qQ 17897.3126 36 38.5 qQ 17897.3028 36 39.5 qQ 17897.2933 36 40.5 qQ 17897.2833 37 32.5 qQ 17897.0870 37 33.5 qQ 17897.0785 37 34.5 qQ 17897.0693 37 35.5 qQ 17897.0597 37 36.5 qQ 17897.0506 37 37.5 qQ 17897.0413 37 38.5 qQ 17897.0314 37 39.5 qQ 17897.0222 37 40.5 qQ 17897.0119 37 41.5 qQ 17897.0011 38 33.5 qQ 17896.8005 38 34.5 qQ 17896.7912 38 35.5 qQ 17896.7820 38 36.5 qQ 17896.7729 38 37.5 qQ 17896.7634 38 38.5 qQ 17896.7534 38 39.5 qQ 17896.7431 38 40.5 qQ 17896.7328 38 41.5 qQ 17896.7226 38 42.5 qQ 17896.7115 39 34.5 qQ 17896.5045 39 35.5 qQ 17896.4954 39 36.5 qQ 17896.4857 39 37.5 qQ 17896.4761 39 38.5 qQ 17896.4658 39 39.5 qQ 17896.4554 39 40.5 qQ 17896.4451 39 41.5 qQ 17896.4349 40 41.5 qQ 17896.1389 APPENDIX II, Continued. 3il0(e) - 3 A x (0,0) R Q J" pn freq. (cm *) 40 42.5 qQ 17896.1286 40 43.5 qQ 17896.1172 40 44.5 qQ 17896.1062 41 36.5 qQ 17895.8893 41 37.5 qQ 17895.8794 41 38.5 qQ 17895.8691 41 39.5 qQ 17895.8588 41 40.5 qQ 17895.8481 41 41.5 qQ 17895.8373 41 42.5 qQ 17895.8261 41 43.5 qQ 17895.8149 41 44.5 qQ 17895.8037 41 45.5 qQ 17895.7919 42 37.5 qQ 17895.5700 42 38.5 qQ 17895.5595 42 39.5 qQ 17895.5486 42 40.5 qQ 17895.5381 42 41.5 qQ 17895.5271 42 42.5 qQ 17895.5157 42 43.5 qQ 17895.5049 42 44.5 qQ 17895.4933 42 45.5 qQ 17895.4808 42 46.5 qQ 17895.4695 43 38.5 qQ 17895.2414 43 39.5 qQ 17895.2305 43 40.5 qQ 17895.2198 43 41.5 qQ 17895.2082 43 42.5 qQ 17895.1977 43 43.5 qQ 17895.1867 43 44.5 qQ 17895.1745 43 45.5 qQ 17895.1616 43 46.5 qQ 17895.1490 43 47.5 qQ 17895.1376 44 39.5 qQ 17894.9041 44 40.5 qQ 17894.8928 44 41.5 qQ 17894.8831 94 A P P E N D I X H , Continued. 3 iT D (e) - SA, (0,0) R Q J" F" freq.^m- 1) 44 42.5 qQ 17894.8709 44 43.5 qQ 17894.8590 44 44.5 qQ 17894.8474 44 45.5 qQ 17894.8355 44 46.5 qQ 17894.8224 44 47.5 qQ 17894.8105 44 48.5 qQ 17894.7972 * i T 0 ( / ) - 3 A (0,0) R Q p J freq.(cm - 1) J 11 pn freq.(cm - 1) 1 3.5 qQ 17908.7247 1 3.5 rP 17907.7272 1 4.5 qQ 17908.8635 1 4.5 qP 17907.8696 1 5.5 qQ 17909.0314 1 5.5 PP 17908.0375 2 3.5 qQ 17908.7920 2 2.5 rP 17906.7749 2 4.5 qQ 17908.8370 2 3.5 qP 17906.8071 2 5.5 qQ 17908.8949 2 4.5 pP 17906.8506 2 6.5 qQ 17908.9694 2 5.5 pP 17906.9073 3 1.5 qQ 17908.7717 2 6.5 PP 17906.9792 3 2.5 qQ 17908.7830 3 1.5 rP 17905.7886 3 3.5 qQ 17908.7991 3 2.5 qP 17905.8004 3 4.5 qQ 17908.8177 3 3.5 PP 17905.8168 3 5.5 qQ 17908.8442 3 4.5 pP 17905.8376 3 6.5 qQ 17908.8781 3 5.5 pP 17905.8649 3 7.5 qQ 17908.9185 3 6.5 PP 17905.8984 4 1.5 qQ 17908.7651 3 7.5 pP 17905.9406 4 2.5 qQ 17908.7691 4 2.5 PP 17904.7960 4 3.5 qQ 17908.7784 4 3.5 PP 17904.8053 4 4.5 qQ 17908.7920 4 4.5 PP 17904.8172 4 5.5 qQ 17908.8067 4 5.5 pP 17904.8334 4 6.5 qQ 17908.8253 4 6.5 PP 17904.8513 4 7.5 qQ 17908.8468 4 7.5 PP 17904.8737 4 8.5 qQ 17908.8734 4 8.5 pP 17904.9008 5 3.5 qQ 17908.7498 5 2.5 PP 17903.7737 5 4.5 qQ 17908.7569 5 3.5 pP 17903.7835 5 5.5 qQ 17908.7665 5 4.5 pP. 17903.7915 95 APPENDIX H, Continued. 3iT0(/) - »A, (o, o) R Q J II pa freq.(cm 1) 5 6.5 qQ 17908.7784 5 7.5 qQ 17908.7920 5 8.5 qQ 17908.8081 5 9.5 qQ 17908.8263 6 3.5 qQ 17908.7114 6 4.5 qQ 17908.7158 6 5.5 qQ 17908.7224 6 6.5 qQ 17908.7299 6 7.5 qQ 17908.7390 6 8.5 qQ 17908.7498 6 9.5 qQ 17908.7618 6 10.5 qQ 17908.7758 7 2.5 qQ 17908.6621 7 3.5 qQ 17908.6645 7 4.5 qQ 17908.6682 7 5.5 qQ 17908.6726 7 6.5 qQ 17908.6779 7 7.5 qQ 17908.6840 7 8.5 qQ 17908.6911 7 9.5 qQ 17908.6992 7 10.5 qQ 17908.7079 7 11.5 qQ 17908.7178 8 3.5 qQ 17908.6116 8 4.5 qQ 17908.6138 8 5.5 qQ 17908.6166 8 6.5 qQ 17908.6201 8 7.5 qQ 17908.6241 8 8.5 qQ 17908.6287 8 9.5 qQ 17908.6342 8 10.5 qQ 17908.6404 8 11.5 qQ 17908.6475 8 12.5 qQ 17908.6552 9 4.5 qQ 17908.5524 9 5.5 qQ 17908.5543 9 6.5 qQ 17908.5565 9 7.5 qQ 17908.5596 9 8.5 qQ 17908.5626 p J freq.