OPTICAL AND INFRARED SPECTRA OF SOME UNSTABLE MOLECULES by JUDITH ANNE BARRY B.S. (Hon.), San Francisco State University, 1981 M.S., San Francisco State University, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT O F THE REQUIREMENT FOR THE DEGREE O F DOCTOR OF PHILOSOPHY in THE FACULTY O F GRADUATE STUDIES Department of Chemistry We accept this thesis as conforming to the required standard THE UNIVERSITY O F BRITISH COLUMBIA 9 November I987 ©Judith Anne Barry, 1987 In presenting degree at this the thesis in University of partial fulfilment of of department this or publication of thesis for by his or that the her representatives. It this thesis for financial gain shall not Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 for an advanced Library shall make it agree that permission for extensive scholarly purposes may be permission. Date requirements British Columbia, I agree freely available for reference and study. I further copying the is granted by the understood that head of copying my or be allowed without my written ii ABSTRACT Some unstable gaseous molecules, cobalt oxide (CoO), niobium nitride (NbN) and aminoborane resolution optical spectroscopy. (NH2BH2), were studied by high A portion of the "red" system of C o O , from 7000 A to 5800 A, was measured using laser induced fluorescence techniques. 6338 A, 6411 Three bands of the system, with origins at A and 6436 A, were rotationally analyzed. levels of these parallel bands are the components of a A j electronic state. 4 that this is the ground state 1.631 A. for metal oxides. the entire and 5/2 spin-orbit Available evidence indicates molecule; its bond length is series of first row diatomic transition The hyperfine structure in the ground state is very small, supporting a CT 8 TI 2 as ob n o*, 3 3 2 electron configuration. has large 2 follow a case (ap) and = 7/2 This work completes the determination of the ground state symmetries assigned of the ft The lower The upper state, positive hyperfine splittings that pattern; it is heavily perturbed, both rotationally vibrationally. The sub-Doppler spectrum of the <x>- A system of NbN was 3 measured hyperfine by intermodulated structure analyzed. fluorescence first order position. 3 techniques, Second order spin-orbit have shifted the o>3- A2 subband 40 c m 3 3 1 and the interactions to the blue of its central The perturbations to the spin-orbit components were so extensive that five hyperfine constants, rather than were required to fit the data to the case (a) Hamiltonian. The three, 3 A- O s system of NbN is the first instance where this has been observed. The magnetic hyperfine constants indicate that all components of iii the A and 0 3 3 spin orbit manifolds may be affected, though the A 3 state interacts most strongly, presumably by the coupling of the A 2 3 component with the A 1 Fermi contact state having the same configuration. interactions in the positive, consistent with a a 8 1 1 3 A configuration. + c) hyperfine constants are negative, configuration. The A and 0 3 3 substates are In the 0 3 The large and state the (b as expected from a 7 t 6 1 bond lengths are 1.6618 A and 1.6712 A, respectively, which are intermediate between those of ZrN and MoN. The Fourier transform infrared spectrum of the V 7 B H 2 wagging fundamental effective of NH2BH2 rotational and was rotationally centrifugal analyzed. distortion A set of constants was determined, but the band shows extensive perturbations by Coriolis interactions with the nearby V 5 and v n fundamentals. A complete analysis could not be made without an analysis of the V 5 - V 7 - V H Coriolis interactions, which very small dipole derivative analysis. is currently not possible because the of the V 5 vibration has prevented its 1 iv TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST O F TABLES vii LIST O F FIGURES viii CHAPTER I. ELECTRONIC TRANSITIONS IN HETERONUCLEAR DIATOMICS I.A. Some Properties of Angular Momenta I.B. Spherical Harmonics and Spherical Tensor Operators I.C. Selection Rules and Hund's Coupling Cases I. D. The Hamiltonian 1 1 4 11 21 I.D.1. Nuclear rotational Hamiltonian 21 I.D.2. Spin Hamiltonian 22 I.D.3. Magnetic hyperfine interactions 25 I.D.3.a. The sign of nuclear coupling constants in transition metal complexes 27 I.D.3.a.i. The sign of the Fermi contact interaction 27 I.D.3.a.ii. The sign of the dipolar nuclear hyperfine interaction I.D.4. The nuclear electric quadrupole interaction I.D.5. A-Doubling 30 31 37 CHAPTER II. THE COMPUTERIZED LASER-INDUCED FLUORESCENCE EXPERIMENTS 40 II. A. Experimental Details 40 II.B. Intermodulated Fluorescence 43 II.C. Computerization 49 V CHAPTER III. ROTATIONAL ANALYSIS O F THE RED SYSTEM O F COBALT OXIDE 52 III.A. Introduction 52 III.B. Experimental 56 III.B.1. Synthesis of gaseous cobalt oxide 56 III.B.2. The spectrum 57 III.C. Analysis 60 111.0.1. Rotational analysis of the 6338 A subband Ill.C.l.a. Rotational constants and hyperfine 60 structure....60 Ill.C.l.b. Perturbations 67 III.C.2. Rotational analysis of the 6436 A subband 70 III. C.3. Rotational analysis of the 6411 A subband 72 III. D. Discussion 78 CHAPTER IV. HYPERFINE ANALYSIS O F NIOBIUM NITRIDE 84 IV. A. Introduction 84 IV.B. Experimental 86 IV. B.1. Synthesis 86 IV.B.2. Description of the <D - A spectrum 3 3 86 IV. C. Non-Linear Least Squares Fitting of Spectroscopic Data....97 IV. D. Results and Discussion 102 CHAPTER V. ROTATIONAL ANALYSIS O F THE v BAND O F 7 AMINOBORANE V. A. Background 113 113 V. B. The Michelson Interferometer and Fourier Transform Spectroscopy 117 V.C. Experimental 126 V.D. The Asymmetric Rotor 127 vi V.E. The Rotational Hamiltonian 134 V.E.1. The Hamiltonian without vibration interaction V.E.2. Coriolis interaction 134 137 V.F. Band Analysis and Discussion 138 Appendix I. NbN 0 - A Correlation Matrix 145 Appendix II. 147 3 3 Transitions of the <D- A System of NbN 3 3 Appendix 11. A. <I>2- Ai 3 147 3 Appendix II.B. <D - A 3 152 3 3 2 Appendix II.C. 0> - A 3 155 3 4 3 Appendix III. Transitions of the V7 band of N H 2 B H 2 160 REFERENCES 170 11 vii LIST OF TABLES 3.1. The most prominent bandheads in the 7000 to 5800 A broadband emission spectrum of gaseous CoO 58 Assigned lines for the 6338 A band of C o O ( A7/2- A7/2) with the lower state combination differences, A2F", in c m 63 3.11. 4 4 1 3.III. Rotational constants for the analyzed bands of the red system of CoO 66 3.IV. Assigned lines from the 6436 A ( A7/2- A7/2) band of CoO....73 3.V. Assigned lines from the 6411 A ( A5/2- As/2) band of CoO....74 3.VI. Ground states and configurations of the first row diatomic transition metal oxides, with the fundamental frequency AG 1/2, B and r for the v"=0 state, and the spin-orbit interval AA for the orbitally degenerate electronic states 83 4.1. 4.II. 4 4 4 4 Molecular constants for the o - A system of NbN 3 3 Rotational constants obtained for the 3 105 o - A system of NbN 3 with the AD and y parameters fixed to zero 110 5.1. Vibrational fundamentals of gaseous NH2 BH2 11 116 5.II. Character table for the C2v point group 128 5.III. Character sets for an asymmetric top rotational wavefunction in the C2v point group 132 5.IV. Molecular constants of the V 7 band of NH2 BH2 11 143 viii LIST OF F I G U R E S 1.1. Polar and Cartesian coordinates 5 1.2. Vector diagram for Hund's coupling case (a) 13 1.3. Vector diagram for Hund's coupling case (b) 15 1.4. Vector diagram for Hund's coupling case (c) 18 2.1. Gaussian inhomogeneously Doppler-broadened velocity population profile 44 Schematic diagram of the intermodulated fluorescence experiment 46 2.2. 2.3. 2.4. 3.1. 3.2. a. The formation of crossover resonances, b. Stick diagram of a spectrum with four of the forbidden transitions that can accompany a AF = AJ = 0 Q transition 48 Schematic diagram of the laser-induced fluorescence experiment interfaced to the PDP-11/23 microcomputer 51 Energy level diagram of a diatomic 3d transition metal oxide..54 Broadband laser excitation spectrum of the three bands of gaseous CoO analyzed in this work 59 3.3. Bandhead of the Q' = Q" = 7/2 transition at 6338 A 3.4. Upper state energy levels of the A /2 - &7/2 6338 A band....68 3.5. A section of the spectrum of the 6338 A band containing A doubling, two avoided crossings, and extra lines 3.6. 3.7. 3.8. 4.1. 4 62 4 7 69 Upper state energy levels of the A7/2 - A /2 6436 A band....71 4 4 7 Bandhead of the Q' = Q" = 5/2 transition at 6411 A Upper state energy levels of the As/2 4 - As/2 6411 4 Broadband spectrum of the 0 - A system of NbN 3 3 75 A band....76 87 ix LIST OF FIGURES (cont.) 4.2. The Q-head regions of the a) 0 2 - A - i , b) 0 - A 2 , and c) 0>4A subbands of NbN 88 3 3 3 3 3 3 3 3 4.3. 4.4. The beginning of the Q head of the <&2- Ai subband 3 90 3 The higher J portion of the 3 > 2 - A i Q head, and the first resolved Q lines 3 3 91 4.5. a) R(1), b) R(2), and c) R(3) of the < D - A subband, illustrating the "forbidden" A F * AJ transitions and the crossover resonances.between the rR and qR lines 92 4.6. a) R2, b) R3 and c) R4 lines of the 0 - A 2 subband of NbN, showing the rR, qR and pR transitions, and the crossover resonances associated with the rR and qR lines 4.6. 4.7. 4.8. 5.1. 5.2. 5.3. 3 3 2 3 1 3 3 d) R5 and e) R6 lines of the <I> - A2 subband of NbN., 3 93 94 3 3 The reversal of hyperfine structure at high J in the 0 4 - A branch 3 3 Partial energy level diagram for NbN 3 Q 95 103 A polychroomatic sugnal in the frequency domain (above) Fourier transformed into the time domain (below) 122 A boxcar function D(x). The Fourier transform of a boxcar truncated interferogram is a spectrum with the line shape function F{D(x)} = 2Lsin(2TtvL)/27rvL 124 The triangular apodization function D(x) (above) produces a spectrum with the line shape function F{D(x)} = 2Lsin(27wL)/(27rvL) (below) 125 2 5.4. Schematic drawing of the OZM NH2BH2 molecule in the x, y, z principal axis system and the a, b, c inertial axis system, showing the C2 ov reflection planes 130 X LIST OF FIGURES (cont.) 5.5. N H 2 B H 2 spectrum of the v band, and the V5 and v n bands with which it undergoes Coriolis interactions 140 7 5.5. Center of the v band of N H 7 1 1 2 BH 2 141 1 CHAPTER I ELECTRONIC TRANSITIONS IN HETERONUCLEAR DIATOMIC MOLECULES I.A. Some Properties of Angular Momenta. In a non-rotating molecule, the angular momentum operators J , S and L have the following diagonal matrix elements: 1 <JQ| J |JQ> =tiQ (1.1) z <SI| S |SI> - til z (1.2) <LA| L |LA> =tiA (1.3) z <JQ| J2|JQ> =fi J(J + 1) (1.4) 2 <SI|S2|SX>=fi S(S + 1) (1.5) 2 <LA| L2||_A> = t i L ( L + 1 ) (1.6) 2 J, S and L are the total, spin and orbital angular momenta, respectively; J , S and L are their respective quantum numbers, and Q, L and A are the projection quantum numbers in diatomic molecules (i.e., along the molecular z axis). The ladder operator L+ of a general angular momentum L has the Cartesian form 2 L± = L ± i L y (1.8) x It has the property of transforming state |L,m> into state | L , m ± 1 > , where m is the quantum number of L. operations are written: <J,Q±1| J For J and S the laddering 1 |Jn> =ti[J(J+1) - Q(Q±1)] /2 1 T <S,I±1| S±|SI>=ti[S(S+1)-I(I±1)] /2 1 (1.9) (1.10) 2 J? in equation (1.9) is not expressed as J± because the commutation relations are different in the space-fixed and molecule-fixed axis systems: 3 J X J Y - J Y J X = Uz J J x y - JyJx = -Uz SPACE (1.11) MOLECULE (1.12) This leads to a sign reversal upon transformation from the spacefixed to molecule-fixed systems (the anomalous sign of i): J±|JM> = fi[J(J+1) - M(M+1)] ' |J,M±1> 1 SPACE (1.13) MOLECULE (1.14) 2 J |JK> =fi[J(J+1) - K(K+1)] /2 |J,K±1> 1 T Although the motion of the electrons about the axis defines a good quantum number A , L itself is not a good quantum number because a diatomic molecule is not a spherical system. Thus L and L do not x y obey the usual operator equations, and L± is left in the form <L+L. + L.L >/2, or <L_L>, with the quantity B<Lj_> appearing on the diagonal + of the rotational Hamiltonian matrix as a minor, constant electronic isotope shift incorporated into the effective vibrational energy. 1 The dot product of two general angular momentum operators A and B is: AB = A B 2 Z + (A+B. + A.B+)/2 (1.15) The addition of angular momenta j i and J2 to form j results in the coupled eigenfunction |jm>: |jm> = I (-i)ji-j2+m V I J T T / ji mirr)2 \mi j 2 j\|jimi> |j m > 2 2 (1.16) m2 -m/ where |ji m 1 > and |J2m > are the uncoupled eigenfunctions, the first 2 term is a phase factor, and V2j+1 is a normalization factor. T h e term in brackets is a coefficient called a Wigner 3-j symbol. definition is given by equation (1.16) rearranged a s : 4 Its 3 / ii J2 j \ I = \mi rri2 -my (^p-i ^ —<jiJ2mim |jm> V2j + 1 2+m 2 (1.17) According to the angular momentum commutation relations for J1J2 and j , 5 the algebraic form for the 3-j symbol is determined by the requirement triangle, that or vector mi + m2 = m and |ji - J2I < j < Gi + J2) (the addition, rule) . 4 If these conditions are not satisfied, the vector coupling coefficient <jiJ2m 1 m2|jm> is 0. 4 Spherical Harmonics and Spherical Tensor Operators. I.B. Spherical harmonics, Y| (0,(p), are m orbital angular eigenfunctions normalized to unity on a unit sphere. they are the eigenfunctions of the differential To be exact operators L corresponding to the eigenvalues 1(1+1) and m : 6 m L Y| (e,<p) = mY| (e,<p) m and L , z (1.17) 2 2 2 7 L Y| (e,(p) = l(l + 1)Y| (e,<p) m momentum (1.18) m The angles 6 and <p are the usual polar coordinates as illustrated in Figure 1.1. The differential where fi = 1, are and L 2 defined in units Z ) 6 L L operators L z = d/id<p (1.19) = -[(sin 9)-l(3/39)(sin 93/39) + ( s i n 9 ) - l 3 / 3 9 ] 2 2 2 (1.20) 2 Expressed in terms of the orbital angular momentum functions of 9 and cp on the unit sphere, a spherical harmonic is: 8 Y| (e,q>) = C|(-1)'+m [(l-m)!/(l+m)!] ' (sin9)m [3/3(cos9)]'+m 1 2 m x (sin9) where 21 (1.21) e <P im ci is a normalization factor: |q| = [(21+1 )!]i/2/( )i/2 2>l! (1.22) 47C Associated Legendre polynomials, P | ( c o s 9), m exploited in quantum spherical harmonics: mechanics because their commonly connection to 6 Y|m(9,(p) = (-) [(2l+1)(l-m)!/4jt(l+m)!]l/ m where of are 2 P|m(cos 9)eimcp (1.23) 6 P| (x) = (1-x2)m/2/ l|! [d'+ /dx'+ ](x -1)1 m m m 2 2 (1.24) When the component m = 0, the spherical harmonic and Legendre polynomial differ only by a constant 9 Y| (9,<p) = [(2I+1)/4TC]1/ P|(cos 9) 2 0 (1.25) 5 Fig. 1,1. Polar and Cartesian coordinates, in which x = rsinBcosq), y = rsinGsincp, z = rcosG. 8 6 The angular derivation of expressions describing the momenta, particularly those for the magnetic quadrupolar irreducible tensor hyperfine spherical operators, manipulation, where interactions, tensors. and follows. necessary in is often A brief the subsequent employed in the Hamiltonian of hyperfine and best approached using explanation expressions Spherical tensor coupling of spherical required for methods are then sections to derive their applied the forms representing the diatomic molecules in the present work. The spherical components of a vector, or first rank operator acting on an angular Cartesian counterparts by: 2 momentum tensor, are related to their A 4 T o(A) = A (1.26) 1 z T ± 1 ( A ) = T ( A ± iA )/V2 1 x (1.27) y A spherical tensor T of rank k is defined as a set of 2k+1 quantities ("components") which transform into one another upon rotation one coordinate system to another and space-fixed axis s y s t e m s ) : 10 T k (for example, molecule- 11 = X T Dpq( )(ccpY) k q between from (1.28) k p P where q and p are the components of the tensor in the molecule- and space-fixed axis systems, respectively, and Dpq (a{3y) is the Wigner k rotation matrix. corresponding transform The angles to the three between a , p and y are successive two coordinate axis the Euler rotations systems. angles required to In spectroscopy, a beam of photons (in the space-fixed axis system) induces a change in the molecule in the molecule-fixed system. Wigner rotation 7 matrices function to project from one axis system to another in order to put the photon beam and the molecules being altered by the photons into the same frame of reference. In the reverse direction, from space- to molecule-fixed coordinates, the relation is: T k = I D k>py)Tk q p pq (1.29) q where the complex conjugation of a rotation matrix is given by 0 M K > P Y ) - (-1) - 0-M,-K (apy) M k The complex conjugation K (1.30) k is required to account for the anomalous sign of i. A Wigner eigenfunctions rotation of J matrix is a matrix describing and J , i.e., 2 a z spherical harmonic transform on coordinate rotation into other functions |jm>: D (ap )|jm> = I |jm >D , Y Premultiplying nVm (])(ap ) Y spherical harmonic functions: m m D^oWy) (1.31) 12 Y angles: 12 due to the orthogonality of D < Q) Dm'm«)(aPy) = <jm'|D(ap )|jm> matrix element collapses |jm>, equation (1.31) by |jm'>* (i.e., <jm'|) and integrating reduces the right hand side to A D how the with one of its projections to a spherical harmonic, which depends (1.32) equal to zero on only two 12 = (-1)P[4TC/(2I+1)]1/2 Y | ( p , a ) p D 'oq(apY) = [4ic/(2l+1 )] ' 1 2 Y| (p, ) q Y SPACE (1.32) MOLECULE (1.33) If both projections are zero, the Wigner rotation matrix collapses to a Legendre polynomial: 9 12 D 'oo(ap ) = P|(cos P) = [47c/(2l+1)]1/2 Y, (p,0) Y 0 (1.34) 8 The Legendre polynomial P|(cos8) is also related to the spherical harmonics by the spherical harmonic addition theorem: P|(C0S 9) = (4n/2l+1) I Y * | ( 9 i , 9 i ) Y| (e q>2) m m m where Y*im(9,cp) = ( - ) Y | . ( 9 , ( p ) . 