(cm J) 5 5.5 PP 17903.8008 5 6.5 PP 17903.8127 5 7.5 pP 17903.8266 5 8.5 pP 17903.8426 5 9.5 pP 17903.8614 6 2.5 PP 17902.7465 6 3.5 pP 17902.7493 6 4.5 pP 17902.7536 6 5.5 pP 17902.7649 6 6.5 PP 17902.7726 6 7.5 pP 17902.7819 6 8.5 pP 17902.7923 6 9.5 pP 17902.8041 6 10.5 pP 17902.8177 APPENDIX II, Continued. 3i7 0(/) - 3 A (o, o) R Q J" pi, 9 9.5 qQ 9 10.5 qQ 9 11.5 qQ 9 12.5 qQ 9 13.5 qQ 10 7.5 qQ 10 8.5 qQ 10 9.5 qQ 10 10.5 qQ 10 11.5 qQ 10 12.5 qQ 10 13.5 qQ 10 14.5 qQ 11 12.5 qQ 11 13.5 qQ 11 14.5 qQ 11 15.5 qQ 18 13.5 qQ 18 14.5 qQ 18 15.5 qQ 18 16.5 qQ 18 17.5 qQ 18 18.5 qQ 18 19.5 qQ 18 20.5 qQ 18 21.5 qQ 18 22.5 qQ 19 14.5 qQ 19 15.5 qQ 19 16.5 qQ 19 17.5 qQ 19 18.5 qQ 19 19.5 qQ 19 20.5 qQ 19 21.5 qQ 19 22.5 qQ 19 23.5 qQ freq. (cm *) 17908.5663 17908.5703 17908.5747 17908.5798 17908.5855 17908.4877 17908.4899 17908.4922 17908.4948 17908.4980 17908.5016 17908.5057 17908.5098 17908.4186 17908.4212 17908.4238 17908.4267 17907.6734 17907.6715 17907.6696 17907.6674 17907.6649 17907.6627 17907.6605 17907.6579 17907.6554 17907.6528 17907.5388 17907.5364 17907.5338 17907.5311 17907.5285 17907.5255 17907.5225 17907.5196 17907.5167 17907.5137 APPENDIX n, Continued. 3JTD(/) - 3 A , (0,0) R Q J" p" freq.(cm *) 20 15.5 qQ 17907.3965 20 16.5 qQ 17907.3937 20 17.5 qQ 17907.3908 20 18.5 qQ 17907.3876 20 19.5 qQ 17907.3843 20 20.5 qQ 17907.3807 20 21.5 qQ 17907.3772 20 22.5 qQ 17907.3736 20 23.5 qQ 17907.3703 20 24.5 qQ 17907.3666 21 16.5 qQ 17907.2464 21 17.5 qQ 17907.2428 21 18.5 qQ 17907.2395 21 19.5 qQ 17907.2360 21 20.5 qQ 17907.2323 21 21.5 qQ 17907.2283 21 22.5 qQ 17907.2251 21 23.5 qQ 17907.2206 21 24.5 qQ 17907.2168 21 25.5 qQ 17907.2126 22 17.5 qQ 17907.0885 22 18.5 qQ 17907.0848 22 19.5 qQ 17907.0812 22 20.5 qQ 17907.0773 22 21.5 qQ 17907.0734 22 22.5 qQ 17907.0691 22 23.5 qQ 17907.0651 22 24.5 qQ 17907.0605 22 25.5 qQ 17907.0562 22 26.5 qQ 17907.0513 23 18.5 qQ 17906.9243 23 19.5 qQ 17906.9205 23 20.5 qQ 17906.9160 23 21.5 qQ 17906.9114 23 22.5 qQ 17906.9073 23 23.5 qQ 17906.9023 23 24.5 qQ 17906.8973 APPENDIX H, Continued. 3JT0(/) - 8 A ( 0 , 0 ) R Q J" freq.(cm 1) 23 25.5 qQ 17906.8926 23 26.5 qQ 17906.8875 23 27.5 qQ 17906.8824 24 19.5 qQ 17906.7516 24 20.5 qQ 17906.7472 24 21.5 qQ 17906.7429 24 22.5 qQ 17906.7382 24 23.5 qQ 17906.7334 24 24.5 qQ 17906.7281 24 25.5 qQ 17906.7228 24 26.5 qQ 17906.7173 24 27.5 qQ 17906.7116 24 28.5 qQ 17906.7056 25 20.5 qQ 17906.5716 25 21.5 qQ 17906.5670 25 22.5 qQ 17906.5619 25 23.5 qQ 17906.5565 25 24.5 qQ 17906.5515 25 25.5 qQ 17906.5460 25 26.5 qQ 17906.5403 25 27.5 qQ 17906.5343 25 28.5 qQ 17906.5284 25 29.5 qQ 17906.5225 26 21.5 qQ 17906.3842 26 22.5 qQ 17906.3790 26 23.5 qQ 17906.3735 26 24.5 qQ 17906.3680 26 25.5 qQ 17906.3620 26 26.5 qQ 17906.3560 26 27.5 qQ 17906.3500 26 28.5 qQ 17906.3443 26 29.5 qQ 17906.3374 26 30.5 qQ 17906.3314 27 22.5 qQ 17906.1832 27 23.5 qQ 17906.1776 27 24.5 qQ 17906.1719 A P P E N D I X . H , C o n t i n u e d . 3 J I 0 ( / ) - 3 A, ( o , o ) R Q J" j?» f req. (cm 1 ) 27 25.5 q Q 17906.1658 27 26.5 q Q 17906.1566 27 27.5 q Q 17906.1595 27 28.5 q Q 17906.1529 27 29.5 q Q 17906.1457 27 30.5 q Q 17906.1394 27 31.5 q Q 17906.1322 28 23.5 q Q 17905.9868 28 24.5 q Q 17905.9810 28 25.5 q Q 17905.9748 28 26.5 q Q 17905.9683 28 27.5 q Q 17905.9620 28 28.5 q Q 17905.9550 28 29.5 q Q 17905.9483 28 31.5 q Q 17905.9336 28 32.5 q Q 17905.9265 29 24.5 q Q 17905.7763 29 25.5 q Q 17905.7700 29 26.5 q Q 17905.7638 29 27.5 q Q 17905.7573 29 28.5 q Q 17905.7500 29 29.5 q Q 17905.7428 29 30.5 q Q 17905.7356 29 31.5 q Q 17905.7282 29 32.5 q Q 17905.7201 29 33.5 q Q 17905.7122 30 25.5 q Q 17905.5579 30 26.5 q Q 17905.5514 30 27.5 q Q 17905.5445 30 28.5 q Q 17905.5376 30 29.5 q Q 17905.5301 30 30.5 q Q 17905.5226 30 31.5 q Q 17905.5148 30 32.5 q Q 17905.5071 30 33.5 q Q 17905.4989 30 34.5 q Q 17905.4907 APPENDIX H, Continued. 3JT0(/) - 3 A (o, o) R Q J" freq.(cm 1) 31 26.5 qQ 17905.3330 31 27.5 qQ 17905.3251 31 28.5 qQ 17905.3180 31 29.5 qQ 17905.3106 31 30.5 qQ 17905.3029 31 31.5 qQ 17905.2946 31 32.5 qQ 17905.2870 31 33.5 qQ 17905.2787 31 34.5 qQ 17905.2700 31 35.5 qQ 17905.2618 32 27.5 qQ 17905.0988 32 28.5 qQ 17905.0916 32 29.5 qQ 17905.0842 32 30.5 qQ 17905.0762 32 31.5 qQ 17905.0681 32 32.5 qQ 17905.0600 32 33.5 qQ 17905.0515 32 34.5 qQ 17905.0429 32 35.5 qQ 17905.0339 32 36.5 qQ 17905.0244 33 28.5 qQ 17904.8572 33 29.5 qQ 17904.8499 33 30.5 qQ 17904.8417 33 31.5 qQ 17904.8334 33 32.5 qQ 17904.8254 33 33.5 qQ 17904.8172 33 34.5 qQ 17904.8079 33 35.5 qQ 17904.7989 33 36.5 qQ 17904.7894 33 37.5 qQ 17904.7801 34 29.5 qQ 17904.6084 34 30.5 qQ 17904.6006 34 31.5 qQ 17904.5920 34 32.5 qQ 17904.5831 34 33.5 qQ 17904.5749 34 34.5 qQ 17904.5652 34 35.5 qQ 17904.5564 APPENDIX EE, Continued. 3 iT 0 (/) - 8 A ( 0 , 0 ) R Q J" freq.(cm *) 34 36.5 qQ 17904.5476 34 37.5 qQ 17904.5376 34 38.5 qQ 17904.5273 35 30.5 qQ 17904.3514 35 31.5 qQ 17904.3429 35 32.5 qQ 17904.3344 35 33.5 qQ 17904.3255 35 34.5 qQ 17904.3160 35 35.5 qQ 17904.3072 35 36.5 qQ 17904.2972 35 37.5 qQ 17904.2877 35 38.5 qQ 17904.2779 35 39.5 qQ 17904.2677 36 31.5 qQ 17904.0865 36 32.5 qQ 17904.0777 36 33.5 qQ 17904.0690 36 34.5 qQ 17904.0602 36 35.5 qQ 17904.0506 36 36.5 qQ 17904.0410 36 37.5 qQ 17904.0311 36 38.5 qQ 17904.0207 36 39.5 qQ 17904.0107 36 40.5 qQ 17904.0000 37 32.5 qQ 17903.8199 37 33.5 qQ 17903.8060 37 34.5 qQ 17903.7963 37 35.5 qQ 17903.7866 37 36.5 qQ 17903.7767 37 37.5 qQ 17903.7665 37 38.5 qQ 17903.7564 37 39.5 qQ 17903.7456 37 40.5 qQ 17903.7355 37 41.5 qQ 17903.7242 38 33.5 qQ 17903.5344 38 34.5 qQ 17903.5251 38 35.5 qQ 17903.5157 38 36.5 qQ 17903.5053 APPENDIX H, Continued. 32I0(/) - 8 A ( 0 , 0 ) R Q J" pn freq. (cm 1) 38 37.5 qQ 17903.4955 38 38.5 qQ 17903.4847 38 39.5 qQ 17903.4742 38 40.5 qQ 17903.4636 38 41.5 qQ 17903.4520 38 42.5 qQ 17903.4402 39 34.5 qQ 17903.2462 39 35.5 qQ 17903.2365 39 36.5 qQ 17903.2264 39 37.5 qQ 17903.2153 39 38.5 qQ 17903.2054 39 39.5 qQ 17903.1946 39 40.5 qQ 17903.1836 39 41.5 qQ 17903.1725 39 42.5 qQ 17903.1610 39 43.5 qQ 17903.1492 40 35.5 qQ 17902.9492 40 36.5 qQ 17902.9381 40 37.5 qQ 17902.9279 40 38.5 qQ 17902.9177 40 39.5 qQ 17902.9088 40 40.5 qQ 17902.8969 40 41.5 qQ 17902.8847 40 42.5 qQ 17902.8731 40 43.5 qQ 17902.8612 40 44.5 qQ 17902.8502 41 36.5 qQ 17902.6441 41 37.5 qQ 17902.6335 41 38.5 qQ 17902.6231 41 39.5 qQ 17902.6128 41 40.5 qQ 17902.6018 41 41.5 qQ 17902.5890 41 42.5 qQ 17902.5774 41 43.5 qQ 17902.5661 41 44.5 qQ 17902.5542 41 45.5 qQ 17902.5411 1 0 3 APPENDIX n, Continued. R 3Hi - 3 A , (o,o) Q J" F" freq.(cm 1) 2 2.5 rQ 1 7 0 5 7 . 5 4 2 6 2 2 .5 qQ 1 7 0 5 7 . 5 0 8 9 2 3.5 rQ 1 7 0 5 7 . 5 3 0 2 2 4 . 5 rQ 1 7 0 5 7 . 5 0 8 9 2 5.5 pQ 1 7 0 5 7 . 3 7 8 5 2 6 .5 qQ 1 7 0 5 7 . 3 7 8 5 3 2.5 rQ 1 7 0 5 7 . 4 6 0 0 3 3 .5 rQ 1 7 0 5 7 . 4 5 4 8 3 4 . 5 rQ 1 7 0 5 7 . 4 4 4 4 3 5.5 rQ 1 7 0 5 7 . 4 2 9 2 3 5.5 pQ 1 7 0 5 7 . 3 7 8 5 3 6 .5 qQ 1 7 0 5 7 . 3 8 0 7 3 6 .5 pQ 1 7 0 5 7 . 3 5 4 8 3 7.5 pQ 1 7 0 5 7 . 3 2 6 9 4 2.5 pQ 1 7 0 5 7 . 3 7 6 1 4 3 .5 qQ 1 7 0 5 7 . 3 7 6 1 4 4 . 5 qQ 1 7 0 5 7 . 3 6 8 3 4 6.5 P Q 1 7 0 5 7 . 3 3 0 1 4 7.5 qQ 1 7 0 5 7 . 3 3 0 1 4 7.5 pQ 1 7 0 5 7 . 3 1 3 4 4 8 .5 qQ 1 7 0 5 7 . 3 1 3 4 5 0 .5 rQ 1 7 0 5 7 . 