6 . 9 , i 3 , i 4 The angles 0 i , 6 2 , 91 and m ( (1-35) 2i m cp are as defined by Fig. 1.1 for vectors n and r2, and 6 is the angle 2 between directions (61,91) and spherical harmonics to eliminate Using Racah's modified (9 ,cp2)2 the factor of [47t/(2l+1)J : 1/2 C|m(9,cp) = [4TC/(2I+1)]1/2 Y| (9 (p) m (1.36) > the spherical harmonic addition theorem b e c o m e s ' 1 3 1 5 P|(C0S 9) = I C*im(9i ,91) C| (e ,q>2) m (1-37) P|(cos 9) = C|(9i ,(pi ) C | ( 9 , c p 2 ) (1.38) m or 16 2 1 4 2 The coupling of two tensor operators to form a compound tensor is similar to the addition of two angular momenta given in equation (1.16):"«0 [Tki(1) ® T k 2 ( 2 ) ] k = £ (-1)ki-k2+q V2k+1 / k i k q V^ 1 k\ 2 ^2 <\) x[rki (1)Jk2 (2)] q 1 (1.39) q 2 Here the tensor T i of rank k i , operating on system (1), is coupled to k tensor T 2 [which operates on system (2)]. k Shorter, alternative ways of denoting a compound tensor are [T (1), T 2(2)] or, for a tensor of kl k rank ki coupled to itself, [T (1,1)], where k = 2 k i . k If two tensors of the same rank k are coupled to give a s c a l a r , i.e., a quantity invariant to a coordinate rotation, the compound tensor of equation (1.39) is also a scalar, or of rank zero. The resulting expression 9 becomes symbol: much simpler and lacks the orientation-dependent 3-j 10 [Tk(1) ® T k ( 2 ) ] ° = (-1) (2k+1)- /2Tk(1)Tk(2) k (1.40) 1 0 where the conventional scalar product T ( i ) T ( 2 ) is given a s : k k Tk(1)Tk(2) = I (-1)q Tk (1) Tk. (2) q q 1 0 ' 1 1 (1.41) q After a compound tensor equation is written which appropriately represents a particular constituent evaluate tensors, the matrix physical interaction the Wigner-Eckart elements T k q of and breaks theorem the is it into its applied to constituent tensors. According to the theorem the matrix elements of a tensor operator are factored into: 1) a 3-j symbol, which contains information on the geometry or orientation of the angular momentum; 2) a reduced matrix element (denoted by double vertical bars), related to the magnitude of the angular momentum but independent of its direction; and 3) a phase factor. Expressed in terms of the eigenfunctions | jm>, where j is the quantum number acted upon by T , m is the k Y projection of j, and Y contains any remaining quantum numbers not of interest in this particular basis, the Wigner-Eckart theorem i s : <YTm | T k , q | jm> = ( - 1 ) 1 ' ^ ' / j' k Y j W j ' l l Tk || j> Y 1 6 (1.42) \ - m ' q m/ Note that the reduced matrix element is independent of m. A reduced matrix element is usually worked out by evaluating the simplest type of matrix Wigner-Eckart theorem. element and then substituting into the For example to obtain <J|| T (J) ||J>, where J 1 refers to a general angular momentum, we calculate the simplest type of matrix element of T ( J ) , namely its q = 0 (or z) component: 1 17 10 <J'M'| T ( J ) MM> = 8 M'5jj'M (1.43) 1 0 M This element is non-vanishing only if J'M' = J M . From the Wigner- Eckart theorem (equation 1.42), M = (-1)J-M / j 1 j \ < j | | -p(J) ||J> \-M 0 UJ Substitution for the 3-j s y m b o l 11 (1.44) produces M « (-1)J-M(_1)J-M M[J(J + 1)(2J + 1)]- <J|| T ( J ) ||J> 1/2 (1.45) 1 Since J and M both have integral or half-integral values, (-1)2(J-M) is 1, which reduces equation (1.45) to: <J ||T (J)|| J> = [J(J + 1)(2J + 1)]i/2 (1.46) 1 An important reduced matrix element is that of the rotation matrix element D.q( )(apy) (cf. equations 1.29 and 1.30): k <J K ||D.q *(apy)||JK> , , = (-1 )J'" '[(2J + 1)(2J' + 1 ) ] k K 1 / 2 / J' k J \ \-K'q (1.47) Kj in which the dot replacing the p indicates that no reduction has been performed with respect to space-fixed dependence on the M quantum number. axes, so there is no Another useful formula gives the matrix elements of the scalar product of two commuting tensor operators (that is, ones which act on different parts of the system) in a coupled b a s i s : 18 <Y'J1 J2 J'M | Tk(1)-Uk(2) |yjiJ JM> = , , , 2 (-1)J1 J2'+J + 5 J M 5 - M / J 12 j l ' l l {k in which T Wigner 6-j <yjl'll M k Tk(1) || Y J1> < Y V | | Uk(2) M h j JY" Y 2 (1.48) 2 acts on ji and U || j > k on j . 2 The term in curly brackets is a symbol, a coefficient which arises in the coupling of three angular momenta, as compared to two in the 3-j s y m b o l . 19 11 I.C. Selection Rules and Hund's Coupling Cases. An electronic transition can occur in a molecule only if there are non-zero matrix elements of the electric dipole moment operator M which allow interaction with electromagnetic probability of such a transition occurring between radiation. The 20 electronic states n and m is proportional to the square of the transition moment, R Rnm where and : (1.49) are the eigenfunctions of states n and m . electric dipole moment nuclei) i s J^n'M^mdT , = n m 2 0 The M for a total of N particles (electrons and 2 1 N M=Zein (1.50) i=1 where e\ is the charge on particle i which has coordinates rj. general case the transition moment In the integral vanishes unless the change in total angular momentum, J , is zero or unity, o r AJ = 0, ± 1 2 2 (1.51) Changes in J of -1, 0 and +1 are denoted by the letters P, Q and R, respectively. The specific selection rules vary depending on the manner in which the spin, orbital and rotational angular momenta are coupled to one another and to the internuclear axis. The angular momentum coupling schemes in diatomic molecules are distinguished by sets of molecule-fixed basis functions called the Hund's coupling cases. main property differentiating the four coupling cases The described below is the number of angular momenta which have well-defined components (quantum numbers) along the internuclear axis. The 12 appropriate coupling case is the one which produces the smallest off-diagonal diagonal matrix elements spectral pattern. elements which for most the rotational closely Hamiltonian, reproduce the or observed The most common cases by far in molecules with no very heavy atoms are cases (a) and (b). Hund's case (a) coupling has the maximum number of well-defined quantum numbers, such that the relations given in equations (1.1), (1.2) and (1.3) basis function for a non-rotating molecule remain v a l i d . - for coupling a case (a) 1 scheme is 23 The therefore |(L)A>|SZ>|Jft>, or |riA;SI,;JftM>, where A, X and ft are the eigenvalues of the z components of L, S and J , with M being the space-fixed analog of ft, and ft = A + X . The semicolon separators indicate 1 products of component wavefunctions. L is incorporated into the label TI for the vibronic state, as it is not a good quantum number (cf. Section I.A). approximation The when case there (a) representation are no strong is a good interactions working in the Hamiltonian which uncouple these angular momenta from the axis. Case (a) occurs where there is a non-zero orbital angular momentum and fairly small spin-orbit coupling, where the coupling of L and S to each other is less important than the coupling of L to the a x i s . 24 The vector diagram for case (a) coupling is given in Fig. 1.2. In case (b) coupling, S is coupled only weakly to the axis, but L remains strongly coupled. (a) state Given a large enough value of J , any case uncouples toward case (b) because as J increases the rotational and spin magnetic moments must ultimately more strongly to one another than L and S are. be coupled Formally it can be said that the rotation (R) has increased to the point where it couples 13 Fig. 1.2. Vector diagram of Hund's coupling case (a). 24 14 to the orbital angular momentum to form a resultant N, causing S to uncouple from L, and therefore from the molecular axis. The effects of rotation become important when BJ becomes large compared to the separations between the spin-orbit components. 1 The transformation of a case (a) situation to case (b) occurs by way of the spin-uncoupling operator, -B(J+S. + J-S+). With its selection rules A S and A A = 0, and AO. = A X = ± 1 , this operator most commonly mixes spin-orbit components of a given 2 S + 1 A state, which is consistent with the physical case (b) phenomenon of uncoupling L from S. 23 The case (b) representation also arises for X states in which there is no orbital angular momentum to couple the spin to the axis. The total angular momentum J in case (b) is thus obtained as: * 2 (1.52) R + L = N; N + S = J instead of the case (a) situation (1.53) R + L +S = J The case (b) basis function, | r i ; N A S J > , is the more physically realistic representation in those cases where the rotational angular momentum N is quantized about the axis, with electron providing only minor corrections to the total energy. spin Its vector diagram appears in Fig. 1.3. When nuclear spin is included in the basis set describing angular momentum coupling in diatomic molecules, the Hund's coupling cases (a) and (b) must be further subdivided. molecules, including those In the majority of diatomic considered in the current work, I is coupled so loosely to the internuclear axis or to S that the dominant coupling is to the rotational angular momentum J , or 15 16 (1.54) J +I =F By analogy with Hund's case (b), those coupling schemes following equation (1.54) are denoted by p* subscripts. coupling cases are called The extended Hund's ap and b p j , corresponding functions |ASXJQIF> and |NASJIF>, to basis respectively.25,26 Coupling schemes in which I is not coupled to J are a , bpN and a bps. with In the a the case ( a ) a a case, nuclear spin is coupled to the molecular axis projection quantum number l , though molecules exhibiting z coupling have never been o b s e r v e d . since nuclear magnetic 27 This is expected moments are on the order of a thousand times smaller than that of the electron, making it unlikely that the dominant nuclear spin coupling will be to the internuclear axis by a magnetic interaction with the electronic momenta. In the bpN and b p s cases I is coupled to N respectively, rather than to J as in case (bpj). and orbital angular and S, Case (bpN) coupling is not expected to be observed, as the magnetic moment of N (composed of R + L) is normally considerably less than that of either J or S, as S has a large magnetic moment and J is the sum of S and L. 27 In Hund's case (bps), I couples to S to form a vector G, which couples to N to form the total angular momentum F: I+S =G G +N = F In a nonrotating momenta are absent, coupling scheme. coupling molecule, where any rotationally case induced angular (bps) will be the dominant case (b) In a rotating case (b) molecule, however, the case that occurs depends on the relative sizes of the coupling of S to I and N: if the I S coupling dominates, the (bps) 17 case occurs. The best condition for a case (bps) molecule is a X state which originates nearly completely from an atomic s orbital. Case (bps) coupling is therefore extensively ScO. ion 2 8 - ' 2 9 rather described in the ground X rare, though it has been state 2 of scandium oxide, This molecule is ideal because the transition 3 0 and closed shell oxygen potentials. This leaves the S c have 2 + widely differing uncontaminated metal ionization by contributions from O " , and the X state far removed from the closed state of non- spherical with which 2 2 symmetry it could m i x . Other 27 molecules that have been observed to conform to case (bps) coupling are the b X and c X states of A I F , and the ground X+ state of L a O . 3 31 3 3 2 2 Note that both of these molecules also adhere to the conditions required for the bps coupling case. Case sufficiently (c) coupling occurs in molecules containing an atom heavy that the spin-orbit interaction which results is so large that electron motion can no longer be defined in either the L or S representations; one of the multiplicity is no longer defined. consequences is that spin This phenomenon is expressed as an axial J ( J ) equal to the sum of L and S, which is then coupled to a R to form the resultant J , as illustrated function for case (c) is therefore defined axial component is are 09BiO ( X I l i / 2 s t a t e ) . 2 2 33 fl. 1 in Fig. 1 . 4 . The basis | r t J ; J Q M > , where the only wella Case(c) molecules observed so far and InH ( I l i 34 24 3 state) . 35 Case (d) coupling is normally only found in molecules where an electron has been promoted principal quantum number n. to a Rydberg orbital with higher The effect of the long distance between 18 Fig. 1.4. Vector diagram for Hund's case (c). 24 19 the electron and the nuclei is that the electron orbital coupled only weakly to the internuclear couple Case axis, but motion is can instead more strongly to the rotational angular momentum, R . > 2 1 2 4 (d) is equivalent to case (b) but with the difference that L is uncoupled from the axis rather than S ; the transition from case (a) A A A A is made by the L-uncoupling operator, -B(J+L. + J.L+) rather than via the S-uncoupling operator. 23 While still in the case (a) or (b) limits, the L-uncoupling operator may induce A-doubling, which lifts the degeneracy of the ± A states. The selection rules for interactions by this operator are AQ = A A = ±1 and A S = 0 . 2 3 The phenomenon of A-doubling is discussed in more detail in the last section of this chapter. appropriate representation when - 2 B J L the energy Case (d) becomes the makes a contribution to levels that is large with respect to the separation of states with differing A . The Hund's coupling cases corresponding to the niobium nitride (NbN) and cobalt oxide (CoO) molecules in this work are most appropriately described by the case (a) and, with higher rotation, case these (b) coupling schemes. cases, (a) and (b): As A and S are defined in both of the following selection rules can be stated for cases 24 A A = 0, ±1 (1.55) AS = 0 (1.56) For case (a), with £ and Cl as good quantum numbers, there are the more specific rules: A Q = 0, ±1 (1.57) A l =0 (1.58) 20 where equation (1.57) follows from equations (1.55) and (1.56). 24 The A S = 0 and A X = 0 rules become less strict as the spin-orbit interaction increases, because the selection rules for the spin-orbit interaction are A Q = 0 with either A A = A X = 0 or AA = - A X = ± 1 . - 3 6 24 In case (b) neither X nor Q. are well-defined, so the 'rotational' selection rule becomes AN = 0, ±1 (1.59) 21 I.D. The Hamiltonian. I.D.1. Nuclear From rotational equation A Hamiltonian _ (1.53) Hamiltonian. it follows that the nuclear rotational A. BR - DR 2 should be written in the form appropriate for 4 case (a) as: Hrot = B(J - L - S ) 2 - D(J - L - S ) where B is the rotational constant, (1.60) 4 and D is the centrifugal distortion constant representing the influence of centrifugal due to rotation on bond length. force Expansion of the B term of equation (1.60) gives A A A A A A A A- A A H = B(J2 + L2 + S2 - 2 J L - 2 J S + 2 L S ) (1.61) Because the x and y components of L are not defined in a nonspherical system, calculations . 1 their effects omitted in subsequent Equation (1.61) therefore simplifies to: H = B[J2 + L.2 + S2 - 2 J L - 2 J S Z The are off-diagonal term, Z Z Z - (J+S. + J . S ) + 2 L S ] + -(J+S. + J . S + ), Z (1.62) Z is the spin-uncoupling operator discussed in Section 1.C. The diagonal and off-diagonal rotational matrix elements are calculated by applying equations (1.1) through (1.10) and equation (1.15) to equation (1.61): < J Q L A S I | H | J n L A S X > = B[J(J + 1) - Q2 S(S +1) - X ( Z + 1)] '2 1 + (1.63) and <JS, Q±1 ,X±1 |H|JSQX> = -B{[(J(J + 1) - Q(Q ± 1)] x[S(S + 1 ) - X ( X ± 1)]} /2 1 (1.64) The D terms are obtained by squaring the matrix of the coefficients of the B terms. 22 I.D.2. Spin Hamiltonian. Spin-orbit coupling can be expressed as the scalar product of the many-electron operators, electronic S and L, spin which and orbital (using angular equations 1.8 momentum and 1.15) is represented in Cartesian form as: H -S = A[(L + i L ) ( S L X = AL S Z y Z - iS )/2 + L S x y Z + (L - i L ) ( S Z x y + iS )/2] x y + A ( L S . + LS+)/2 (1.65) + where A is the spin-orbit coupling constant. Neglecting the terms off-diagonal in L, equation (1.65) can be shortened t o : H .s = A L S L z 31 (1.66) z which has the selection rule A S = 0, and produces diagonal matrix elements of A A S . The dipolar spin-spin interaction can be represented by the classical Hamiltonian for two bar magnets, or dipoles, n : H= aiiu )/(r 2 1 2 ) 3 - 3(m-r )(H2-ri2)/(ri2) 12 in which ri2 is the vector between dipoles u i and u 2 5 3 7 ( - ) 1 67 , or ri - T2- The magnetic dipole of spin S is u = -gu S (1.68) B where g is the dimensionless electronic g factor and LIB is the Bohr magneton (the unit on an electronic magnetic moment, equal to efV2m where e and m are the charge and mass of the electron, respectively). The dipolar interaction 38 in terms of two electron spin vectors separated by vector r is therefore: H -s = (g P /r ){Si-S 2 2 3 s 2 - 3(si-r)(s r)/r2} 2 (1.69) Considering only the q = 0 terms (i.e., neglecting the components q = ±1 and ± 2 ) , the interaction reduces t o : 37 H - s = (g P /r ){S (i)S (2)(3cos20 2 s 2 3 z z 12 - 1) 23 - (S.(DS + (2) + S (1)S.(2))(3 2ei2 + COS - 1)/4} (1.70) Averaging over all orientations of ri and xi and expressed in terms of a total spin S , equation (1.70) becomes: H s - s = ( g W / r ) [ 3 S 2 - S 2 - (S.S+ + S+S.)/2] 3 z z = (9 |iB /r )[2S - (S + iS )(S 2 2 z = (gW/r )(3S x 3 or in terms z x - S2) (1.71) zero-field 2A ), 3 8 H -s = 2 X ( 3 S s primary y of the spin-spin coupling constant X (or splitting parameter The - iSy)] 3 Z - §2)/3 (1.72) spin-spin interaction originates from two mechanisms: contribution to X is from the dipolar interaction the of two unpaired spins, but there is also an effect due to second order spinorbit coupling, which may in fact be considerably larger: x = ass aso Ci .73) + Second order perturbation theory applied to the interaction produces a spin-spin interaction as follows. order contribution of the spin-orbit 39 interaction spin-orbit The second in single particle terms is: E ( ) = I [E 2 so TI'A'S' - E - ' s r X ^ A l a i l i l n ' A ^ X<T A |ajf |TiA> , l l A S T l , 1 A 1 i , j j x X<SX|Si|ST><ST|Si|SX> The (1.74) term summing over X ' produces the dipolar spin-spin term < S X | s S j | S X > , as well as other matrix elements not of interest here r because they are off-diagonal in A . The dipolar spin-spin interaction matrix elements are obtained by applying equations (1.2) through (1.5) to equation (1.72): 24 <jni_ASI| H -s |Jfll_ASI> = 2X[X - S(S + 1 )/3] (1.75) 2 s The states they mix have A X (=AA) and A S = 0, ± 1 , ± 2 4 0 Centrifugal distortion corrections to the spin-orbit and spin-spin interactions—Ao and XQ, respectively-must also be considered. Terms containing the parameters AQ and XD are therefore added to the rotational Hamiltonian (equation 1.60) as follows: 41 Hrot = BR2 - D R + A R 2 L S + 2X R2(3S - S ) / 3 (1.76) 2 4 D Z Z D Z Since the products of the operators in the AQ and XJJ terms are not Hermitian, a Hermitian average must be taken by symmetrizing the products with the anticommutator. for the AD and Xo The diagonal matrix parameters therefore follow the elements rotational constant B, but are multiplied by the elements for the spin-orbit and spin-spin interactions, respectively. The off-diagonal elements do likewise, except that since there are no off-diagonal terms in A or X, the factor for these interactions becomes the average of the two A diagonal elements. As before, the operator R 2 is simplified by omission of the x and y components of -2J-L + 2L-S + L . 2 The spin-rotation operator, the dot product of the spin and rotational angular momenta, is written in Cartesian form a s : A- A- A. A- H -R = y(J " L - S ) S (1.77) S Neglecting L+terms, equation 3 1 (1.77) produces the expanded Hamiltonian: H -R = Y[J S - L S S + ( J S . + J-S )/2] (1.78) <jni_ASX| H -s |Jnl_ASX> = y[X - S(S + 1)] (1.79) 2 Z Z Z - S 2 z + + with diagonal elements: 2 s and off-diagonal elements equal to those given in equation (1.64), but replacing B with -y/2. 25 I.D.3. Magnetic hyperfine interactions. The magnetic hyperfine interactions include all interactions of the nuclear spin, I, with the other angular momenta in the basis set, which for the case(a) moments interact basis are J , L and S . weakly with the Nuclear rotational magnetic giving rise to a scalar interaction term written: (1.80) where ci denotes the interaction constant. 2 =J 2 moment 25 H|.j = cil-J F magnetic From equation (1.54), + 21-J + I (1.81) 2 so that the IJ interaction can be expressed in terms of F as: H|.j - q(F - J 2 2 - i )/2 (1.82) 2 The matrix elements can be obtained directly from equation (1.4) as: < A S I J Q I F | HI.J |ASIJQIF> = C|[F(F + 1) - J(J + 1) - l(l + 1)]/2 (1.83) The interactions of electronic and nuclear spins are represented by the Hamiltonian: 26 Hi.s = b l S + c l S z (1.84) z with b = aF - c/3 where aF and c are the isotropic hyperfine directly constants, proportional The former to the quantity of electron nucleus, while between l interaction and S z (Fermi-contact) respectively. spinning z (1.85) the dipolar, and dipolar interaction density or bar magnet, at the interaction is the same as given in equation (1.67). of nuclear spin with the electronic orbital is The magnetic moment is a scalar product of I and L which is treated in the same manner as the L S (1.66). interaction described by equations (1.65) and The resulting Hamiltonian is t h e r e f o r e : 26 31 26 -ai L HI.L z (1.8.6) z in which a is the interaction constant. The b term of equation (1.84) is expressed in spherical tensor form as: H i s = bTl(l)Tl(S) To derive the matrix elements of (1.87) the interaction, I is first uncoupled from J by application of equation (1.48): <nASIJQIF| T1(I)T1(S) h A S T J Q I F > = , , , [l(l + 1)(2I + 1)]1'2 < n A S L J Q | | V(S) I h ' A S T ' J ' ^ ^ (1.88) where the [(l(l + 1)(2I + 1 ) ] term is the reduced matrix element of 1/2 T (l) according to equation (1.46). 1 By projecting the reduced matrix element in equation (1.87) from the space-fixed axis system to the molecule-fixed system, using Wigner rotation matrices as equation (1.47), the general matrix element can be expressed a s : in 3 1 <iiASUOIF| HT.S |ri ASTJ n IF> = , , , (-1)>+J'+F/F J |) [l(l + 1)(2I + 1)(2J + 1)(2J' + 1)]i/2 I ( - 1 ) J - « / J 1 J ' \ \ l I J'J q \-Q q Q'J ( - 1 ) S - I / S 1 S ' \ Z <S||Tl(S)||S'><TiAS|bih AS > , X The c l S 2 z and a l L z z (1.89) , Hamiltonians are treated by the same method. Evaluation of the 3-j and 6-j symbols with the appropriate A- A A A f o r m u l a e - , yields the matrix elements for bl-S, c l S 5 4 2 z except that the only matrix elements z and a l L , written for the a constants are those diagonal in A and X , respectively. A A z z and c The resulting matrix elements employed in the hyperfine analysis of NbN are as follows: 27 <JIFQIM| Hut |JIFQIM> . Qh R(J)/[2J(J + 1)] (1.90) <JIFftXM| H f |J-1,IFflIM> = h -h(j2-Q2)1/2p(J)Q(J)/[ J(4j2-1)1/2] (-|. 1) 2 9 <JIFQXM| Hhf |JIFQ±1,X±1,M> = b[(J+Q)(J±Q+1 )]1/2R(J)V(S)/[4J(J+1)] (1.92) <JIFQXM| Hhf_|J-1,IFQ±1,I±1,M> = +b[(J*Q)(J+n+1 )]1/2P(J)Q(J)V(S)/[4J(4J2-1 )1/2] (1.93) where the following abbreviations have been used: R(J) = F(F + 1) - J(J + 1) - l(l + 1) (1.94) P(J) = [(F - I + J)(F + J + I + 1)]l/2 (1.95) Q(J) = [(J + I - F)(F - J + I + 1)]i/2 (1.96) V(S)-[S(S + 1 ) - I ( I ± 1 ) ] l / 2 (1.97) The constant b is that given in equation (1.84), while h is used in the diagonal elements in order to incorporate the a, b and c constants into one: h = aA + (b + c)X I.D.3.a. (1.98) The sign of nuclear hyperfine coupling constants in transition metal complexes. I.D.3.a.i. The sign of the Fermi contact interaction. For an isotropic (Fermi contact) interaction involving only pure s electrons, the isotropic hyperfine constant aF is positive because the magnetic field generated at the nucleus by the interaction is in the same direction contributions to the as the electronic isotropic spin. hyperfine However, interaction negative occur when there are open shell d or p electrons which polarize s electrons in inner (filled) orbitals via an exchange interaction which promotes an 28 electron from an inner s orbital to an outer empty one. For 4 3 example, a ground electronic configuration with a single unpaired 3d electron, ¥ can mix with excited 0 = states (3s+)(3s-)(3d+) resulting from the promotion of an electron from a 3s to 4s orbital to produce the three functions: 43 ¥ 1 = (4s+)(3s-)(3d+) ¥2 = (3s+)(4s-)(3d+) ¥3 = (3s+)(4s+)(3d+) This is known as a configuration interaction, in which the ground and excited states possess different spin distributions yet form the basis for the same irreducible representation, in keeping with the requirement remains First order yielding an that the energy perturbation of the system theory expression for constant. 44 is applied to describe the mixing, the hyperfine contribution due to configuration interaction that is a function of the product of the ns and ms orbitals evaluated at the nucleus [ns(0)ms(0)], times an exchange integral J(ms,3d,3d,ns), divided by the energy separation between the ms and ns orbitals: 3 X=8TCS °° X [ns(0)ms(0) x J(ms,3d,3d,ns)]/(E -E ) m n (1-99) n-1 m=4 The quantity x ' s independent of c h a r g e 4 3 and is related to the isotropic Fermi contact coupling constant, aF, b y : a where g e and g n F = (2/3)geLi gnHnX B 4 4 (1.100) are the electronic and nuclear g factors and LIB and L i are the Bohr and nuclear magnetons. n The quantity [ns(0)4s(0)]/(E4- 29 E ) for the n • 1, 2, 3 s orbitals of the neutral atoms of the first row n transition metals from V to Cu was found to increase by about 20% across the series. The exchange integrals varied in the opposite sense, though more gradually, decreasing by an overall 14% from V to C u . 4 3 An alternative approach to the configuration interaction (CI) core (or spin) conceptualize differs from but CI configuration orbitals. polarization, is not a treatment which as in that the which theoretically orbitals originates from sound. wavefunction for the to core easier to This theory 4 4 to spin-dependent one-electron is therefore of the amount of spin density of each sign. configurations be involved belong The resulting hyperfine interaction independent may is represent polarization a single a function CI requires two spin- the model wavefunction. is a The spin-polarized unrestricted Hartree-Fock function (UHF) where UHF differs from the conventional, or restricted, Hartree-Fock one-electron wavefunctions are the orientation of the s p i n . 4 4 function in that the trial not required to be independent of The radial functions whose spins are being polarized, corresponding to spin up and spin down, differ from one another because they couple differently with the unpaired d or p electrons. The resulting hyperfine interaction is negative because the polarized spin has the opposite sense to the unpaired electron which induces the polarization. 44 30 I.D.3.a.ii. The sign of the interaction. The sign and dipolar nuclear of the dipolar magnitude hyperfine hyperfine interaction depends on the number and type of open shell d and p electrons. interaction constant for such an electron in orbital r\ i s 4 5 Cj = 3g UBg un<il|r- (3cos2e - 1)/2fo> (1.101) 3 e where 6 is the n angle between the The nucleus and the ith unpaired electron at a distance r; closed shell electrons do not contribute to <3cos 9 - 1>. Using for sake of illustration the ground electronic 2 4 X " state of V O , with the configuration ( a 7 i a 5 ) , there are three 2 4 1 2 n non-bonding o interaction. If 1 n 8 2 the open shell electrons contributing assumption is made that the to the interacting electrons are metal centered, the hyperfine constants a r e : (A i s 0 ) V 0 « (1/3)(A| )4so 46 (1-102) 80 (A ip)vo= (2/3)(A p)3d6 d IS (1-103) di where these A parameters are related to aF, b and c by: Aiso = A i + Adip = aF A ± = b = a F (1.104) - c/3 (1.105) Adip = c/3 (1.106) A|| = b + c (1.107) Combining equations (1.101), (1.103) and (1.106), the expression for c becomes: c = 3g UBgnM2/3)<3d5|r-3.(3cos 8 - 1)/2|3d8> 2 e (1.108) Using the algebreic expression for the spherical harmonic Y20 (see Section I.B) , the matrix element portion of equation (1.108) can be 47 written in terms of the n, I and m quantum numbers as: <nlm)r-3.(3cos e - 1)/2|nlm> = (1/2)<lm|3cos e - 1 |lm><nl|r-3|nl> 2 2 31 1(1+1) 3m2- (1.109) <r-3> | n (2l-1)(2l+3) For a 8 orbital, equation (1.109) reduces to (2/7)<r > i, producing a 3 n value for c (in c n r ) 1 of 4 6 c = -(4/7)g LiBgn^n<r- >3d/hc (1-110) 3 e When an electron is promoted from the 4so to 4pa orbital to produce the C Z * excited state, all three electrons contribute to the dipolar 4 term and c becomes (in cm- ): 1 c = 3g UBg M(2/3)<r- -(3cos2e - 1)/2> 8 3 e 3d n + (1/3)<r- (3cos2e-1)/2> ]/hc 3 4pa c = g ^BgnM-(4/7)<r- >3dS 3 e + (2/5)<r- > 3 4pa ]/hc (1.111) Using this method the different values for c corresponding to the various possible electron configurations of an electronic state can be estimated, which assists in the assignment of an electronic state. I.D.4. The The nuclear electric quadrupole interaction. nuclear electric quadrupole interaction involves two second rank tensors, representing the electric field gradient and the nuclear quadrupole moment. A simple quadrupolar Hamiltonian is method by which to derive the with the use of spherical harmonics and Legendre polynomials. To obtain the Hamiltonian for the electrostatic interaction of the nuclear quadrupole moment with the electric field gradient at the nucleus, a multipole expansion is made for the scalar coupling of the charges of the nucleons with those of the electrons. A multipole 32 expansion is a spherical harmonic expansion (or Legendre polynomial expansion) where the values of I in the spherical harmonic Y | m are referred to as monopole, dipole, quadrupole and octopole for I = 0, 1, 2 and 3 . By Coulomb's l a w , the electrostatic Hamiltonian is 4 8 49 H = Ieq /R n n which describes the interaction (1-112) n between n nucleons with charge q and an electron with charge e, with an electron-nucleon of R . n separation The electrostatic potential at the electron is n V =Iq /R n The distance R n (1.113) n is the resultant of the two vectors originating from the nuclear center to the nth nucleon (r ) and to the electron n with the angle between vectors r cosines and R denoted by 9 . n (R), The law of n gives the relation between R , r , R and 0 : 5 0 n Rn = (R 2 + r n n - 2Rr cose ) 2 n n = R[1 + ( r / R ) n 2 1 / 2 n - 2(r /R)cose ] n (1.114) 1/2 n By the generating function for Legendre polynomials , 51 [1 - 2 ( r / R ) c o s 9 n equation as: + (r /R) ] 2 n 1 / 2 n = I P|(cos0 )(r /R) n (1.115) 1 n (1.113) can be written in terms of a Legendre polynomial 5 2 V = X I P|COS(0n)q rn /R'+ l=0 n l (1.116) 1 n Each Legendre electronic and harmonic addition polynomial nuclear tensor theorem), (1.116) the multipole represents the operators scalar (from the producing from equations expansion: 48 52 product of spherical (1.112) and 33 A Hmultipole = e v * = H ( - 1 ) [ I (e/R' ) 1=0 m e +1 m C| (e (pe) x I qni-n' C|,. (0 (pn)] m ei M ni (1-117) n where the summations over e electrons and n nucleons represent terms in electronic (8 ,(pe) and nuclear (0 ,<pn) angular coordinates, e n respectively. The first term in this expansion which is non-vanishing describes the quadrupolar interaction. The I = 0 term can be represented by Z e V , or the Coulombic interaction between the nuclear charge and 0 the electrons, and is included in the electronic Hamiltonian. 53 The dipole term, I = 1 , is the product of the electric dipole moment of the nucleus, which is zero, and the electrostatic field of the electrons, which is invariant over the nuclear volume and therefore produces the no interaction. interaction 53 The I = 2 quadrupole term, however, is of the nuclear electric quadrupole moment, Q , with the electric field gradient ( V E ) experienced by the nucleus due to the charge distribution of the electrons. For those nuclei possessing a quadrupole moment, then, the quadrupolar Hamiltonian is the scalar product of these two tensor quantities: H Q = 54 -T2(VE)T2(Q) (1.118) where the minus sign is present due to the negative charge of the electron. The quadrupole moment is a measure of how spherical the nucleus is, as indicated by the value of the nuclear spin, I. nuclear charge 3z 2 distribution - (x2 + y2 + z ) angular coordinate). 55 spherical symmetry or 3cos e n 2 2 a v e from a v e The deviation of 1 (where 8 N is given by: is the nuclear This value is non-zero if I is greater than 34 1/2, which odd nucleons (i.e, differences in the number of neutrons with respect to protons). The mechanism is dictated giving by the rise to number specific of values of I is imperfectly understood, though it seems to approximate the same shell model that applies to electrons. Thus, zero spin results from spin-pairing if the number of protons (Z) equals the number of neutrons (N), and predictions for I can usually be made for nuclei possessing odd N or Z based on the number of particles occupying open s h e l l s . 