3 2 1 1 5 1.5 rQ 1 7 0 5 7 . 3 2 1 1 5 1.5 P Q 1 7 0 5 7 . 3 1 3 4 5 2 .5 qQ 1 7 0 5 7 . 3 1 3 4 5 2 .5 pQ 1 7 0 5 7 . 3 0 8 9 5 3.5 qQ 1 7 0 5 7 . 3 0 8 9 5 3.5 pQ 1 7 0 5 7 . 3 0 2 8 5 4 . 5 rQ 1 7 0 5 7 . 3 1 3 4 5 4 . 5 qQ 1 7 0 5 7 . 3 0 2 8 5 4 . 5 pQ 1 7 0 5 7 . 2 9 5 0 5 5.5 qQ 1 7 0 5 7 . 2 9 5 0 5 5.5 pQ 1 7 0 5 7 . 2 8 7 6 5 6.5 qQ 1 7 0 5 7 . 2 8 7 6 5 7.5 qQ 1 7 0 5 7 . 2 7 8 5 p J freq. (cm 1) 2 3.5 rP 1 7 0 5 5 . 5 4 2 0 2 4 .5 qP 1 7 0 5 5 . 4 7 5 5 2 4 . 5 PP 1 7 0 5 5 . 3 6 5 9 2 5.5 qP 1 7 0 5 5 . 5 3 0 2 2 5.5 PP 1 7 0 5 5 . 3 9 2 6 2 6.5 pP 1 7 0 5 5 . 4 3 1 3 3 1.5 rP 1 7 0 5 4 . 4 5 7 2 3 2.5 qP 1 7 0 5 4 . 4 3 8 8 3 3.5 qP 1 7 0 5 4 . 4 4 3 9 3 4 .5 qP 1 7 0 5 4 . 4 4 8 8 3 4 .5 pP 1 7 0 5 4 . 4 0 9 8 3 5.5 PP 1 7 0 5 4 . 4 0 7 5 3 6.5 pP 1 7 0 5 4 . 4 0 4 6 3 7.5 PP 1 7 0 5 4 . 3 9 9 1 4 3.5 qP 1 7 0 5 3 . 4 0 7 9 4 4 . 5 PP 1 7 0 5 3 . 3 8 7 8 4 5.5 PP 1 7 0 5 3 . 3 8 2 9 4 6.5 pP 1 7 0 5 3 . 3 7 6 9 4 6.5 qP 4 7.5 PP 1 7 0 5 3 . 3 7 0 2 4 8 .5 PP 1 7 0 5 3 . 3 6 2 2 5 2.5 PP 1 7 0 5 2 . 3 4 6 1 5 3.5 pP 1 7 0 5 2 . 3 4 2 7 5 4 .5 pP 1 7 0 5 2 . 3 3 9 5 5 5.5 qP 1 7 0 5 2 . 3 4 8 5 5 5.5 PP 1 7 0 5 2 . 3 3 5 1 5 6.5 qP 1 7 0 5 2 . 3 4 6 1 5 6.5 pP 1 7 0 5 2 . 3 2 9 9 5 7.5 qP 1 7 0 5 2 . 3 4 2 7 5 7.5 PP 1 7 0 5 2 . 3 2 4 1 5 8.5 qP 1 7 0 5 2 . 3 3 9 5 5 8.5 PP 1 7 0 5 2 . 3 1 7 8 5 9.5 PP 1 7 0 5 2 . 3 1 0 6 6 1.5 pP 1 7 0 5 1 . 2 8 2 8 6 4 . 5 PP 1 7 0 5 1 . 2 7 6 0 104 APPENDIX H, Continued. 3II, - *Aa (o, o) R Q P J' / pi, freq. (cm 1) J " p" freq.(cm 1) 5 8.5 qQ 17057.2679 6 4.5 qP 17051.2828 5 9.5 qQ 17057.2559 6 5.5 PP 17051.2721 6 2.5 qQ 17057.2322 6 6.5 qP 17051.2796 6 2.5 pQ 17057.2286 6 6.5 PP 17051.2679 6 3.5 qQ 17057.2286 6 7.5 qP 17051.2760 6 3.5 pQ 17057.2245 6 7.5 pP 17051.2637 6 4.5 rQ 17057.2322 6 8.5 qP 17051.2721 6 4.5 qQ 17057.2245 6 8.5 PP 17051.2586 6 4.5 pQ 17057.2199 6 9.5 pP 17051.2532 6 5.5 rQ 17057.2286 6 10.5 pP 17051.2471 6 5.5 qQ 17057.2199 6 5.5 pQ 17057.2140 6 6.5 rQ 17057.2245 6 6.5 qQ 17057.2140 6 7.5 qQ 17057.2079 6 8.5 qQ 17057.2010 6 9.5 qQ 17057.1930 6 10.5 qQ 17057.1838 7 4.5 qQ 17057.1338 7 5.5 qQ 17057.1304 7 6.5 qQ 17057.1263 7 7.5 qQ 17057.1214 7 8.5 qQ 17057.1163 7 9.5 qQ 17057.1108 7 10.5 qQ 17057.1047 7 11.5 qQ 17057.0976 8 7.5 qQ 17057.0215 8 8.5 qQ 17057.0175 8 9.5 qQ 17057.0130 8 10.5 qQ 17057.0088 8 11.5 qQ 17057.0039 8 12.5 qQ 17056.9981 105 APPENDIX II, Continued. R Q P J" freq^cm-1) 4 J" F" freq.(cm-1) J freq.(cm 1 ) 3 3.5 rR 17419.6158 3 1.5 rQ 17415.7841 3 2.5 qP 17412.7182 3 3.5 qR 17419.6077 3 1.5 qQ 17415.7803 3 3.5 rP 17412.6687 3 4.5 rR 17419.5034 3 2.5 rQ 17415.7275 3 3.5 qP 17412.6432 3 4.5 qR 17419.4938 3 2.5 qQ 17415.7186 3 3.5 PP 17412.6237 3 5.5 rR 17419.3645 3 2.5 pQ 17415.7100 3 4.5 rP 17412.5781 3 6.5 rR 17419.1973 3 3.5 rQ 17415.6461 3 4.5 PP 17412.5210 3 7.5 rR 17419.0008 3 3.5 qQ 17415.6348 3 5.5 qP 17412.4271 4 2.5 rR 17420.5528 3 3.5 pQ 17415.6230 3 5.5 PP 17412.3955 4 2.5 qR 17420.5485 3 4.5 rQ 17415.5398 3 6.5 qP 17412.2849 4 3.5 rR 17420.5006 3 4.5 qQ 17415.5253 3 6.5 PP 17412.2462 4 3.5 qR 17420.4951 3 4.5 pQ 17415.5108 3 7.5 PP 17412.0737 4 4.5 rR 17420.4329 3 5.5 qQ 17415.3900 4 3.5 qP 17411.5579 4 4.5 qR 17420.4262 3 6.5 qQ 17415.2278 4 3.5 PP 17411.5475 4 5.5 rR 17420.3501 3 7.5 qQ 17415.0385 4 4.5 rP 17411.5126 4 5.5 qR 17420.3421 4 0.5 rQ 17415.6461 4 4.5 qP 17411.4968 4 6.5 rR 17420.2523 4 2.5 rQ 17415.5894 4 4.5 PP 17411.4834 4 6.5 qR 17420.2432 4 2.5 qQ 17415.5840 4 5.5 qP 17411.4220 4 7.5 rR 17420.1395 4 2.5 pQ 17415.5785 4 5.5 pP 17411.4060 4 7.5 qR 17420.1299 4 3.5 rQ 17415.5398 4 6.5 pP 17411.3143 4 8.5 rR 17420.0126 4 3.5 qQ 17415.5322 4 7.5 