55 By convention, the nuclear electric quadrupole moment is defined classically a s 1 1 Q = ej(3z2 - r2)p(r)dx (1.119) where p(r) is the nuclear charge density, and dx denotes over the nuclear becomes: volume. Quantum mechanically the definition 52 Q = e-lXq r 2(3cos20 n n The integration quantum mechanical n observable n - 1) (1.120) corresponding to equation (1.120) is the nuclear quadrupole moment, Q, defined by convention as 5 4 Q - <l,mi=l| Q |l,mi=l> (1-121) A. The definition of Q was made prior to the invention of spherical tensors and therefore expressions lacks the factor of 1/2 needed for the P2(cos9) - T ( X ) = (3cos e - 1)/2; Q was also defined 2 2 0 without the electron charge e. The spherical tensor definition is therefore T2 (Q) = eQ/2 0 with the corresponding scalar quantity (1.122) 35 eQ/2 = <l,mi=l|T2 (Q)||,mi=l> (1.123) 0 The quadrupole tensor, from equation (1.117), is of the form T2(Q) = Iq r2 C2(e ,(pn) n n The (3 V/3z2) (1-124) n n electric field gradient (EFG) evaluated at the nucleus, has the spherical tensor form (from equation 1.117) of: 2 0 l -T2(VE) = ZeR-3C2(e ,cpe) (1 -125) e e with the corresponding field gradient coupling constant defined as q = <j,mj=J|(a2V/az2) |J,mj=J> (1.126) 0 where (d \lldz ) 2 required 2 0 = eR-3(3cos6 - 1). e Thus, with the factor of 1/2 by the spherical harmonic definition of the quadrupole moment, the E F G tensor can be expressed as: -T2 (VE) = q/2 (1-127) 0 To derive the matrix elements for the quadrupolar interaction (equation 1.116), equation (1.48) is applied to evaluate the scalar coupling of two commuting tensor operators in a coupled basis (I must be unravelled from J ) : ^ ' A ' l S T i J ' n ' I F I H Q |nA;SX;JQIF> = (-1)J+I+F5 F/F I F 12 Then project J'^Tl'A'jJ'Q'H -T2(VE) ||riA;JO><l|| T2(Q) |||> (1.128) J 1/ T (VE) 2 from space- to molecule-fixed axes with equation (1.29): ^ • A ' j J ' Q ' H ^ V E J I h A i J ^ = X<J'n'|| D2. *( pY) ||J«><Ti'A ||-T2 (VE)||TiA> , q a q q = X(-1) '- '[(2J+1)(2J'+1)]1/2/j' J Q q The last term of equation 2 \a' q J W A ' H -T2 (VE) | h A > a) q (1.129) (1.128) is evaluated with the Wigner- Eckart theorem, in conjunction with equation (1.123): 36 <l,mi=l| T 2 ( Q ) ||l,mi=l> = eQ/2 « (-1)'-'/ I 2 l\<l|| T2(Q) ||l> 0 V-l Substituting for the 3-j s y m b o l 57 (1.130) 0 \) and solving for the reduced matrix element gives <l|| T2(Q) ||l> = eQ!2( I 2 V-IO I/ = eQ/2 [(2l+1)(2l+2)(2l+3)/2l(2l-1)]l/2 In terms of the molecule-fixed T ( V E ) 2 (1.131) tensor in equation (1.129), the coupling constant q is defined by the diagonal reduced element of T2(VE): <A||-T2 (VE) ||A> = q/2 (1.132) 0 A first order approximation was made in the current study to neglect the ± 1 and ± 2 components of T ( V E ) , that is, to exclude 2 quadrupole matrix elements combination of equations off-diagonal (1.128), (1.129), in Q. (1.131) Appropriate and (1.132) therefore yields the matrix elements ^ ' A ' j S T l J ' n ' I F I -T2(VE)T2(Q) |riA;SI;JftlF> = (1/4)eqQ(-1)J+'+F ff I \2 J J j [(2l+1)(2l+2)(2l+3)/2l(2l-1)] /2 ,N 1 I / x l ( - 1 ) J ' - « [ ( 2 J + 1 ) ( 2 J + 1 ) ] l / 2 / J' 2 J \ q \-Q' q , , ClJ (1.133) From the triangle condition for a 3-j symbol, which states that the third J value must not lie outside the sum and difference of the first two J v a l u e s , the 3-j symbol in equation (1.133) requires A J to be 18 0, ± 1 or ± 2 . From equation (1.133) and these selection rules, the specific matrix elements employed in this work are as follows: 37 <JIFQIM| H Q |JIFQIM> = eQq[3fl2-J(J+1 )]{3R(J)[R(J)+1 ]-4J(J+1 )l(l+1)} 81(21-1 )J(J+1 )(2J-1 )(2J+3) (1.134) <JIFnlM| H Q |J-1,IFQIM> = -eQq3n[R(J)+J+1](j2-fl2)l/2P(J)Q(J) 2J(2J-2)(2J+2)(2I-1 )(4J2-1) 112 (1.135) <JIFQIM| H Q |J-2,IFQIM> = eQq3[(J-1)2-fl2]1/2(j2-fl2)l/2p(J)Q(J)P(J-1)Q(J-1) 41(21-1 )4J(J-1 )(2J-1 )[(2J-3)(2J+1 )]1 /2 The terms R(J), P(J), Q(J), P(J-1) and Q(J-1) (1.136) are as in equations (1.93), (1.94) and (1.95). I.D.5. A - D o u b l i n g . The phenomenon of A-doubling results from the breakdown of the Born-Oppenheimer electronic which and occurs approximation, nuclear when motion.26 molecular which allows the separation of it is the lifting of +A d e g e n e r a c y rotation interferes with the well- defined quantization of the z component of electronic orbital angular momentum about the molecular axis. rotational Hamiltonian components of the The operators in the spin and responsible for A-doubling are the x and y electronic orbital angular momentum which produce matrix elements off-diagonal in A . In the rotational A A Hamiltonian, this is the L-uncoupling operator, - 2 B J L . spin-interaction terms of the Hamiltonian, is used, yielding the complete A-doubling the Among the spin-orbit Hamiltonian: operator 57 V = - 2 B J L + Xaji-Si The A-doubling interaction operators is treated by degenerate (1-137) perturbation t h e o r y , which for A states must be taken to fourth order in order 5 8 38 to connect |A = 2> to |A = -2> via states with A = 1 and 0 (i.e., IT. and X states). For this reason the interaction is smaller than that in n states, since the mixing of |A = 1> and |A = -1> states requires only second order perturbation contains those terms theory. The unperturbed 57 adhering to the Hamiltonian Born-Oppenheimer approximation which are diagonal in A and independent of the orbital degeneracy. fourth-order out the The perturbation can be treated through the use of a effective Hamiltonian, which is obtained by subtracting unperturbed energy from the complete Hamiltonian expression to leave an effective Hamiltonian which operates only on the vibronic state of interest, |l k> 5 7 0 ' 5 9 Heff< > = P V ( Q o / a ) V ( Q / a ) V ( Q o / a ) V P - P V ( Q o / a 2 ) V P V ( Q o / a ) V P o 4 0 0 0 0 0 - P V ( Q o / a 2 ) V ( Q o / a ) V P V P o - PoV(Qo/a)V(Qo/a2)VPoVPo 0 0 + P V(Qo/a3)V P V P V P 0 The operator P , extending 0 0 over the 0 (1.138) 0 k-fold degeneracy of l , is 0 defined as Po = I | l k x l k | k 0 (1.139) 0 while (Qo/a )= I I | l k x l k | / ( E - E ) l=l k n 0 (1.140) n n 0 where I denotes any vibronic state with energy E|, E 0 is the energy of state <l k|, and k labels all rotational, spin and electronic quantum 0 numbers in a vibronic state l The (H ot) 2 A r form 0 or I. Hamiltonian in equation (1.137) has 2A+1 terms of the form n ( H . . ) , where n ranges from zero to 2 A . n s 0 it is written: 57 In the case (a) 39 HL.D..A = m ( S 4 + S.4)/2 - n ( S 3 j + + S.3J.)/2 A + A + 0 (S 2j 2 A + + q (j 4 A where the notation on to m Mulliken A and are S 0 + .141) included to be consistent with the 2 A r o with ( H . . ) . + (1 Christy accompanies ( H t ) 2 A A §.2j.2)/2 - p ( S J 3 + SJ.3)/2 + j.4) + factors of 1/2 of parameter + + + 60 for II states. Thus the A A . PA is with ( H t ) " ( s . o . ) and so 2A 1 H ro The number of those parameters that can be determined equals the spin multiplicity up to a maximum of 5. 4 q In a state, for example, only four of the five parameters are included in the A-doubling matrix elements, with m A excluded because the spin-orbit interaction need not be extended to fourth order. In a 4 A state where there are four Q, substates, the terms appear in the 4 x 4 matrix as + different parity terms which split a given level into two parity, labelled e +(-1)J- and / k and f. By convention, the e levels of levels have levels have parity -(-1)i- , where k is 1/2 and k 0 for half-integer and integer values of spin, r e s p e c t i v e l y . ' 62 magnitude 4 61 4 of the A-doubling observed in this work in the The A7/2- A 7 / 2 transition of C o O ranged from 0.2 to 1.2 c m - , while that in 1 the n 3 0 state of NbN is on the order of six wavenumbers. 40 CHAPTER II THE COMPUTERIZED LASER-INDUCED FLUORESCENCE EXPERIMENTS II.A. Experimental The Coherent Details. laser excitation Radiation model experiments were performed CR-599-21 using a scanning single frequency (standing wave) dye laser, pumped by a Coherent Radiation model lnnova-18 argon ion laser operated at a wavelength of 514 nm and a power of 2.0 to 3.5 normally 100 to 150 selecting dye. 6 3 W. Output power from the mW. portions of the dye laser was The tunability of the laser comes from broad fluorescence band of an organic Two dyes were employed for both the cobalt oxide (CoO) and niobium nitride (NbN) studies. For maximum output at 590 nm (ranging from 570 to 620 nm or 17540 to 16130 cm- ), the dye used 1 was rhodamine 6G (Exciton Chemical Co.), with the structure -OC H 2 63 5 ^v-CHj ^C H 2 made to a concentration of 2 x 1 0 the lower energy regions, the M in ethylene glycol. -3 dye DCM methyl-6-p-dimethylaminostyryl-4H-pyran, Co.) To reach (4-dicyanomethylene-2- from Exciton Chemical was dissolved in 3:7 benzyl alcohol to ethylene glycol to form nearly saturated 2.5 x 10" 514 5 nm, DCM's maximum 3 M solutions. output power At a pump wavelength of occurs at 640 nm, and 41 broadband laser operation (16670 to 14390 cm" ). range 600 leads to bubble formation, so the solution was minimize bubbling by running the dye tubing through a of dry ice mixed to 695 nm The benzyl alcohol required to dissolve the 1 DCM occurs over the with a 1:3 solution of water cooled to -30 ° C slush to CaCl2- All chemicals were used as obtained. A small fraction of the output beam was diverted to an iodine absorption Another or emission fraction Fabry-Perot was cell sent interferometer for to a with absolute Tropel a 299 frequency fixed-length MHz calibration. semiconfocal free-spectral range, providing a common ladder of frequency markers against which the sample and iodine spectra could be referenced. The beam containing the majority of the output power was passed down the axis of the stream fluorescence (LIF) photomultiplier tube of sample detected at molecules, with the right angles to the longitudinal laser-induced beam equipped with a high transmittance with low a pass optical filter to reduce scattered light, and powered by 300 to 500 V from a high voltage power supply. Phase-sensitive detection was acheived with a Princeton Applied Research (PAR) model 128A lockin amplifier receiving chopped sample and reference signals, with the reference beam supplied by a Spectra-Physics model 132 Lablite HeNe gas laser. The resolved fluorescence experiments were performed with a 0.7 m Spex Industries model 1702 spectrometer which dispersed the spectrum onto the detector elements of a microchannel-plate intensified array detector (PAR model 1461), mounted at the output end of the spectrometer. The spectral window of the array detector was calibrated with a Burleigh model WA-20VIS wavemeter. 43 II.B. Intermodulated Fluorescence. A laser-induced fluorescence transition has a Gaussian velocity population profile forming an because of the Doppler effect, inhomogeneously broadened line, the freqeuncy absorbed by molecules moving away from the light source appears to be lower than that absorbed profile by molecules moving toward (zero velocity) shifted; that is, the the transition it. At the frequency Q center molecules have zero velocity with respect to 65 (or the is not Doppler- the light wave with which the molecules i n t e r a c t . ' free of "sub-Doppler") spectroscopy, two beams) with frequency w propagate the sample gas molecules. travelling 66 In Dopplerwaves in opposite directions (laser through Molecules moving with velocity v along the axis of the laser beams absorbs radiation from one beam at a frequency ft = co(1 + v/c), and from the other beam at ft = co(1 - v/c). These opposite Doppler shifts cause each beam to depopulate portion of the lower state velocity profile symmetrically about profile center at v = c(ft ± co)/ft (see Fig. 2.1). a the This depletion is termed "burning a Bennet hole", creating a homogeneous profile in the lower s t a t e . approaches the As the laser is scanned, and the laser frequency 66 non-Doppler-shifted resonance frequency, the two Bennet holes converge until they meet at the center, or zero velocity (see Fig. 2.1). a The resulting lower state population depletion causes corresponding depletion in the intensity profile of the fluorescence, called a "Lamb dip". Intermodulated fluorescence (IMF) is a technique which enables relatively small Lamb dips to be detected against the large Dopplerbroadened profile so that they are directly measured as spectral 44 F i g . 2.1. G a u s s i a n inhomogeneously Doppler-broadened velocity (v ) population (n) profile, showing two B e n n e t holes (solid lines) which c o n v e r g e at zero velocity (dotted line) to form a L a m b dip in the profile of intensity v e r s u s laser tuning f r e q u e n c y . z 6 6 45 peaks. The two laser beams are modulated (i.e., chopped to produce certain phase trains) with frequencies amplifier, with the phase sensitive frequency of fi + f , fi and detector f. The 2 referenced passes only (fi + f2)-modulated 2 laser beams 66 a A schematic used to obtain the niobium sub-Doppler spectra is illustrated in Fig. 2.2. counterpropagating to input signals, such as those occurring when two Bennet holes m e e t diagram of the IMF experiment lock-in nitride In practice, the two must be slightly misaligned from one another to avoid feedback into the laser. A LIF signal normally arises from Bennet holes caused by allowed A F = A J transitions meeting at the velocity profile center. However, Lamb dips also originate from holes burned by "forbidden" A F # A J transitions meeting at the center. Since the selection r u l e s 24 on F and J are A F = 0, ± 1 and AJ = 0, ± 1 , transitions with A F = A J ± 1 and ± 2 are also possible. For a Q transition, with A J = 0, the F selection rule requiring that A F = 0, +1 allows the transitions rQ (AF = AJ + 1), qQ (AF = AJ) and pQ (AF = A J - 1). If A J = +1, A F = +1, 0 and -1 corresponds to the transitions rR, qR and pR (or A F = A J , AJ - 1 and AJ - 2). The same occurs for P branches where A F = A J , AJ + 1 and A J + 2 lines (pP, qP and rP) occur. observed only at low These satellite branches are values of J because the transitions is proportional to the angle between F. 6 7 intensity of the the vectors J and When J and F are large with respect to I this angle approaches zero, and only A F = AJ transitions are observed. 9/2 for the nuclear spin of Nb allows A F * A J The large value of transitions to be seen at higher values of J than is normally possible. Accompanying a pair of A F = AJ and A F = AJ ± 1 transitions, or a 46 Discharge in flow system Chopper Tunable dye laser Refe 'ence signal PMT Lock-in Fabry - - Perot spectrum PDP-11/23 Microcomputer \, 3-pen chart recorder < ^ c 11 PMT y calibration ' Interpolation markers Fig. 2.2. Schematic drawing of the intermodulated fluorescence experiment used in this laboratory. The discharge cube where the sample and laser light are combined is shown in the top left corner. 47 A F = A J ± 1 and A F = AJ ± 2 pair, may be a "crossover resonance" occurring exactly mid-way between the two. Such a phenomenon requires that the two transitions sharing a common level lie within the same Doppler profile. spectra of the nearly hyperfine components. Crossover resonances occur in the coincident transitions of closely IMF spaced The means by which crossover resonances are generated is depicted in Fig. 2.3, with a schematic stick drawing of the resulting spectrum. AF=AJ=0 b) X X 1 1 AJ-I 1 AF= AJ | I 1 + 1 Fig. 2.3. a) The formation of crossover resonances (Fi + A and F + A-i) as the result of allowed AJ = A F transitions (Ai and A ) occurring within the same Doppler-broadened velocity profile as forbidden AJ * A F transitions (Fi and F ). The diagram shows the laser scanning toward the nonDoppler-shifted A F = AJ transition (occurring at A i + A ) and beyond toward higher frequency to the A F = AJ + 1 transition (Fi + F ). If the F's and A's are exchanged, the first central Lamb 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 = AJ = 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. 2 2 2 2 2 2 49 II. C. Computerization. Part of the work for this thesis involved computerizing all stages of the Doppler-limited and intermodulated fluorescence (subDoppler) LIF experiments RSX-11M operating on a PDP-11/23 system. These microcomputer with an stages included: 1) laser scanning, and data acquisition and storage; 2) peak finding; and 3) frequency calibration. All of the software Each stage comprises a separate program. was written with F O R T R A N - 7 7 except for the laser scanning and data acquisition, programmed in M A C R O . The PDP-11 computer is structured such that space for executable code is quite limited. overlaid. allows This constraint required that the three programs be Overlaying is a method of memory the sum of the individual management subroutines to memory limitations of the computer. far exceed A remainder resides in the the When an overlaid program is executed, only a portion of the subroutines are sent into while the which relatively memory, limitless disk space. set of overlay directives is written which describes the program in terms of a calling "root" segment and any number of subprogram "branch" segments; the branches may themselves call "subbranches". The computer uses these directives to build the task file such that during program execution the memory space at any given time is occupied only by the root segment and the branch being called at that time. Since the main responsibility of the root is to call subroutines in the branches, the root is made as short as possible to allow most of the program execution. software to remain disk-resident throughout 50 The heart of the first program is the M A C R O routine which orchestrates laser scanning and data acquisition via its control of the following hardware • The peripheral devices: 16-bit digital-to-analog converter (D/A), which sends a voltage ramp to the laser so that it scans a range of up to 1.4 c m - . 1 • The containing 4-channel, two registers to status register (CSR) sample, 12-bit iodine and analog-to-digital converter process incoming data. The (A/D), control receives the voltages (data points) from the interferometer detectors. The buffer preset register receives the point from the C S R , stores it temporarily, delivers it both to the 12-bit output storage buffer for transfer to disk. D/A and to the then appropriate The sample spectrum is signal averaged over four points prior to transfer to the buffer. • Three 12-bit D/A's, which send the three data points to the chart recorder for a hardcopy of the spectra. • The real-time (crystal-oscillator) peripheral devices are interrupt-driven rate producing a resolution clock, by which the above to operate at a user-chosen compatible with the lock-in constant and the frequency range scanned by the laser. time The use of interrupts ensures that the task will be serviced by the computer's central processing unit exactly as dictated by the clock. The interfacing of computer schematically in Fig. 2.4. scanning is complete, and experiment is illustrated Upon return from the M A C R O routine after the three spectral vectors are stored in unformatted files with the first record of the sample file serving as a housekeeping experimental record parameters. containing spectral identification and 51 PROGRAM 12-bit A/D: 16-bit D/A Interface 12-bit D/A Chan 1 Chan 2 0 1 2 BNC Connections Fabry-Perot Interferometer Si-diode Detector Fig. 2.4. Schematic diagram of the laser-induced fluorescence experiment and how it is interfaced to the PDP-11/23 microcomputer. 52 CHAPTER III ROTATIONAL ANALYSIS OF THE RED SYSTEM OF COBALT OXIDE III.A. Introduction. In German-occupied Belgium during World War II, Malet and Rosen observed a number of electronic bands of gaseous cobalt oxide (CoO) 5000 and 10000 A using the exploding wire technique. between The lower state v"=0 and v"=1 vibrational frequency (i.e., the separation of 69 the vibrational states) was found to be 840 c n r , and this 1 state was assumed to be the ground electronic state. The next spectroscopic experiments on C o O came years later, in 1979. The first was a low resolution infrared spectrum of C o O (with ± 0 . 2 c m - 1 line precision) obtained with a microwave discharge source, giving a vibrational frequency constant e A.70 (B ) of 842.2 cm- , 1 an equilibrium rotational of 0.522, and an equilibrium bond length (r ) of e The absence of a Q branch in the spectrum led to the tentative assignment of a X ground s t a t e , though the possibility was not 24 ruled out that the spectrum was that of a low-lying excited A 1.60 matrix isolation infrared study followed state. 70 shortly afterwards , in 71 which cobalt from a cobalt cathode sputtering source and oxygen were codeposited at low temperature argon. (14 K) into a solid matrix of The ground state vibrational frequency was measured in this work to be 846.4 c m " . 1 In the next year, matrix isolation electron spin resonance (ESR) studies of a large group of transition containing molecules with including CoO. matrices and 7 2 the high spin multiplicities were metal- reported, In spite of high concentrations of C o O within the expertise of the laboratory in conducting 53 experiments of this type, no C o O ESR signal was observed. CoO was therefore concluded to possess an orbitally degenerate ground state, because orbital degeneracy in linear molecules (in matrices of low enough temperature that only the ground state is populated) causes a g tensor anisotropy so large that the spectrum is spread out over such a large magnetic field that it cannot be observed. The E S R spectrum of a paramagnetic 1 state, on the other hand, will possess little or no g anisotropy and will exhibit only a small deviation from the free electron value, g of the overall orbital deduced from the frequency, n B e = 2.0023, due to the spherical symmetry angular relation momentum. ' 7 2 The value of g is 7 3 hv = giiBH, where v is the resonance the Bohr magneton, H the applied magnetic field, and h Planck's constant; the g anisotropy is taken as g_i_ - gn. No further work has been published on C o O since this E S R study, leaving the ground state of the molecule to be the only one of the first row transition metal oxides yet to be established. Field-free atomic orbitals of a diatomic transition metal molecule are split by the axial field of the other atom, as shown in Fig. 3.1. From the electron configurations of manganese, iron and nickel monoxides, MnO (4SO)1 (3d5) (3d7i) FeO (4SC)1 (3d5) (3d7i) NiO (4sa) 2 2 3 2 2 (3d8) (3dn) 4 2 , it can be seen that there are two possible candidates for the ground electronic state of C o O . 4so If the seventh valence electron occupies the orbital, the spin multiplicity and direct products given by the resulting o 8 r c configuration produce a 2 3 2 4 A j electronic s t a t e ; if 74 54 M orbitals MO orbitals 0 orbitals 4po4p 4ptr ~7 "*" = / / / 3 d * / 3 d 3drr / / / _4j j,' V --- // 3d* N \ \ 4s<* , \x \ x \ \ \ 2pff \ \ \ \ \ A \ \ v \ \. 2p == 2 pa \ \ \ \ \_ 2s<J 2s Fig. 3.1. Relative orbital energies of a diatomic 3d transition metal oxide. The ordering of the 3d5 and 4sa molecular orbitals is variable. 92 55 instead it fills up the 8 orbital, a uncertain orbital 75 ordering 4 X~ state 4so orbital self-consistent-field calculations ground s t a t e . symmetry the X" left the problem in the hands configuration MCSCF) of 4 76 4 0 with However, respect to the of the theoreticians. complete active the 3d8 Multi- space (CAS on FeO were extrapolated to C o O to predict a Weltner, however, first predicted a state of A based on trends predicted a results. in the other TM o x i d e s , then 7 7 ground state based on E S R experiments . 72 later It was from this stage of development that the current study proceeded. 56 III.B. Experimental III.B.1. Synthesis of gaseous cobalt oxide. Cobalt oxide was made in a Broida-type oven a s s e m b l y follows: as an alumina crucible containing cobalt metal powder (Fisher Scientific C o . ; 0.14% tungsten 74 basket. Ni, 0.11% Fe) was heated resistively The basket was enclosed in a radiation in a shield comprising an inner ceramic sleeve enveloped by an outer copper sleeve and fitted lid, with zirconia felt packed very tightly around the basket. To produce cobalt oxide (CoO) in quantities sufficient for measurable fluorescence, temperatures point of the alumina crucible (1920 approaching the °C) were excess of cobalt's melting point of 1495 ° C . required, melting well in C o O was formed in the gaseous stream of vaporized cobalt atoms, argon carrier gas and molecular oxygen at a pressure of roughly 1 Torr, with a ratio of approximately 150(±15):1 argon to oxygen. Fluorescence, however, occurs only in the presence of laser excitation, which is as with NiO in which only the ground state is populated by the reaction of metal and O 2 . 7 8 Unlike the production of C u O 7 9 , which is more efficient with N 2 O than O2, no C o O fluorescence was observed using N 2 O as the oxidant. The requirement of high temperature drastically hampered the efficiency of C o O synthesis in two ways. First, there was aluminate) extensive formation of Thenard's B l u e 8 0 (cobalt deposits on the crucible and on the surface of the liquid cobalt; this phenomenon was also reported in 1966 by Grimely and coworkers who heated solid C o O in an alumina cell to high Second, produces the an reaction alloy of that cobalt renders vapor the with basket the very temperatures . 81 tungsten basket susceptible to 57 cracking, with breakage occurring after at most three heatings of a basket assembly. III.B.2. The spectrum. The laser excitation spectrum of gaseous C o O was over the range of 7000 to 5800 A at Doppler-limited described in Section II.A. It is evident that the further to both higher and lower energies. to those we have sometimes varied dramatically the superior sensitivity measured, between provided by the additional bands were observed. measured from without the portion a broadband intracavity of the The laser assembly), spectrum rotationally range from 15450 to 15790 c n r 1 system 69 though extends correspond in the intensities the two techniques. With LIF method, a number of most prominent spectrum are resolution, as The bands observed by Malet and Rosen with the exploding wire technique frequency investigated listed analyzed (i.e., ones, as one in Table obtained 3.1. The thus far covers the (6470 to 6335 A), which includes three red-degraded bands whose heads lie at 15778 c m - (6338 A), 1 15598 c m - (6411 1 A) and 15538 c m - 1 (6436 A). spectrum of this region is shown in Fig. 3.2. The broadband 58 Table 3.1. The most prominent bandheads in the 7000 to 5800 A broadband emission spectrum of gaseous C o O . Values are accurate to roughly ± 3 c m - , with band strength denoted by: s = strong, m = medium, w = weak. 1 Wavelength group 5920 A 6120 A 6320 A 6650 A 6900 A Wavenumber 16916 m 16366 w 15832 w 15296 vw 14704 w and intensity 16846 s 16322 s 15778 s 15228 w 14477 m 16256 m. 15597 w 15036 m 14469 s 16088 w 15538 m 15004 s 0"= 7/2 5/2 7/2 Fig. 3.2. Broadband laser excitation spectrum of the three bands of gaseous C o O analyzed in this work (linewidths are on the order of 1 cm- ). 1 60 III.C. Analysis. III.C.1. Rotational analysis of the 6338 A subband. Ill.d.a. Rotational constants and hyperfine structure. The strongest band, at 6338 A, was the only band of the three for which a complete analysis was possible, given the available The line assignments, listed in Table 3.II, state combination wavelength-resolved verify that certain differences. For were made using lower added assurance, fluorescence experiments lines possessed data. were common upper some performed levels. to For example, if a pair of lines with a common upper level, such as Q(J") and R(J"-1), result are excited, the fluorescence pattern of the R line excitation will be identical produced as a to that obtained from the Q line, barring changes in the scattered laser light at the excitation wavelength. Lower between value, state combination differences lines with a common upper state thereby structure: providing information on measure differences that differ in their J " the lower state energy 1 AiF"(J) = R(J) - Q(J +1) = Q(J) - P(J + 1) (3.1) A2F"(J) = R(J - 1) - P(J + 1) (3.2) From the definitions of R, Q and expressions, it can be shown t h a t P, and from the energy level 83 AiF"(J) = 2B"(J + 1) - 4D"(J + 1) A F"(J) = (4B" - 6D")(J + 1/2) - 8D"(J + 1/2) 2 (3.6) 3 3 (3.7) The lowest Q line of this band was assigned as J ' = J " = 7/2, using the average of the A-|F" combination differences from the first R and P lines, and a rough estimate of 0.5 c m - 1 for the value of B. The 61 possible electronic states corresponding to a value of ft of 7/2 are 4 A and 0 , but only the A state has an electronic configuration that 2 4 can reasonably be expected to belong to the ground state. The three subbands analyzed in the current work demonstrate that the most intense C o O transitions are those with ft" = 7/2. must come from state. the lowest the spin-orbit component of the ground Since the spin-orbit manifold must be inverted for its lowest energy component to be 7/2, The Presumably these the electronic state is assigned as A j . 4 relatively low intensity of the Q lines transition vibrational as parallel, or level can definitely ft' 1 (see Fig. 3.3), = ft" = 7/2. The fluorescence excited: strong fluorescence was observed 851.7 c m - the Q(3.5) experiments where the transition, but nothing to the blue. isotopic labelling studies, such as with C o 1 8 available on the upper state quantum number. to the red of the lying vibrational 6338 A lower be assigned as v" = 0, resolved band indicates levels; on this basis, the line was to the red of In the absence of 0, 83 state based on Q(3.5) 1 identifies no information is Extensive structure that there are lower upper state vibrational quantum number is suggested to be at least two. The lower state rotational constants B and D were calculated by least squares from the equation (3.7). Table 3.II A F " combination 2 The A F " ( J ) 2 difference combination differences formula are in given in along with the assigned lines of the 6338 A band. Using these B" and D" values to calculate the lower state energy levels, the upper state energy levels were calculated; then a least squares fit to the expression E(J) = T + BJ(J + 1) - DJ2(J + 1)2 0 (3.8) to «sr' m LAW U 3.5 6.5 5.5 8.5 10.5 12.5 7.5 3.5 4.5 5.5 I I 4.5 R I i III 15.5 10.5 8.5 6.5 Q Fig. 3.3. Bandhead of the Q' = Q" = 7/2 transition at 6338 A, exhibiting the broadening due to hyperfine interactions and the weak Q branch signifying a parallel transition. The weak background is reproducible. 63 Table 3.11. Assigned lines from the 6338 A band CoO with the lower state combination differences, A asterisk denotes a blended line. J" R ( 4 A 7 / 2 ) of in c n v . An A 7 / 2 4 1 2 F H , A F" P Q - 2 103 iO-C 3.5 157747* 15771.060 45 15774.7* 15770.2* 15766.64* 5.5 15774509 15769.2* 15764737 6.5 15774110 15768.002 15762.704 14.015 7.5 15773.526 15766.64* 15760.494 16.014 -2 8.5 15772.760 15765.024 15758.096 18.011 -6 9.5 15771.810 15763.259 15775.515 20.013 -5 10.5 15770.668 15761.291 15752.747 22.015 -3 11.5 15769.343 15759.158 15749.791 24021 2 12.5 15767.828 15756.829 15746.647 26.024 6 13.5 15766.126 15754316 15743.319 28.017 145 15764236 15751.615 15739.811 30.019 2 15.5 15762.159 15748.715 15736.107 32.018 2 16.5 15759.892 15732.219 34011 -3 17.5 15757.445 15728.147 36.012 18.5 15754806 15723.880 38.006 19.5 15751.994 15719.439 40.005 1 15714.801 42.000 0 20.5 21.5 15749.350 15747.947 15748.821 15749.039 15745.600 15745.906 15709.994 I -l 1 -2 43.995 15745.720 22.5 15742.273 15742.345 15705.347 15703.946 45.987 -3 15704829 15705.045 23.5 15738.696 15738.769 15699.615 15699.914 47.984 1 15699.735 245 15734923 25.5 15735.005 15694290 15694360 49.980 4 15730.951 15731.051 15688.720 15688.786 51.975 6 26.5 15726.783 15726.911 15682.948 15683.030 53.968 8 27.5 15722.428 15722.592 15767.986 15677.080 55.956 6 285 15717.883 15718.106 15670.826 15670.954 57.938 -2 64 Table 3.11. continued. J" 1 P R A F" 2 103 O-C 29.5 15713.130 15713.500 15664491 15664654 59.929 30.5 15709.131 15708.185 15657.951 15658.178 61.908 -9 0 15707.960 31.5 15703.047 15703.110 15651.226 15651.589 63.903 0 32.5 15697.705 15697.888 15645.226 15644.282 65.890 l 15644.057 33.5 15692.171 15692.441 15637.158 15637.220 67.870 -4 34.5 15686424 15686.786 15629.837 15630.015 69.861 3 35.5 15680.487 15680.945 15622.311 15622.580 71.836 -5 36.5 15674.251 15674893 15614.591 15614.950 73.820 -2 37.5 15667.933 15668.678 15606.670 15607.122 75.787 -15a 38.5 15661.347 15662.251 15598.464 15599.106 77.783 39.5 15654.547 15665.638 15590.150 15590.896 79.752 -8 40.5 15647.498 15648.844 15581,594 15582.500 81.738 2 41.5 15640.453* 15641.831 15572.810 15573.900 83.713 1 42.5 15634643 15563.785 15565.131 85.684 -2 43.5 15627.272 15554769 15556.159 87.641 •-183 44.5 15619.712 15547.002 45.5 15611.954 15537.657 89.615 -158 91.591 -98 46.5 15604.010 15528.121 47.5 af\lot included in the least squares fit. 15518.405* 1 65 was used to obtain B' and D' from the unperturbed levels with J ' = 5.5 to 19.5. upper The results appear in Table 3.III. Note since the state B value is only 81% of the lower state, by the relation r"/r' = (B7B") / 1 electronic 2 the C o O bond length increases by a full 10% upon excitation. The hyperfine structure in C o O arising from the of 7/2 that follows the case (ap) decrease with pattern increasing where rotation, approximation by equation (1.90): 5 9 the Co nuclear spin hyperfine described to widths a first 85 E fs - ft[aA + (b + c)X](1/J){[F(F+1) - J(J+1) - 1(1+1 )]/2(J+1)} (3.9) h (In case (bpj) coupling the hyperfine widths are independent of N for each of the spin components.) The hyperfine splitting in the P lines is found to be wider than that in R lines of the same J " , while P and R lines possessing the same upper state J are of comparable widths. Since a upper comparison of P and R lines of the same J" demonstrate state properties, while those with equal J ' represent the lower state, it can be seen that the hyperfine interactions produce larger splittings in the upper state than in the lower state. equation (3.9) it can components of a rotational higher values of F. low J lines, for also Partially instance component is on the be line seen that the eight From hyperfine will be more widely spaced at resolved hyperfine splittings in some P(5.5), show that the high frequency side. highest F value This ordering of the hyperfine components shows that the change in the Fermi contact parameter, b' - b", is positive. 84 66 Table 3.III. Rotational constants for the analyzed bands of the red system of C o 0 . a B 10?D Upper Levels: 6338 A, Q = 7/2 15772.513 + 3 6411 A, ft = 5/2 a+15594.974 ± 2 6436 A, Q= 7/2 15535.77 0.40531 ± 9 6.4 ± 19 0.0038 0.42503 ± 24 27 ± 7 0.0049 0.422 b 4 Lower Levels (X Aj): 4 Q = 5/2 a 0.5026 ± 9 3.6 ± 14 0.0024 Q = 7/2 0 0.50058 ± 4 6.50 + 15 0.0031 a 6 V a l u e s in c m * , with error limits of three standard deviations in units of the last significant figure, a • AA - 244 c m - . N o least squares fit; see text. 1 1 b 67 111.0.1 .b. Perturbations. A plot of the upper state energy levels as a function illustrates the perturbations in the upper states. appears to be free of A-doubling perturbations, since the lower (cf. state Section of J(J + 1) The lower 1.B.6) combination state and other differences are entirely regular: the two A2F"(J) values, given by the two A-doubling components, are equal to within experimental error. Figure 3.4 shows that upper state A-doublings begin at J ' = 21.5, and that in some places extra transitions occur; the section of spectrum in Fig. 3.5 illustrates these perturbations. securely identified combination The because they differences as give the extra exactly main lines the and lines could be same A2F"(J) their relative intensities are in the same ratio. Two avoided crossings can be seen in Fig. 3.4: a strong one, where both of the A components are perturbed, at J ' = 30.5 - 31.5, and a weaker one where the lower A component is mildly perturbed at J ' = 37.5. differently, Since the avoided crossings affect the A - c o m p o n e n t s the alternatively perturbing has a very state is orbitally non-degenerate, large A-doubling of its own. The or state perturbing the J ' = 22.5 level appears to have a relatively small A or Q-doubling. as (A-doubling exhibited by a X state is referred to here Q-doubling .) 85 The state responsible for all of the above perturbations could conceivably be a single case (a) 4 X state. The small Q - d o u b l i n g occurring near J ' = 22.5 could arise from the X3/2 component, while 4 the considerably larger Q-type component 8 5 splitting associated with the is capable of affecting upper state levels that are 4 Xi/2 1 5 7 7 3 • 4 0 0 8 0 0 1200 J(J 1) + Fig. 3.4. Upper state energy levels of the A z / 2 - Aj/2 6338 A band, scaled by subtracting the quantity 0.405J(J + 1) - 6.4x10- J (J + 1)2, plotted against J(J + 1). 