qP 17411.2327 5 2.5 rR 17421.4363 4 3.5 pQ 17415.5253 4 7.5 PP 17411.2110 5 2.5 qR 17421.4346 4 4.5 rQ 17415.4752 4 8.5 PP 17411.0948 5 2.5 pR 17421.4330 4 4.5 qQ 17415.4655 5 2.5 qP 17410.5130 5 3.5 rR 17421.4015 4 4.5 pQ 17415.4570 5 2.5 PP 17410.5079 5 3.5 qR 17421.3978 4 5.5 rQ 17415.3958 5 3.5 rP 17410.4884 5 4.5 rR 17421.3569 4 5.5 qQ 17415.3846 5 3.5 qP 17410.4799 5 4.5 qR 17421.3521 4 5.5 pQ 17415.3744 5 3.5 pP 17410.4736 5 5.5 rR 17421.3022 4 6.5 rQ 17415.3023 5 4.5 rP 17410.4480 5 5.5 qR 17421.2964 4 6.5 qQ 17415.2890 5 4.5 qP 17410.4382 5 6.5 rR 17421.2378 4 6.5 pQ 17415.2778 5 4.5 pP 17410.4300 5 6.5 qR 17421.2310 4 7.5 rQ 17415.1942 5 5.5 qP 17410.3879 5 7.5 rR 17421.1639 4 7.5 qQ 17415.1799 5 5.5 PP 17410.3778 5 7.5 qR 17421.1565 4 7.5 pQ 17415.1667 5 6.5 qP 17410.3283 5 8.5 rR 17421.0806 4 8.5 qQ 17415.0565 5 6.5 pP 17410.3167 5 8.5 qR 17421.0724 5 0.5 qQ 17415.5177 5 8.5 qP 17410.1836 5 9.5 rR 17420.9885 5 1.5 qQ 17415.5033 5 8.5 pP 17410.1687 106 APPENDIX H, Continued. 3 U a - *A3 R Q J' ' pn 5 2.5 qQ 5 2.5 P Q 5 3.5 rQ 5 3.5 qQ 5 4.5 qQ 5 5.5 qQ 5 5.5 pQ 5 6.5 qQ 5 6.5 pQ 5 7.5 qQ 5 7.5 pQ 5 8.5 qQ 5 9.5 qQ 5 9.5 pQ 6 1.5 qQ 6 2.5 rQ 6 2.5 pQ 6 3.5 qQ 6 3.5 pQ 6 4.5 qQ 6 5.5 rQ 6 5.5 qQ 6 6.5 qQ 6 6.5 pQ 6 7.5 rQ 6 7.5 qQ 6 7.5 pQ 6 9.5 qQ 6 9.5 pQ 7 3.5 qQ 7 4.5 qQ 7 5.5 qQ 7 5.5 pQ 7 6.5 qQ 7 7.5 pQ o,o) p freq. (cm *) J" pn freq.(cm 1) 17415.4782 5 9.5 PP 17410.0825 17415.4752 6 2.5 qP 17409.4222 17415.4486 6 2.5 pP 17409.4194 17415.4434 6 3.5 qP 17409.3989 17415.3990 6 3.5 PP 17409.3944 17415.3456 6 4.5 qP 17409.3690 17415.3386 6 4.5 PP 17409.3635 17415.2829 6 5.5 qP 17409.3325 17415.2738 6 5.5 PP 17409.3260 17415.2097 6 6.5 qP 17409.2898 17415 .2017 6 6.5 pP 17409.2820 17415.1287 6 7.5 qP 17409.2408 17415.0385 6 7.5 PP 17409.2317 17415.0271 6 8.5 qP 17409.1857 17415.3958 6 8.5 PP 17409.1754 17415.3793 6 9.5 pP 17409.1135 17415.3744 6 10.5 pP 17409.0456 17415.3530 7 2.5 PP 17408.3246 17415.3491 7 3.5 PP 17408.3063 17415.3214 7 5.5 qP 17408.2595 17415.2890 7 5.5 pP 17408.2547 17415.2829 7 6.5 qP 17408.2275 17415.2383 7 6.5 PP 17408.2217 17415.2322 7 7.5 qP 17408.1906 17415.1942 7 7.5 pP 17408.1842 17415.1871 7 8.5 qP 17408.1493 17415.1800 7 8.5 pP 17408.1419 17415.0650 7 9.5 qP 17408.1032 17415.0565 7 9.5 PP 17408.0952 17415.2541 7 10.5 qP 17408.0530 17415.2322 7 10.5 pP • 17408.0441 17415.2022 7 11.5 PP 17407.9888 17415.1733 17415.1691 17415.1256 APPENDIX II, Continued. 3 i 7 3 - 3 A3 (o, o) R Q J' freq.(cm l) 7 8.5 qQ 17415.0875 7 8.5 pQ 17415.0816 7 9.5 qQ 17415.0395 7 9.5 pQ 17415.0330 7 10.5 qQ 17414.9872 7 11.5 qQ 17414.9304 8 3.5 qQ 17415.1458 8 4.5 qQ 17415.1287 8 5.5 qQ 17415.1055 8 6.5 rQ 17415.0844 8 6.5 qQ 17415.0800 8 7.5 qQ 17415.0501 8 8.5 qQ 17415.0168 8 9.5 qQ 17414.9801 8 10.5 qQ 17414.9392 8 11.5 qQ 17414.8952 8 12.5 qQ 17414.8481 9 4.5 qQ 17415.0118 9 4.5 pQ 17414.9906 9 5.5 qQ 17414.9943 9 6.5 qQ 17414.9741 9 7.5 qQ 17414.9500 9 8.5 qQ 17414.9236 9 9.5 qQ 17414.8946 9 10.5 qQ 17414.8623 9 11.5 qQ 17414.8273 9 12.5 qQ 17414.7894 9 13.5 qQ 17414.7491 10 5.5 qQ 17414.8678 10 6.5 qQ 17414.8516 10 7.5 qQ 17414.8325 10 8.5 qQ 17414.8114 10 9.5 qQ 17414.7876 10 10.5 qQ 17414.7616 10 11.5 qQ 17414.7332 10 12.5 qQ 17414.7027 APPENDIX H, Continued. 3 i J 3 - * A, (o, o) R Q J" pa freq.