4 4 7 2 o Fig. 3.5. A section of the spectrum of the 6338 A band containing A doubling, two avoided crossings, and extra lines. The extra R(30.5) line to the blue of the A-doubled R(30.5) lines corresponds to the anomalous point in Fig. 3.4 near J(J + 1) = 1000. All lines, down to the weakest, are reproducible, though relative intensities between lines at either end of the spectrum may not be accurate since the spectrum is compiled from several laser scans. 70 widely spaced in J , analogous to the situation observed in Fig. 3.4. III.C.2. Rotational analysis of the 6436 A subband. The fairly intense 6436 A band is another Q' = Q" = 7/2 transition whose lower state is the same as that of the parallel 6338 A band, as the lower state combination differences of the two bands are equal to within experimental error. only 237 cm" 1 below that of the Because the upper level lies 6338 A upper state, and the frequency separating the strong groups of subbands (cf. Table 3.1) is on the order of 600 c m - , it cannot electronic state as the 6338 A band. is considerably belong to the 1 wider than in the Also, the hyperfine 6338 A available to say what this upper structure subband, which also points to a different upper electronic state. information same There other is not enough electronic state is. Although its high intensity suggests that it is another A7/2 - A 7 / 4 4 2 transition, there are other channels through which intensity can be derived. In the very dense, perturbed "orange" system of FeO, for example, transitions to the electronic states high vibrational levels of various acquire considerable intensity the upper state of the s y s t e m . by interacting lower with 86 The upper state energy levels are plotted as a function of J(J + 1) in Fig. 3.6, up to the limit of our analysis thus far at J ' = 26.5. A- doubling is first observed at J ' = 20.5, very much like the 6338 A band upper levels which are first seen to split at J' = 21.5. Perturbations in the upper state have scattered the levels to such a degree that a good least squares fit to the upper state constants was not possible, though a value of B' could be estimated (see Table -i r 15538 E-red c m " 1 15537 + + + 15536 -J 2 01 0 I l_ 6 0 0 4 0 0 J(J 1) + Fig. 3.6. Upper state energy levels of the A z / 2 - A 7 / 2 6436 A band, scaled by subtracting the quantity 0.42J(J + 1), plotted against J(J + 1). 4 4 72 3.III). lines assigned in the 6436 A band are compiled in Table The 3.1V. Rotational analysis of the 6411 A subband. III.C.3. The ft = ft" = 5/2 subband whose head lies at 6411 1 A is much weaker than the other two subbands, and is also badly perturbed, which has precluded analysis beyond J ' = 20.5. so far are listed in Table 3.V. All the lines assigned The transition was assigned as ft' = ft" = 5/2 by the methods used previously for the 6338 A subband, and it appears that the lower state is the ft = 5/2 spin-orbit component of the ground electronic state. The crowded head region of the band is shown in Fig. 3.7. The perturbations in the ft' = 5/2 upper state are illustrated by the plot of the scaled upper state energy levels as a function of J(J + 1) in Fig. 3.8. The A-doubling is much larger than in the upper levels of the ft = 7/2 bands, with the splitting first discernible at Doppler- limited resolution at J ' = 10.5. At J ' = 16.5 one of the A - c o m p o n e n t s is drastically pushed to lower energy, and no further J ' levels could be assigned. 20.5. The The other component also disappears abruptly at J ' = suddenness with which the branches break off is surprising, because there is no appreciable loss of intensity before the rotational structure ceases. This fragmentary behavior has been observed before, for example in the 5866 A band of FeO where the structure disappears suddenly at 1 to the b l u e . 86 The 6411 J ' = 15, and then reappears 12 cmA upper level in C o O has obviously suffered a massive perturbation near J ' = 20.5. branches extensive resume will require To find where the wavelength resolved 73 Table 3.1V. Assigned lines from the 6436 A ( A7/2- A7/2) band of CoO, in c m - . 4 4 1 J" R P 7.5 15537.998 15524.45 8.5 15537.505 15522.347 9.5 15536.866 15519.996 10.5 15536.061 15517.490 11.5 15535.096 15514.839 12.5 15533.976 15512.039 13.5 15532.692 15509.082 14.5 15531.241 15505.950 15.5 15529.644 15502.682 16.5 15527.897 15499.226 17.5 15525.955 15495.625 18.5 15523.820 15491.877 19.5 15521.589 15521.967 15487.959 20.5 15519.589 15519.536 15483.815 21.5 15516.412 15517.029 15479.586 15479.959 22.5 15513.383 15514.393 15475.130 15475.539 23.5 15511.448 15511.650 15470.420 15471.035 24.5 15508.438 15508.784 15465.395 15466.408 25.5 15461.468 15461.670 26.5 15456.463 15456.809 27.5 15451.345 15451.840 74 Table 3.V. Assigned lines from the 6411 A band ( A5/2- As/2) of C o O , in c n r . An asterisk denotes a blended line. 4 4 1 R J" P Q 2.5 15597.270* 15594.293 3.5 15597.577* 15593.751 4.5 15597.730* 15593.067 15589.2* 5.5 15597.730* 15592.23* 15587.557 6.5 15597.577* 15591.183 15585.670 7.5 15597.270* 15590.039* 15583.654 8.5 15596.194 15588.739 15581.492 9.5 15596.194 15579.076 15596.240 10.5 15595.432 15595.512 15576.718 11.5 15594.523 15594.647 15574.076* 15574.125 12.5 15593.461 15593.618 15571.307 15571.389 13.5 15592.254 15592.708 15568.389 15568.512 14.5 15590.896 15591.441 15565.323 15565.493 15.5 15589.390 15588.739 15562.106 15562.559 16.5 15587.729 15582.734 15559.270 17.5 15585.958 15555.222 15554.561 18.5 15583.958 15551.558 19.5 15581.845 15547.736 20.5 - 15543.770 21.5 15539.646 P(39.5) 6338 A band III i g i III III 1 1 1 1 55 85 105 1251 145 1 45 25 35 R Q 55 i 1 I U II 155 • 1 165 i II 175 85 45 Fig. 3.7. Bandhead of the ft' = ft" = 5/2 transition at 6411 A. The band is extensively overlapped by other bands, as evidenced by the dense collection of unassigned lines. 65 15598 15595 100 200 3 0 0 4 0 0 J(J 1) + Fig. 3.8. Upper state energy levels of the As/2 - A s / 6411 A scaled by subtracting the quantity 0.42J(J + 1), plotted against J(J + 1). 4 4 2 77 fluorescence measurements in the surrounding region. Such studies must be postponed until we develop a less cumbersome method by which to synthesize gaseous C o O . The upper and lower calculated in the same in Table 3.III. manner Kratzer's the equilibrium as for the 6338 A constants, band, are given relationship , 83 D for state B and D rotational values = 4Be /co 3 e (3.10) 2 e of the rotational constants and the vibrational frequency (co ) can be approximated for the v = 0 level by e D = 4B /AGi/2 3 0 2 0 (3.11) Using equation (3.11) to calculate an approximate value for D", it is found to be about 60% larger than the observed value. 78 III. D. Discussion. Of the two possible ground electronic state configurations for CoO, 4 £ - (a7t 5 ) or A ( o J t 5 ) , evidence has been presented in the 2 4 4 2 2 3 rotational analysis of the excitation spectrum of gaseous C o O which strongly supports that the ground state is vibrational frequency spectroscopy in of 846.4 low-temperature matches the value of 851.7 fluorescence work. cm (14 cm- 1 4 phase is The fundamental measured - 1 K) matrix by infrared isolation closely 71 obtained from this laser induced Since the ground electronic state should be the only one populated at 14 K, and a 5.3 c n r gas Aj. not unreasonable, this shift from the solid to 1 suggests that the lower electronic state of the three bands studied here is the ground state. The matrix isolation electron spin resonance s t u d y 72 which could not produce a signal from C o O eliminates the possibility for the ground state, taking this absence of a result as valid. condition under which an orbitally non-degenerate 4 £ - as The only electronic state with case (a) coupling can produce no E S R signal when isolated in a low-temperature matrix is if it possesses an odd spin with the ft = 0 level the only one populated. multiplicity The band intensities support an inverted order for the spin-orbit manifold since the ft' = ft" = 7/2 bands are strongest, followed by ft' = ft" = 5/2. The rotational analysis of two same electronic determine the interval, AA. because the state true provides B value For molecules spin-orbit and ft spin-orbit components the an information estimate in which interaction spin is very for of the required to the uncoupling large, the spin-orbit is small effective B value for a given spin-orbit component differs from the true B value 79 A A by an amount that depends on the spin-uncoupling operator, - 2 B J S . A second order perturbation treatment of two O substates separated by A A and connected by this operator produces the relation: Beff.ft = B(1 +2BI/AA) Solving equation 24 (3.12) (3.12) simultaneously for both A A and the true B value for the v" = 0 level, using the effective B Q = 7 / 2 and BQ=5/2 values in Table 3.Ill, gives B = 0.5037; A A - -244 c n r (3.13) 1 The spin-orbit coupling interval A A is not expected to be accurate to better than 10%, as equation (3.12) does not take into account the centrifugal distortion corrections to A and X, called AD and Xo Section I.B.3). FeO 7 5 (cf. For example, the initial estimate of |AA| made for was 180 c m - , based on the approximation in equation (3.12), 1 yet the value was later f o u n d 87 to be 190 c m " . 1 The definition of B, as a function of the mean value of the bond length r during the vibration, is 8 3 B = (h/87i cu)<r > 2 where (3.14) 2 |i is the reduced mass of the molecule. equation With the B value in (3.13), the bond length in the zero point vibrational level is calculated from equation (3.14) to be: r (X Aj) = 1.631 4 0 (±0.001) A (3.15) The 10% increase in bond length to 1.80 A upon electronic excitation to the upper A j state is quite large compared to transitions in the 4 other first row diatomic transition metal oxides. The A n 4 <- X X " 4 transition of V O produces a 7% i n c r e a s e ; A ! , <- X F I r and B I I r < 45 X n 5 r in C r O give 2-1/2 5 5 5 and 5-1/2% i n c r e a s e s ; the 90 parallel transition of MnO at 6500 A shows a 4% 8 £ + <_ 6 £ + i n c r e a s e ; but 91 80 various subbands of the orange system of FeO do show bond length increases of up to as m u c h a s 1 1 % , and a state perturbing the MnO 8 7 A 1 + state has a bond 10% longer than that of the ground s t a t e . 6 91 The magnetic hyperfine structure and spin-orbit coupling constant can be used to give information about the excited states as well as the ground state. The insignificant hyperfine structure in the ground state is consistent with the lack of unpaired s electron density in the 4 A o rc 8 2 2 configuration. 3 The upper state configuration can be assigned as a 7 i 5 a * for three reasons: 2 1) the large, a strong indicate positive Fermi hyperfine contact electrons (cf. Section I.B.3). in a diatomic transition 3 splittings interaction cm-" . 1 4 to open state shell s metal oxide it usually shows up clearly in Most states with unpaired s electrons have positive values for ap: aF for S c O 2 due upper When an unpaired s electron is present the Fermi contact parameter. for VG-45 o 8 in the £ - - +0.02593 c m * ; a 1 F a L = 7 3 2 for M n O +0.0667 c m - ; aF + 9 0 1 o 8 n 6£+ .» +0.0151 2 2 An exception is the ground state of C u O , which has a large, negative Fermi contact parameter shell so electrons. 79 in spite of the presence of open Three configurations are believed to make significant contributions to the Ti\ ground state: 2 C u ( 3 d ° ) 0-(2p5), + 1 C u ( 3 d ° 4 s ) 0(2p ), and 1 4 Cu(3d 4s4p) 0(2p ) 9 4 Only the last one has open shell metal-centered orbitals which will participate significantly in the hyperfine interactions. molecular orbitals, this configuration is proposed to b e : In terms of 7 9 3da 8 7i (Cu), 4sa(Cu) + 2pa(0), PTC(CU) + 2p7i(0) 1 4 4 81 The wavefunction can therefore be expressed as a linear combination of Slater determinants (showing only the unpaired electrons for clarity): V( rii) 2 = (1A/6){2|do(a) po(a) pit(P)| - |do(a) pa(P) pw(o)| -|do(P) pa(a) pn(a)\} (3.16) The authors propose that the negative terms in the wavefunction are responsible for the negative value for aF of -0.0139 c m - . The C X " 1 4 state of V O , with a 3 d 8 o * configuration, is an example where the 2 promotion of an electron from s a to a non-s type a orbital produces a negative value for the I S a interaction constant of -0.00881 c m - , as 1 result of spin polarization. 45 The a * orbital is believed to be a linear combination of 3do, 4sa and 0(2pa). 2) the fact that the Q. = 7/2 and Q. = 5/2 subbands lie very close in the spectrum shows that the spin-orbit intervals A A " and A A ' are nearly equal. o7t 6 a* 2 '6' 3 The 4 A states of the configurations C T C 8 2 2 3 and will have orbital angular momentum coming only from the hole, so that they should have roughly the same spin-orbit couplings. 3) Following from 2), the negative sign of A also suggests a 8 hole, or 8 3 configuration. The o - 7 t 8 c * configuration can give rise to 19 electronic states 2 from the orbitals. 3 different 74 arrangements of the electrons within The result will be a dense collection of states ranging up to S = 5/2 and A = 4, among which are, for example, a with the configuration a(T)jc(TT)8(TiT)a*(T) A 6 melange are the a (T )n ( t i )8 (T 11 ) a * ( T ), state. expected to interact 4 r state and a As the states comprising such a strongly with one another, this 82 could explain the extensive perturbations experienced by the upper states of C o O investigated here. As discussed in Section III.C.I.c, the only perturbing state for which we have clear evidence appears to be a Z 4 2 state, arising possibly from a a 7 i 8 a * 2 3 configuration, or IxlAx Ax2l=4£. 2 Now that the ground state configuration of C o O has been determined in this work, the entire series of first transition metal oxide ground states is now established. states and some major molecular constants of the metal monoxides appear in Table 3.V. row diatomic The ground 3d transition Although many more excited states of cobalt oxide remain to be discovered, the most interesting results for the immediate future would be the direct measurements of the spin-orbit coupling intervals, and sub-Doppler measurements of the hyperfine structure. However, the experiments would require a more efficient means of generating C o O than has been used so far. 83 Table 3.VI. Ground states and configurations of the first row diatomic transition metal oxides, with the fundamental vibrational frequency A G 1/2, B and r for the v" = 0 state, and the spin-orbit interval A A for the orbitally degenerate electronic states. The AA value for C o O has not been established with certainty. Ground state 2£+ ScO TiO 3A VO 4 CrO Electron configuration r I5pi a oS MnO 6£+ FeO 5AJ CoO 4 B (cm- ) r (A) 964.65 0.51343 1.668 1000.02 0.53384 2 1 0 1.623 AA - Ref 29,30 101.30 89 1001.81 0.54638 1.592 - 45,88 o8 7t 884.98 0.52443 1.621 63.22 90 832.41 0.50122 1.648 871.15 0.51681 1.619 0 § 2 00H n 2 2 Aj O283TC2 851.7 0.5037 NiO 3£- c28 7c 825.4 0.505 CuO 2 629.39 0.44208 U \ 0 1 o82 2 r AG-|/ (cm- ) 4 2 a 8 7i3 2 4 8 0 1.631 - 91 -189.89 87 (-240) this work 1.631 - 1.729 -277.04 78 92,93 84 CHAPTER IV HYPERFINE ANALYSIS OF NIOBIUM NITRIDE IV.A. Introduction. Niobium nitride (NbN) is an exemplary molecule in which to study hyperfine interactions magnetic moment radioactive atom. proportionately informative in diatomic (JIN) of 9 3 Nb molecules, because the exceeds that of any nuclear other non- The magnetic hyperfine structure which results is large analysis. and well-resolved, Following the allowing precise, initial observation of NbN in 1969 by Dunn and R a o , the first low resolution hyperfine analysis 9 4 of the 3<X>-3A system was performed in 1975 with a grating spectrograph. by Femenias ej.ai The study produced values for 9 5 the magnetic hyperfine constants a, b and c which suggested that the excited 3 state < X > makes hyperfine structure. a non-negligible contribution the The spectra also exhibited line broadening at very high J values, indicating either A-doubling in the A 3 transition to from case (ap) to (bpj) state or a coupling with increasing rotation. In the meantime, the fundamental frequencies of the ground states of N b N and N b N were measured to be 1002.5 c m 1 4 1 5 by IR spectroscopy in a 14 K argon matrix. published a culminating they number in the of 1986 papers on the 3 0 - and 974 cm" 1 A Russian group 96 3 1 system A publication by Pazyuk e _ L a i 1 0 0 proposed a set of rotational, centrifugal distortion 9 7 . 9 8 - 9 9 , , in which and spin- orbit coupling constants (B, D and A), and an energy level scheme for the system. drastically However, the spin-orbit splittings for both states were miscalculated, and the ordering of the spin-orbit 85 manifolds was inverted, observed near 5600 satellites, rather due to their A as $ 3 3 - 3 interpretation A and 3 3 o 3 they spin-orbit A 2 than as parts of the n - A system to which actually belong. they In 1979, an optical emission study measured eight subbands belonging to five systems, including 3 < E > - the upper and lower state B values for e a c h . grating spectrograph analysis of the 3o- 3 A resolution (±0.01 Their work produced for the (0,0) band, 3 A , and determined Most recent was a 1 0 1 system performed by the same investigators involved in the preliminary higher - 2 of bands 1975 study, but at a c m - line position), and up to J " = 8 8 . 1 the following set of 1 0 2 molecular constants in units of cm" with the uncertainty in the last 1 digit given in parentheses: T X3A fixed to 0 A3o 16504.938(3) The central A subband 3 investigations 8 A 0 shift B 10 D 105A 7 D 183.0(2) -33.1(2) 0.50144(4) 4.56(6) =-4 241.6(1) 7.39(2) 0.49578(4) 4.88(6) =-4 parameter because 8 accounts for the shift in the 3<j> - of second order described in the current 3 spin-orbit work resolution laser spectroscopy performed on NbN. mark effects. The the first high 86 IV.B. Experimental. IV.B.1. Synthesis of gaseous niobium nitride. Niobium nitride was formed in a flow system by reacting the vapor from a sample of warmed niobium (V) chloride (=80 ° C ) with nitrogen. The approximately nitrogen 1:18 (v/v) was at entrained with argon 1 Torr pressure. in a A few ratio of centimeters upstream from the fluorescnce cell, the vapor was passed through a 2450 MHz microwave discharge (powered by a Microtron model 200 microwave generator). To obtain intermodulated fluorescence spectra, two nearly coincident laser beams were passed in opposite directions across the lavender-colored flame of the discharge, with the fluorescence detected at right angles to the beams through a deep red low pass filter to the photomultiplier tube, as described in Section II.A. IV.B.2. Description of the 0 - A 3 3 spectrum. Broadband spectra of the three subbands of the G> - A system of 3 NbN are illustrated in Fig. 4.1. 3 The middle spin-orbit component, <E>3- A2, is shifted to higher energy rather than being equidistant 3 3 between the presumably outer due to subbands, and intensity is also stealing by an considerably unseen state. vibrational sequences are plainly visible, up to (v',v") = (5,5) 3 weaker, The in the 0>4- A3 subband. 3 At sub-Doppler resolution, the variation in hyperfine structure between the three subbands is apparent from the Q head regions shown in Fig. 4.2. The hyperfine interaction in the <E>3- A2 3 3 subband is much less pronounced than that in the other two because 3fl> - Ai 3 2 50 cm* i 1 1 3<I>3- A2 3 16145 cm-1 16543 c m 1 Fig. 4.1. Broadband spectrum of the 3<J>-3A system of NbN, obtained with the intracavity assembly removed, using the dye rhodamine 6G. Note that the vibrational sequence of the 3 O - 3 A subband is visible up to (v\v") = (5,5). 4 3 a) Fig. 4.2. The Q heads of the a) 3 < D 2 - 3 A i , b) 3 < D 3 - 3 A 2 , and c) 3 < D 4 - 3 A 3 subbands of NbN. CD 89 the value of X in both states is zero. hyperfine splitting In the <X>4- A3 subband 3 is considerably larger the 3 than that in 3 <J>2- A-|, 3 since Q is three times as large in the former subband (cf. equations 1.90 and 1.98). The assignment of the densely overlapped 3<j>2- Ai Q 3 head is shown in Figs. 4.3 and Fig. 4.4. 3<x>2- Ai 3 subband, illustrated completely (cf. The low-J R branches of the in Fig. 4.5, are exemplary for their resolved A F * A J transitions and crossover resonances Section II.B for a discussion of these transitions). hyperfine pattern is quite different in the central subband: The at J " = 2 the high F component is on the low frequency side, but at J " = 3 the hyperfine values structure with the reverses order and continues on at highest F component at high higher J frequency. The development of this <X>3- A2 R branch hyperfine structure is shown 3 3 in Fig. 4.6. As the rotation of the molecule increases, spin-uncoupling is observed in the Q branches of the outer two subbands as a reversal in the hyperfine structure: the hyperfine splitting narrows with increasing J until the components collapse into a spike; then they reverse their order and widen with increasing rotation (see Fig. 4.7). Therefore hyperfine structure which begins with its components increasing in F toward increasing frequency reverse to an order in which the F values decrease with frequency. The reversal in the Q branches occurs at J = 27 and J = 38 in the <l>2- Ai and <I>4- A3 3 subbands, transition respectively. is less The sensitive to hyperfine the 3 structure effects of diagonal matrix elements are independent b and c. 3 3 in the <E>3- A2 rotation, 3 3 since The Q branch of its o -»• 3 2 Fig. 4.3. The beginning of the Q head of the <D2- Ai subband. Each A F = 0 line is connected to the A F = +1 lines with the same F" value by a thick horizontal line. Components of the Q(7) and Q(8) lines are also present in this region, but are not labelled. 3 3 CO o Fig. 4.4. The higher J portion of the 0 > 2 - A i Q head, and the first resolved Q lines. The crossover resonances are not labelled. 3 3 CO 92 4.5 a) 6.5 0.05 cm-1 5.5 b) R(1) 4.5 4 • 3.5 • • * 2.5 7.5 c) 6.5 5.5 4.5 3.5 r r C C r I ic r C CC c Fig. 4.5. a) R1, b) R2, and c) R3 lines of the 3 o - A i subband, illustrating the "forbidden" A F * A J transitions (• for qR, * for pR) and the crossover resonances (c) between the rR and qR lines. Each A F - A J transition (•) is labelled with the lower state F value, with the corresponding satellite transitions following it to the red (right) in the order: c (if seen), • , * (if seen). The scale shown in (b) is the same for all spectra. 3 2 93 Fig. 4.6. a) R2, b) R3 and c) R4 lines of the 3<D -3 subband of NbN, showing the rR, qR and pR transitions and the crossover resonances associated with the rR and qR lines (denoted by c.o.). 3 A2 94 qR d) 7 5 2.5 rRII l l l l l | l ; 0.01 I cnr 1 1 CO. 9 5 e) rR| 55 -is | | | | 0.01 cnr 1 Fig. 4.6. d) R5 and e) R6 lines of the d>3-3A2 subband of NbN; the labelling follows that of Fig. 4.6 a, b and c. 3 Q(35) Q(33) Q(31) Q(29) P(8) tliJlL Q(37) Q(39) 0.3 cm* 1 1 l P(9) 16848.5275 cm" Q(41) P(11) Q( 3) 4 I 1 16851.5933 cm" Fig. 4.7. The reversal of hyperfine structure at high J in the <I>4- A3 Q branch, caused by the effects of spin-uncoupling. Actual reversal occurs in the line of maximum intensity, Q(38). 3 1 3 CO cn 96 this subband therefore narrows up to about J = 12, and then remains nearly constant in width up to the limit of our data at J = 27. 97 IV. C. Non-Linear Least Squares Fitting of Spectroscopic Data. In order to acquire the best set of molecular constants in a Hamiltonian, one must iteratively improve an estimated set of constants until a satisfactory fit of the observed data is obtained. In approaching the non-linear type of Hamiltonian typically describing a spectroscopic problem, the Hamiltonian is divided into its two constituents: the coefficients containing number dependence, and the molecular constants, o r H = X X H m=1 m X m the quantum 1 0 3 (4.1) m is the mth parameter (or molecular constant) out of a total of p parameters, and H m is the "skeleton matrix" containing the quantum number dependence of the mth parameter. Hamiltonian may be expressed a s : 1 0 1/2 For example, a simple n 1 0 3 (J + 1/2)2 - 2 0" 2 -[(J + 1/2)2 _1]1/2 + B H-Tr 0 -[(J + 1/2)2 - -|]1/2 0 -1/2 1 (J 1/2)2 + The matrix of eigenvalues (or energy levels) E Of the Hamiltonian is obtained by diagonalization with the eigenvectors U: UtHU = E (4.2) U is a unitary matrix such that the adjoint of U (U^, or the conjugate of the transpose U ) equals the inverse of U (U- ). T 1 The combination of equations (4.1) and (4.2) allows the HellmannFeynman theorem to be employed, which s t a t e s : aE /3X = m 104 fa *(dWdX)*¥ dx m m (4.3) 98 For a single matrix element Feynman theorem b e c o m e s : ii of parameter m, the Hellmann- 103 [UT(aH/3X )U]ii = 3Ei/aX = Bj (4.4) m m Using equation (4.1), equation (4.4) can also be written as: Bim - [UTH U]ii (4.5) m The Hellmann-Feynman derivatives B j m form the derivatives matrix, B, which give the dependence of the energy on variations in the parameters. To apply this relation to an iterative solution of unknown molecular parameters, equation single energy level, E j c a l c (4.2) is expressed in terms of a : Ejcalc = (UtHU)n (4.6) Substituting equation (4.1) into equation (4.6) gives P E p i c = X X (UtH U)ii m m (4.7) m=1 With the relations in equations (4.4) and (4.5), the energy can be written: P Epic = I X B m i m (4.8) m=1 To express equation (4.8) in terms of transitions rather than energy levels, the upper and lower state eigenvalue vectors (E 1 and E") are subtracted to give y, and B' and -B" are combined into one derivatives matrix B. Equation (4.8) therefore transforms t o : y =BX where 1 0 5 (4.9) y is the vector of calculated transitions, B is the matrix of known derivatives, and X is the vector of estimated parameters. If 99 there are N transitions and p parameters to be determined, y has length N, B is a matrix of size N by p, and X has length p. To obtain X , both sides of equation (4.9) are multiplied by ( B B ) - B : T 1 T (BTB)-1(BTB)X = ( B B ) " B y T 1 X = (B B) T _ 1 T B y (4.10) T In a problem where the estimated parameters X are iteratively improved, we calculate parameter c h a n g e s A X . rather than X itself. Equation (4.10) is therefore expressed a s : 1 0 6 AX = ( B B ) " B A y T where 1 (4.11) T A y is the vector of residuals (i.e., the observed transitions minus the calculated). The fitting process begins with a set of estimates for the molecular constants, which are used to generate calculated transitions (ycaic) a n c j {heir residuals (Ay). The set of corrections to the constants, given by equation (4.11), is added to the initial iteration. sets estimates to provide improved constants for the The process is repeated, of calculated transitions, magnitude iteratively next producing improved residuals and constants of the residuals is reduced to a satisfactory until the level, for example, to the vicinity of the experimental precision. The least squares program for the <X> - A system of NbN was 3 3 written in F O R T R A N 77 by the author, except for U B C Amdahl library routines for diagonalizing parameter Hamiltonian changes from matrices and inverting matrices, the Hellman-Feynman for the <D and A states 3 dimension of (21 + 1)(2S + 1), or 30. 3 and calculating derivatives. have a The maximum The 30 x 30 matrices (one for each F) were diagonalized in two steps. In the first step, only the rotational part of the Hamiltonian was diagonalized, in ten separate 100 J submatrices. In the second step, the entire matrix (rotational and hyperfine) was diagonalized. ordering of eigenvalues diagonalization to spin-orbit from step one preserve the original basis functions. the Two steps were employed because the was matching used in the second of eigenvalues with the This is possible because the separation of components is large compared to the perturbation made by the hyperfine interactions. Analogous to the common formula for the standard deviation, s = [I(xobs . calc)2/ ]l/2 x (4.1 ) n 2 the weighted least squares standard deviation is obtained f r o m : n a = [ I (yjQbs . calc)2Wjj/(n-m)] / i=1 1 1 0 5 (4.13) 2 yj where n is the number of independent measurements, m the number of unknowns to be estimated, n-m the degrees of freedom, and Wjj the diagonal element of the weight matrix for point i . estimates 1 0 5 To determine of the precision of the estimated constants, a variance- covariance matrix @ is calculated b y : 1 0 5 0 = c (BTB)2 A diagonal element (4.14) 1 0jj is called the variance (not to be confused with the variance that is the square of the standard deviation, G ). 2 The square root of &\\ gives the estimated molecular constant i. covariances. standard error, or precision, of The off-diagonal elements ©jj are Both the variances and covariances are only estimated values, because they depend on the precision of the measurements, a . 2 The goodness of the structure of the model lies in ( B B ) - . T 1 101 Normalization correlation of the variance-covariance the matrix, C, with elements cjj = eij/(eiiejj)i/2 where Cjj = 1 for i ~ j, and (-1 < Cjj < +1) ( for i * j. C of the precision of the measurements since o out. matrix gives Therefore the off-diagonal 2 4 . 1 5 ) is independent has been cancelled elements represent the interdependence of the molecular constants on one another, for a given data set. A value for CJJ that closely approaches unity indicates that constants i and j cannot be determined independently. 102 IV.D. Results and Discussion. Initial line assignments were facilitated grating spectrograph work of Dunn e l s i of the P, Q and R rotational lines. 1 0 2 by the unpublished , who listed the positions Initial attempts to obtain a least squares fit to the hyperfine constants in a case (a) basis (i.e., as they were presented in Sections I.D.3 and I.D.4) did not succeed, because the hyperfine constants required to fit the three subbands are not consistent with one another. and In the light of this observation, the unequal first order spin-orbit spacings, it was concluded that the various substates are perturbed differently by second order spin-orbit interactions. this interaction 109 According to the AQ = 0 selection rule for , the electronic states perturbing the <J> substate 3 include ^3,4, T4, 0>3, A2,3 and A2. The A substates can interact 1 with 3 1 3 1 3 0 2 , 3 , <J>3, A 2 , T11,2 and n 1. 1 1 3 1 The 0 1 and A 1 states isoconfigurational with G> and A are expected to be the closest of 3 3 these states to d> and A , and therefore the ones most responsible 3 3 for the perturbations (see Fig. 4.8). central spin-orbit The effect would be to shift the components, <X>3 and A 2 , to lower 3 3 energy. However, the hyperfine constants suggest that there could also be second order spin-orbit interactions occurring with the other members of the manifolds, though we can say nothing about their relative sizes. The o - A 3 3 system of NbN is the first observed instance of a molecule represented by Hund's case (a) which requires modifications to the Hamiltonian because of extensive second order spin-orbit interactions. This phenomenon can be considered a slight tendency toward the case (c) coupling s c h e m e . The 1 0 9 molecular constants obtained for the o - A system of NbN 3 3 103 Hs.oO) •10 817*1 — H .o.( ) 2 s in 3n .4 --742.2 .30 — < •in \ o CD 00 CD -3n CO in CD in CD C2 1L+ 2 - ^ 1A ai5i , _._. 3 3A $383.4 1 — Fig. 4.8. Partial energy level diagram for NbN. The figure is not to scale, but illustrates the relative ordering of states, except in the case of the low-lying configurations o and 0 8 where the ordering is uncertain. 2 104 are given in Table 4.1. The unequal perturbations in the <D and A 3 spin-orbit manifolds means structure, only h constants in the matrix elements diagonal in ft and £ can be determined, the that, in rather constants (cf. equations 1.90 and 1.98). the than magnetic 3 hyperfine individual a, b and c The h constants, subscripted by their X values, are as follows: h-i = aA - b - c = aA - (b + c)-i (4.16) ho = aA (4.17) h+i = aA + b + c = a A + ( b + c)+i (4.18) In an unperturbed system, the average of h-i and h+i equals ho; that is, (b + c)-i and (b + c)+i in equations (4.16) and (4.18) are equal. This is far from the case in the <x>- A system of NbN, where (b + 3 3 c)+i is 39% smaller than (b + c)-i in the A state, and 10% 3 in 0>. 3 It was also found, in the 3 A state, that two distinct b constants are required in the <X=-1|X=0> and <X=0|Z=+1> elements (referred to here as b.-i/o larger and brj/+i, matrix respectively). Therefore, a total of five magnetic hyperfine constants are required to fit the data, rather than the usual three: h-i, ho, h+i, b-1/0 and bo/+i replace a, b and c. It is clear that the perturbations in the A state are 3 much more pronounced than those in G>. The A b+i/o value is 34% smaller than 3 3 b o / - i , comparable to the 39% difference between the and (b + c)+i constants. 3 A (b + c)-i In the upper state, however, two distinct b values off-diagonal in X are not necessary: two 3 attempts to distinguish 0 b constants produced values that were very highly correlated (-0.998) and with standard errors so high that the constants were indeterminable. It is evident, then, that the A state lies closer to 1 105 Table 4.1. Molecular constants for the 3C>-3A system of NbN.a O A To 16518.509(1) A 247.4116(5) 191.7038(8) B 0.495814(4) 0.501465(4) D 0.4943(4) x 10-6 0.4622(2) x 10-6 X -16.817(2) 3.430(2) y 0.011(2) -0.0217(6) -0.58(2) x 10-4 -0.105(3) x 10-3 XD -0.150(6) x 10-4 -0.1314(6) x 10-3 h-1 0.0633(2) -0.0616(3) ho 0.0411(4) 0.0458(5) h+1 0.0168(2) 0.1112(3) b -0.02(1) - - 0.085(5) A D b-1/0 bo/+i e Qqo 2 0 0.056(5) fixed to zero -0.39(8) x 10-2 Derived constants: (b+c).i -0.0222(4) 0.1074(6) (b+c)+i -0.0246(5) 0.0654(6) a 0.000547 Values are in c m - . The numbers in parentheses are three times the standard errors of the constants, in units of the last significant figure. The standard deviation of the transition measurements is given by a. The magnetic hyperfine constants, h, (b + c) and b, are explained in the text. 3 1 106 the A state than 0> does to <X>. Note from Fig. 4.8 that the ordering 3 1 3 of states in the 8rc manifold is contrary to that dictated by Hund's rule 1 1 0 , which would place the higher multiplicity <I> (and therefore closer to the 0 TI state 1 3 constant c cannot be extracted state). The dipolar since separate below hyperfine b constants are required for the three substates. The (b + c) and b constants clearly support the 5 s a 4 d 6 1 4drc 5 1 3 A configurations for the A and 0 1 3 3 states, respectively. 