(cm 1) 10 L3.5 qQ 17414.6700 10 14.5 qQ 17414.6351 11 6.5 qQ 17414.5509 11 7.5 qQ 17414.5366 11 8.5 qQ 17414.5202 11 9.5 qQ 17414.5034 11 10.5 qQ 17414.4828 11 11.5 qQ 17414.4618 11 12.5 qQ 17414.4393 11 13.5 qQ 17414.4153 11 14.5 qQ 17414.3899 11 15.5 qQ 17414.3627 12 7.5 qQ 17414.3763 12 8.5 qQ 17414.3627 12 9.5 qQ 17414.3475 12 10.5 qQ 17414.3311 12 11.5 qQ 17414.3133 12 12.5 qQ 17414.2944 12 13.5 qQ 17414.2740 12 14.5 qQ 17414.2527 12 15.5 qQ 17414.2298 12 16.5 qQ 17414.2058 13 8.5 qQ 17414.1897 13 9.5 qQ 17414.1769 13 10.5 qQ 17414.1629 13 11.5 qQ 17414.1479 13 12.5 qQ 17414.1318 13 13.5 qQ 17414.1145 13 14.5 qQ 17414.0961 13 15.5 qQ 17414.0768 13 16.5 qQ 17414.0563 13 17.5 qQ 17414.0350 14 fl.5 qQ 17413.9914 14 10.5 qQ 17413.9794 14 11.5 qQ 17413.9664 14 12.5 qQ 17413.9525 APPENDIX H, Continued. 3 iT 3 - 3 A3 (o, o) R Q J" p„ freq. (cm *) 14 13.5 qQ 17413.9380 14 14.5 qQ 17413.9221 14 15.5 qQ 17413.9057 14 16.5 qQ 17413.8881 14 17.5 qQ 17413.8700 14 18.5 qQ 17413.8505 15 10.5 qQ 17413.7810 15 11.5 qQ 17413.7702 15 12.5 qQ 17413.7581 15 13.5 qQ 17413.7454 15 14.5 qQ 17413.7318 15 15.5 qQ 17413.7174 15 16.5 qQ 17413.7024 15 17.5 qQ 17413.6865 15 18.5 qQ 17413.6699 15 19.5 qQ 17413.6525 16 11.5 qQ 17413.5587 16 12.5 qQ 17413.5482 16 13.5 qQ 17413.5372 16 14.5 qQ 17413.5254 16 15.5 qQ 17413.5130 16 16.5 qQ 17413.5001 16 17.5 qQ 17413.4863 16 18.5 qQ 17413.4719 16 19.5 qQ 17413.4570 16 20.5 qQ 17413.4414 17 12.5 qQ 17413.3239 17 13.5 qQ 17413.3144 17 14.5 qQ 17413.3041 17 15.5 qQ 17413.2934 17 17.5 qQ 17413.2700 17 18.5 qQ 17413.2575 17 19.5 qQ 17413.2446 17 20.5 qQ 17413,2312 17 21.5 qQ 17413.2169 18 13.5 qQ 17413.0767 APPENDIX H, Continued. 3JIa - »A 3 (o, o) R Q J" F» freq.(cm *) 18 14.5 qQ 17413.0678 18 15.5 qQ 17413.0583 18 16.5 qQ 17413.0484 18 17.5 qQ 17413.0381 18 18.5 qQ 17413.0271 18 19.5 qQ 17413.0158 18 20.5 qQ 17413.0036 18 21.5 qQ 17412.9916 18 22.5 qQ 17412.9790 19 14.5 qQ 17412.8087 19 15.5 qQ 17412.7999 19 16.5 qQ 17412.7910 19 17.5 qQ 17412.7849 19 18.5 qQ 17412.7814 19 19.5 qQ 17412.7715 19 20.5 qQ 17412.7611 19 21.5 qQ 17412.7506 19 22.5 qQ 17412.7389 19 23.5 qQ 17412.7279 20 15.5 qQ 17412.5454 20 16.5 qQ 17412.5370 20 17.5 qQ 17412.5290 20 18.5 qQ 17412.5210 20 19.5 qQ 17412.5121 20 20.5 qQ 17412.5030 20 21.5 qQ 17412.4938 20 22.5 qQ 17412.4842 20 23.5 qQ 17412.4740 20 24.5 qQ 17412.4637 21 16.5 qQ 17412.2602 21 17.5 qQ 17412.2529 21 18.5 qQ 17412.2462 21 19.5 qQ 17412.2384 21 20.5 qQ 17412.2302 21 21.5 qQ 17412.2221 21 22.5 qQ 17412.2136 21 23.5 qQ 17412.2048 APPENDIX n, Continued. 3JT3 - 3 A 3 (o, o) R Q J" freq.(cm 1) 21 24.5 qQ 17412.1959 21 25.5 qQ 17412.1865 22 17.5 qQ 17411.9630 22 18.5 qQ 17411.9567 22 19.5 qQ 17411.9502 22 20.5 qQ 17411.9433 22 21.5 qQ 17411.9359 22 22.5 qQ 17411.9285 22 23.5 qQ 17411.9209 22 24.5 qQ 17411.9127 22 25.5 qQ 17411.9046 22 26.5 qQ 17411.8964 23 18.5 qQ 17411.6530 23 19.5 qQ 17411.6472 23 20.5 qQ 17411.6412 23 21.5 qQ 17411.6352 23 22.5 qQ 17411.6285 23 23.5 qQ 17411.6217 23 24.5 qQ 17411.6151 23 25.5 qQ 17411.6077 23 26.5 qQ 17411.6011 23 27.5 qQ 17411.5933 24 19.5 qQ 17411.3302 24 20.5 qQ 17411.3244 24 21.5 qQ 17411.3194 24 22.5 qQ 17411.3143 24 23.5 qQ 17411.3079 24 24.5 qQ 17411.3021 24 25.5 qQ 17411.2960 24 26.5 qQ 17411.2898 24 27.5 qQ 17411.2831 24 28.5 qQ 17411.2767 25 20.5 qQ 17410.9943 25 21.5 qQ 17410.9900 25 22.5 qQ 17410.9849 25 23.5 qQ 17410.9801 25 24.5 qQ 17410.9750 APPENDIX H, Continued. 3JT a - sA3(o,o) R Q J" F" freq.(cm 1) 25 25.5 qQ 17410.9696 25 26.5 qQ 17410.9642 25 27.5 qQ 17410.9583 25 28.5 qQ 17410.9528 25 29.5 qQ 17410.9471 26 21.5 qQ 17410.6463 26 22.5 qQ 17410.6423 26 23.5 qQ 17410.6378 26 24.5 qQ 17410.6336 26 25.5 qQ 17410.6292 26 26.5 qQ 17410.6244 26 27.5 qQ 17410.6197 26 28.5 qQ 17410.6146 26 29.5 qQ 17410.6097 26 30.5 qQ 17410.6046 27 25.5 qQ 17410.2743 27 26.5 qQ 17410.2700 27 27.5 qQ 17410.2658 27 28.5 qQ 17410.2606 27 29.5 qQ 17410.2575 27 30.5 qQ 17410.2534 27 31.5 qQ 17410.2475 28 24.5 qQ 17409.9086 28 25.5 qQ 17409.9053 28 26.5 qQ 17409.9018 28 27.5 qQ 17409.8986 28 28.5 qQ 17409.8948 28 29.5 qQ 17409.8916 28 30.5 qQ 17409.8879 28 31.5 qQ 17409.8843 28 32.5 qQ 17409.8806 29 25.5 qQ 17409.5220 29 26.5 qQ 17409.5195 29 27.5 qQ 17409.5164 29 28.5 qQ 17409.5133 29 29.5 qQ 17409.5102 29 30.5 qQ 17409.5068 29 31.5 qQ 17409.5041 29 32.5 qQ 17409.5012 29 33.5 qQ 17409.4982 113 APPENDIX II, Continued. R 3 n a -Q J' freq.(cm - 1) 3 5.5 pQ 17365.9706 3 6.5 qQ 17365.8284 4 5.5 qQ 17365.9706 4 5.5 P Q 17365.9598 4 6.5 rQ 17365.8915 4 6.5 qQ 17365.8767 4 6.5 P Q 17365.8646 4 7.5 qQ 17365.7689 4 7.5 pQ 17365.7540 4 8.5 qQ 17365.6472 4 8.5 pQ 17365.6307 5 4.5 qQ 17365.9706 5 4.5 pQ 17365.9646 5 5.5 rQ 17365.9259 5 5.5 qQ 17365.9175 5 5.5 P Q 17365.9095 5 6.5 qQ 17365.8552 5 6.5 pQ 17365.8463 5 7.5 qQ 17365.7840 5 7.5 pQ 17365.7735 5 8.5 qQ 17365.7035 6 1.5 rQ 17365.9493 6 1.5 qQ 17365.9493 6 2.5 qQ 17365.9309 6 3.5 qQ 17365.9069 6 4.5 rQ 17365.8810 6 4.5 qQ 17365.8767 6 4.5 pQ 17365.8707 6 5.5 rQ 17365.8441 6 5.5 qQ 17365.8381 6 5.5 pQ 17365.8325 6 6.5 rQ 17365.8008 6 6.5 qQ 17365.7939 6 7.5 rQ 17365.7509 6 7.5 qQ 17365.7428 p J a pn freq. (cm i ) 3 2.5 rP 17363.3548 3 2.5 qP 17363.3331 3 3.5 rP 17363.2891 3 3.5 qP 17363.2605 3 3.5 PP 17363.2382 3 4.5 qP 17363.1678 3 4.5 PP 17363.1386 3 5.5 rP 17363.0929 3 5.5 qP 17363.0514 3 5.5 pP 17363.0159 3 6.5 qP 17362.9134 3 6.5 pP 17362.8708 3 7.5 pP 17362.7019 4 0.5 rP 17362.2773 4 1.5 rP 17362.2602 4 1.5 qP 17362.2520 4 2.5 rP 17362.2302 4 2.5 qP 17362.2190 4 3.5 pP 17362.1609 4 4.5 qP 17362.1124 4 4.5 pP 17362.0978 4 5.5 qP 17362.0398 4 5.5 pP 17362.0218 4 6.5 qP 17361.9541 4 6.5 PP 17361.9331 4 7.5 qP 17361.8556 4 7.5 PP 17361.8312 4 8.5 PP 17361.7178 5 0.5 qP 17361.1587 5 1.5 rP 17361.1492 5 1.5 qP 17361.1447 5 2.5 qP 17361.1217 5 2.5 pP 17361.1161 5 3.5 qP 17361.0896 5 3.5 pP 17361.0829 114 APPENDIX H, Continued. 3IT 3 - *A3 ( 1 , 1 ) R Q P J" F" freq.(cm *) J II pn freq.(cm *) 6 7.5 pQ 17365.7354 5 4.5 rP 17361.0609 6 8.5 qQ 17365.6860 5 4.5 qP 17361.0487 6 8.5 pQ 17365.6777 5 5.5 qP 17360.9996 6 9.5 qQ 17365.6225 5 5.5 pP 17360.9886 6 9.5 pQ 17365.6146 5 6.5 qP 17360.9413 6 10.5 qQ 17365.5533 5 6.5 pP 17360.9286 6 10.5 pQ 17365.5433 5 7.5 qP 17360.8745 7 2.5 qQ 17365.8060 5 7.5 PP 17360.8600 7 3.5 qQ 17365.7891 5 8.5 qP 17360.7997 7 4.5 qQ 17365.7656 5 8.5 PP 17360.7834 7 4.5 pQ 17365.7617 5 9.5 PP 17360.6988 7 5.5 qQ 17365.7376 6 3.5 qP 17359.9996 7 6.5 rQ 17365.7097 6 3.5 pP 17359.9946 7 7.5 rQ 17365.6729 6 4.5 qP 17359.9703 7 7.5 qQ 17365.6668 6 4.5 PP 17359.9641 7 9.5 qQ 17365.5768 6 5.5 qP 17359.9346 7 10.5 rQ 17365.5359 6 5.5 pP 17359.9269 7 10.5 qQ 17365.5251 6 6.5 qP 17359.8926 7 11.5 qQ 17365.4690 6 6.5 pP 17359.8838 8 3.5 rQ 17365.6608 6 7.5 qP 17359.8445 8 3.5 qQ 17365.6577 6 7.5 pP 17359.8346 8 4.5 qQ 17365.6406 6 8.5 qP 17359.7905 8 5.5 qQ 17365.6182 6 8.5 PP 17359.7795 8 6.5 rQ 17365.5975 6 9.5 qP 17359.7306 8 6.5 qQ 17365.5929 6 9.5 pP 17359.7186 8 7.5 rQ 17365.5693 8 7.5 qQ 17365.5637 8 7.5 pQ 17365.5591 8 8.5 qQ 17365.5307 8 9.5 qQ 17365.4942 8 9.5 pQ 17365.4887 8 10.5 qQ 17365.4544 8 10.5 pQ 17365.4480 8 11.5 qQ 17365.4111 8 11.5 pQ 17365.4043 8 12.5 qQ 17365.3642 APPENDIX H, Continued. 3JT3 - »A3 ( 1 , 1 ) R Q J' freq.(cm *) 8 12.5 pQ 17365.3572 9 5.5 qQ 17365.4816 9 6.5 rQ 17365.4651 9 6.5 qQ 17365.4614 9 6.5 pQ 17365.4575 9 7.5 qQ 17365.4384 9 8.5 rQ 17365.4167 9 8.5 qQ 17365.4142 9 8.5 pQ 17365.4080 9 9.5 qQ 17365.3834 9 9.5 pQ 17365.3782 9 10.5 qQ 17365.3517 9 10.5 pQ 17365.3467 9 11.5 qQ 17365.3172 9 11.5 pQ 17365.3121 9 12.5 qQ 17365.2800 9 12.5 pQ 17365.2744 9 13.5 qQ 17365.2402 9 13.5 pQ 17365.2342 10 6.5 qQ 17365.3148 10 7.5 qQ 17365.2937 10 8.5 qQ 17365.2726 10 9.5 qQ 17365.2490 10 10.5 qQ 17365.2234 10 11.5 qQ 17365.1954 10 12.5 qQ 17365.1652 10 13.5 qQ 17365.1328 10 14.5 qQ 17365.0981 11 6.5 qQ 17365.1454 11 7.5 qQ 17365.1301 11 8.5 qQ 17365.1126 11 9.5 qQ 17365.0933 11 10.5 qQ 17365.0721 11 11.5 qQ 17365.0490 11 12.5 qQ 17365.0242 11 13.5 qQ 17364.9973 A P P E N D I X H , Continued. sJT3 - 3 A 3 ( i , i) R Q J' freq. (cm J) 11 14.5 qQ 17364.9688 11 15.5 qQ 17364.9385 12 7.5 qQ 17364.9484 12 8.5 qQ 17364.9343 12 9.5 qQ 17364.9180 12 10.5 qQ 17364.9005 12 11.5 qQ 17364.8813 12 12.5 qQ 17364.8604 12 13.5 qQ 17364.8380 12 14.5 qQ 17364.8142 12 15.5 qQ 17364.7891 12 16.5 qQ 17364.7623 13 8.5 qQ 17364.7386 13 9.5 qQ 17364.7250 13 10.5 qQ 17364.7101 13 11.5 qQ 17364.6939 13 12.5 qQ 17364.6763 13 13.5 qQ 17364.6575 13 14.5 qQ 17364.6375 13 15.5 qQ 17364.6163 13 16.5 qQ 17364.5938 13 17.5 qQ 17364.5701 14 9.5 qQ 17364.5147 14 10.5 qQ 17364.5020 14 11.5 qQ 17364.4883 14 12.5 qQ 17364.4731 14 13.5 qQ 17364.4573 14 14.5 qQ 17364.4401 14 15.5 qQ 17364.4220 14 16.5 qQ 17364.4027 14 17.5 qQ 17364.3827 14 18.5 qQ 17364.3615 15 10.5 qQ 17364.2767 15 11.5 qQ 17364.2648 15 12.5 qQ 17364.2520 15 13.5 qQ 17364.2384 APPENDIX H, Continued. 3 i 7 3 - 3 A3 (x, 1 ) R Q J' F" freq. (cm 1) 15 14.5 qQ 17364.2236 15 15.5 qQ 17364.2081 15 16.5 qQ 17364.1916 15 17.5 qQ 17364.1742 15 18.5 qQ 17364.1561 15 19.5 qQ 17364.1369 16 11.5 qQ 17364.0240 16 12.5 qQ 17364.0128 16 13.5 qQ 17364.0011 16 14.5 qQ 17363.9884 16 15.5 qQ 17363.9749 16 16.5 qQ 17363.9608 16 17.5 qQ 17363.9456 16 18.5 qQ 17363.9300 16 19.5 qQ 17363.9136 16 20.5 qQ 17363.8965 17 12.5 qQ 17363.7565 17 13.5 qQ 17363.7460 17 14.5 qQ 17363.7351 17 15.5 qQ 17363.7235 17 16.5 qQ 17363.7113 17 17.5 qQ 17363.6981 17 18.5 qQ 17363.6845 17 19.5 qQ 17363.6703 17 20.5 qQ 17363.6556 17 21.5 qQ 17363.6401 18 13.5 qQ 17363.4745 18 14.5 qQ 17363.4650 18 15.5 qQ 17363.4548 18 16.5 qQ 17363.4440 18 17.5 qQ 17363.4326 18 18.5 qQ 17363.4208 18 19.5 qQ 17363.4082 18 20.5 qQ 17363.3955 18 21.5 qQ 17363.3819 18 22.5 qQ 17363.3679 A P P E N D I X H , C o n t i n u e d . S J T 3 - 3 A 3 ( 1 , 1 ) R Q J f req. ( cm 1 ) 19 14.5 q Q 17363.1774 19 15.5 q Q 17363.1678 19 16.5 q Q 17363.1591 19 17.5 q Q 17363.1493 19 18.5 q Q 17363.1386 19 19.5 q Q 17363.1281 19 20.5 q Q 17363.1168 19 21.5 q Q 17363.1050 19 22.5 q Q 17363.0929 19 23.5 q Q 17363.0801 20 15.5 q Q 17362.8660 20 16.5 q Q 17362.8575 20 17.5 q Q 17362.8488 20 18.5 q Q 17362.8399 20 19.5 q Q 17362.8303 20 20.5 q Q 17362.8204 20 21.5 q Q 17362.8100 20 22.5 q Q 17362.7992 20 23.5 q Q 17362.7883 20 24.5 q Q 17362.7766 21 16.5 q Q 17362.5392 21 17.5 q Q 17362.5317 21 18.5 q Q 17362.5236 21 19.5 q Q 17362.5156 21 20.5 q Q 17362.5069 21 21.5 q Q 17362.4980 21 22.5 q Q 17362.4885 21 23.5 q Q 17362.4790 21 24.5 q Q 17362.4689 21 25.5 q Q 17362.4586 22 17.5 q Q 17362.1987 22 18.5 q Q 17362.1916 22 19.5 q Q 17362.1844 22 20.5 q Q 17362.1767 22 21.5 q Q 17362.1692 22 22.5 q Q 17362.1609 APPENDIX n , Continued. *II3 - 3A3 ( 1 , 1 ) R Q J freq. ( c m 1 ) 22 23.5 q Q 17362.1524 22 24.5 q Q 17362.1436 22 25.5 q Q 17362.1344 22 26.5 q Q 17362.1252 23 18.5 q Q 17361.8429 23 19.5 q Q 17361.8363 23 20.5 q Q 17361.8312 23 21.5 q Q 17361.8233 23 22.5 q Q 17361.8164 23 23.5 q Q 17361.8086 23 24.5 q Q 17361.8010 26 25.5 q Q 17360.6691 23 25.5 q Q 17361.7932 23 26.5 q Q 17361.7852 23 27.5 q Q 17361.7766 26 26.5 q Q 17360.6640 26 27.5 q Q 17360.6585 26 28.5 q Q 17360.6527 26 29.5 q Q 17360.6474 26 30.5 q Q 17360.6418 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0060385/manifest

Comment

Related Items