1 and The (b + c) and b values are large and positive, indicating that the dominant mechanism for the coupling of electronic and nuclear spins is the Fermi contact interaction. This is consistent presence of an unpaired s a electron, as in the s a d 8 1 of A . The 0 3 3 1 with the configuration (b + c) and b constants are negative, and small compared to those in 3 A. This is characteristic of a hyperfine interaction which occurs because of spin polarization in orbitals having nodes at the nucleus, such as The difference TC 5 . 1 by electrons 1 between the Fermi contact and spin polarization hyperfine constants in NbN is similar to that found in the V O states 4po 3d8 1 2 C I\ 4 4sa 3d8 45 1 The ratio of A ( b + c ) e / ^ ( b + c ) 3 3 a v a v e 2 X Z " and 4 = -3.7, while b ( X ! " ) / b(C Z-) = -3.1. 4 4 The quadrupole coupling constant for the lower state is -3.9 (+.8) x 10" c m - , while that of the upper state was fixed to zero after it 3 1 was found to be too small to be determined. e Qq 2 2 4 0 The sign of the <J> state 3 is consistent with the quadrupole moment for ecm . 2 9 3 N b of -2 x 10- The upper and lower state constants for the interaction of nuclear spin and rotation (ci) were fixed to zero, as they were found to be on the order of - 1 0 -5 to - 1 0 - 6 c m - , almost 1 completely 107 correlated ( . 9 9 9 ) , and with standard errors as large as the values themselves. transition It metal is the usual case for diatomics containing a for ci to be too small to be determined (see for example references 3 1 , 4 5 , 7 9 and 9 1 ) . In the rotational part of the Hamiltonian, the A, B and D constants are very well determined in spite of the high correlations between A' and A" ( . 9 9 8 5 ) and B ' and B " ( . 9 9 5 ) . carrying allow information B and D to about the The high rotational lines spin-uncoupling operator, be determined individually, rather determining their differences, B ' - B " and D' - D". -2BJS, than simply Since all three subbands were fitted simultaneously, and B was extracted with good precision, A could also be determined. This is possible since A, B and the effective B values for each subband are related b y : B ff,n = B(1 + 2 B I / A A ) e 24 (4.19) From the B values, the bond lengths are calculated to be: r r There have mononitrides. been ( A) = 1.6618 A 3 0 0 = 16712 A ( 3 < D ) very few rotational studies of transition metal Aside from the current work, the known bond lengths (r , in A) are: 0 TiNln X I 1.583 2 A n 2 r B I 1.646 X I 1.696 2 ZrN^ 2 2 B I 1.740 A n 2 1.702 x I" 1.634 2 M O N 1 1 3 1.597 4 108 A*n 1 -654 The 3d transition metal monoxide series isovalent with ZrN (and TiN), NbN, MoN is S c O , TiO, V O , whose ground state bond lengths go as 1.668 A <\ 3 1.623 A89 and 1.592 Here the bond length A*5. decreases with each additon of a bonding 8 electron. The A and 0 3 3 NbN bond lengths show that the nitrides are consistent with this trend, with values intermediate between those of ZrN and MoN. The very large spin-spin interaction constants X (equations and 1.73) are caused by contributions from the second order spin- orbit interactions which induce the substantial shift of the subband. 1.72 3 G > 3 - 3 A 2 The centrifugal distortion correction to X, however, is considerably larger than its expected value of Xo - X(Ao/A). The reason for this probably lies in the fact that we have not yet made direct measurements of the spin-orbit intervals. In this context, then, the centrifugal distortion correction constants A D and XQ are essentially fudge factors which enable the least squares fit to converge to a minimum lying within a broad minimum which contains the true molecular constants. So although this set of constants is an internally consistent one which fits the data, once the derived A values are replaced by direct measurements the constants change slightly to enable the fit to converge to the true, minimum. may nearby With the data we now possess, however, the A D and Xo values given in Table 4.1 are necessary to obtain a fit. To demonstrate this fact, a fit of the rotational constants was made in which Xo and y were fixed to zero, and all hyperfine constants were fixed at the values determined in this work. The initial values for the floated constants were taken from the grating 109 spectrograph work of Dunn e _ t a i (see p. 8 5 ) , with the exception of 102 AQ which was given an initial value of zero; the parameter 8 in their work is equal to - 2 X . The fit converged to a standard deviation of 0 . 0 0 1 3 cm 8 - 1 , which is about incorporates XD and y. 2 . 5 times higher than the fit which As expected, the final set of constants (Table 4.II) is very similar to those determined by the grating spectrograph analysis, with the exception of To, which was found from LIF data to be 1 3 . 5 c m - 1 higher than that from the grating work. The residuals contain systematic errors in the positions of the rotational lines, as compared to the constants. The random residuals generated systematic errors and by the full higher standard set of deviation reflect the inability of the model to fit the data without XD, AD y. However, other than feature as stated above, the resulting rotational constants, B and D, are only effective of this fit is that the and first ones. order Another important spin-orbit coupling constants A' and A" are 1 0 0 % correlated, as are the second order spin-orbit parameters 4.II). X' and X." (see the correlation matrix in Table This is a direct reflection of the fact that the coupling constants are derived rather than measured. spin-orbit As a result, only the difference A X can be determined, rather than separate X' and X" values. For these reasons, a fit excluding XD and y may produce a set of rotational constants that more accurately represents the real situation, though the addition of XD and y creates a model which is able to fit the data. It is worth noting that in a purely case (a) basis, y, AD and XD are correlated such that only two of the three can be determined. 41 In the 3 O - 3 A system of NbN this correlation is broken 110 Table 4.II. Rotational constants obtained for the 3d>.3A system of NbN with the XD and y parameters fixed to zero, and the hyperfine constants fixed to the values in Table 4 . l . The correlation matrix follows the constants. a O 16518.4653(2) 242.59(8) 0.495796(8) 0.5005(7) x 10-6 -3.70(8) -0.484(5) x 10- To A B D X 0 184.5(1) 0.501447(8) 0.4685(4) x 10-6 16.53(8) -0.793(8) x 10-4 4 A a D a A 0.00138 T h e format of the table follows that of Table 4.I. Correlation To A' To 1.0000 B' D' X' Matrix X 0.0996 0.0820 -0.4333 0.3401 1.0000 -0.0473 0.1235 0.5124 1.0000 -0.0973 0.0934 1.0000 0.0366 1.0000 A' B' D' A ' D X" 0.0995 0.1580 -0.2227 0.3408 To A' 1.0000 -0.0335 0.3977 0.5115 B' -0.0474 0.9936 -0.0789 0.0928 D* 0.1233 -0.1874 0.4224 0.0360 0.5124 0.1025 -0.2270 1.0000 X,' A ' -0.0643 -0.2737 -0.0883 ••0.0659 A" 1.0000 -0.0336 0.3976 0.5116 B" 1.0000 -0.0839 0.1020 D" 1.0000 -0.2275 X" 1.0000 A" D A " D B" D" A " D 0.1715 0.3233 -0.3322 -0.2099 0.0874 0.8660 0.3233 -0.2929 0.0982 0.0875 1.0000 AD 0.1538 -0.0645 -0.3204 -0.4503 -0.0664 1.0000 111 to some extent by the high J data where there as a distinct tendency towards case (b) (see the correlation matrices in Appendix I and Table 4.II). For the future, a direct measurement of the spin-orbit intervals must be made. forbidden The most likely method for doing this is to locate "spin-orbit satellite" transitions which disobey the (a) selection rule A £ = 0 (equation 1.57). very weak, case Since these transitions are resolved fluorescence experiments can be performed to enhance the signal. To record the spectrum of a 3<X> -3A 2 example, an allowed 3 < D 2 - 3 A I transition is excited. line, for 2 The resulting emission spectrum of the satellite transition is recorded over a long exposure detector. time using the microchannel-plate intensified array The lines which hold the most promise for producing spin- orbit satellites are high J lines affected by spin-uncoupling, since the A L =0 selection rule weakens with increasing rotation. However, it is also important that the excited line be strong, so a compromise must be made between high J and line strength when choosing lines for excitation. Other important tasks are to locate the singlet states which interact with the A 2 and <x>3 spin-orbit components, and to search 3 for the expected a or 5 > . 1 ( 1 X + ) 3 2 1 Z + state to determine if the ground state is The ordering of the 0 8 states ( A and A ) and the o 3 1 depends on the relative ordering of the centered molecular orbitals (see Fig. 3.1). 4 S C T 2 and 3d5 3 A state metal- Diatomic transition metal oxides and fluorides isoelectronic with NbN demonstrate that these orbitals lie very close to one another. predict in NbN whether the A 3 or Z + state will be lower in energy. 1 R Therefore one cannot readily 112 For example, the d -transition metal monoxide series, consisting of 2 titanium oxide (TiO), zirconium oxide (ZrO) and hafnium oxide (HfO), is variable in this respect. 1 A state lying 3500 cm" ground state 1 1 3 1 TiO has a A 3 above t h a t which lies 1650 is believed to have a X+ 1 cm- 1 1 5 . r ground s t a t e , with the However, ZrO has a below the 1 1 1 4 3 A r state 1 1 6 1 . X + HfO ground state also, but with the 0 8 states further removed from the ground state than those in ZrO due to the greater ligand field splitting between the o and 8 orbitals in H f O . In the d -transition 1 fluoride ScF 3 A 1 + have a 1 £ 1 determined 1 1 8 . Tantalum of NbN, is predicted from nitride relative 1 1 9 (TaN), (LaF), + and the 5d matrix isolation studies to ground state, though the possibility of entirely ruled o u t . the and lanthanum fluoride ground states, while the ordering of Z in LaF is not k n o w n counterpart then, monofluoride series, comprising scandium (ScF), yttrium fluoride (YF) and Y F have Z 1 1 5 r metal 1 1 4 3 A has not been To identify the ground state of NbN securely, position of the experimentally. 1 X + and 3 A r states must be 113 CHAPTER V ROTATIONAL ANALYSIS OF THE V y - F U N D A M E N T A L OF AMINOBORANE, N H B H 2 V.A. 2 Background. This work examines the B H 2 out-of-plane wagging fundamental of aminoborane (NH2BH2), the simplest alkene in the B=N homologues of the hydrocarbons. Long before N H 2 B H 2 was studied experimentally, its small size and the interest in B-N compounds led to extensive acceptor theory theoretical nature studies of the calculations 120 B-N of it. In particular, the donor- bond atttracted attention, as Huckel done in 1964 predicted that the bond moment was in the direction B to N rather than the reverse, as required by formal valence charge theory. distributions, These electronic preliminary structures calculations, covering and geometries for a number of B-N compounds, were followed by C N D O (complete neglect of differential overlap) 1 2 1 and ab initio 1 2 2 ' 1 2 3 - calculations 1 2 4 predicting these and other properties such as the dipole force constants, barriers to rotation and stabilities. extreme instability at practical difficulties for theoreticians' the room experimentalists predictions. symmetrical temperature, cleavage to Aminoborane's however, verify moment, or imposed refute the It's first synthesis was in 1966 of vacuum cycloborazine sublimed from pyrolyzed at 1 3 5 ° C , where N H 2 B H 2 and other decomposition products could be trapped in a liquid nitrogen cold trap, and then identified by mass spectroscopy 1 2 5 . The aminoborane was found to decomposed spontaneously after warming to room temperature. have In 114 1968, gaseous aminoborane and diborane (B2H6) were observed by molecular beam mass spectroscopy as products of the spontaneous decomposition of temperature. When Kwon and McGee performed both pyrolysis and 126 solid ammonia borane ( N H 3 B H 3 ) at room radiofrequency discharge experiments on borazine (the BN analog of benzene), N H 2 B H 2 a n d B 2 H 6 w e r e again the p r o d u c t s . 127 They were recovered in a -168 ° C trap, then separated by vacuum distillation of diborane from aminoborane at -155 ° C . At this temperature, small amounts of both evaporation and polymerization observed. Polymerization temperatures above this, and The pronounced Pusatcioglu et a l becomes is fairly instability 1 2 6 build the of of N H 2 B H 2 were dominant process significant at -130 monomeric °C. aminoborane at 1 2 7 led in 1977 to investigate the possibility of using NH2BH2 to thermally pyrolyzed gaseous ammonia stable inorganic borane, polymers. condensed the They monomeric N H 2 B H 2 product at 77 K, then allowed it to polymerize as it warmed. In 1979 a microwave spectrum of N H 2 B H 2 was obtained, using a sample formed from the reaction of 5-10 and diborane at 500 ° C . least-squares thereby fit were establishing 1 2 9 Molecular constants calculated by a consistent the mTorr each of ammonia with symmetrical a planar structure aminoborane, rather than the asymmetrical N H 3 B H . configuration, NH2=BH2 for Perhaps the most important outcome of this work was the determination of the dipole moment to be 1.844 D in the direction from N to B, as opposed to the theoretical predictions of B to N . 1 2 0 ' 1 2 1 The assumption of an N B direction for the dipole moment was based on the observation that the dipole moment of N H 2 B H 2 is 0.751 D smaller than that in B H 2 B F 2 . 115 The same group recently reported microwave spectra of five isotopic species of N H 2 B H 2 , improving the constants and geometric parameters obtained in the previous s t u d y . 130 Recently, at the University of British Columbia, the first gas phase Fourier measured. transform infrared spectrum of aminoborane was The synthesis combined the solid-state and vapor- 1 3 1 phase ammonia borane pyrolysis techniques. Solid NH3BH3was heated to about 70 ° C in a flow system maintained at approximately 200 microns, and the vapors produced were passed through a furnace at about 400 ° C , to pyrolyze unreacted sublimed sample. aminoborane's eleven infrared (IR) active were recorded at medium resolution (0.05 band at 1337 c m cm- ). 1 fundamentals present that 1 time the bands of all of the IR 1 3 2 work - active resolution 1 though V5 is vanishingly weak because its dipole appears completed cm- ), with the V4 A-type have been recorded at U B C at 0.004 c m - (see Table 5.1), derivative vibrations being also recorded at very high resolution (0.004 1 Since fundamental Nine of 1 3 3 ' is transform IR study to 1 3 4 a be very has been , with the remainder currently underway. The contribution small. to Some the high analysis resolution Fourier of aminoborane, being the rotational analysis of the C-type V7 fundamental whose origin is at 1004.7 c m - . 1 116 Table 5.1. Vibrational fundamentals of gaseous N H 2 B H 2 1 1 Symmetry Ai A 2 Bi cm- V1 3451 NH symmetric stretch v 2495 BH symmetric stretch V3 1617 NH2 symmetric bend v 1337.474 2 4 V5 1145 V6 837 V7 V8 B 1 2 3 2 Type of motion 1 1 1004.6842 612.19872 BN stretch BH2 symmetric bend Torsion (twist) BH2 wag N H 2 wag 3533.8 NH asymmetric stretch vio 25643 BH asymmetric stretch V11 1122.2 NH 2 rock V12 742 BH 2 rock V9 Reference (131) Reference (133): Reference (134) vs (1,0) band; reference (132): vs (2,0) band 117 V.B. The Michelson Interferometer and Fourier Transform Spectroscopy. The infrared interferogram transformed with a B O M E M was DA3.002 associated software (version 3.1). recorded and Fourier Michelson interferometer and Three sources of infrared light are available depending on the wavelength region desired: a quartz- halogen lamp for the near IR and visible regions, a globar for the mid-IR, and a mercury-xenon lamp for the far IR. After first being filtered and focused at an aperature, the infrared light passes to a collimating mirror beamsplitter, and where it is reflected as a is divided in two. parallel One beam beam to a continues through to a fixed mirror, while the other is reflected onto a mirror moving at constant velocity. As one of the beams has a fixed path length and the other a constantly varying one, the recombination of the beams at the beamsplitter produces a resultant of sinusoidal waves that are out of p h a s e . 1 3 5 absorbed measured by the sample is The portion of the resultant not at the detector as the interferogram. The point along the moving mirror's travel at which the moving zero fixed and path sinusoidal difference waves mirrors into interferogram two light the exactly (ZPD)--should phase, producing a maximum in the Because are in with amplitude. interference equidistant-called principle constructive it all the interference 135 patterns producing the result from the optical path difference beams, bring the is essential that constant intervals of mirror displacement. signal infrared between sampling occur the at This is achieved in the BOMEM DA3 spectrophotometer by a He-Ne laser. Operating at 632.8 118 nm, or 15796 c m " , the laser provides an extremely 1 base of 31,592 cycles per cm of mirror t r a v e l . 136 precise time The cycles, called fringes, trigger spectral sampling at a frequency normally equal to one sample/laser fringe, though the rate can be increased to up to eight times the provided by prevents destructive transition temperature this laser fringe laser excellent: interference frequencies, dependent resulting uncertainty is frequency. and its its by two thermal fluctuations in the The in the phase coherence single-mode other operation closely stabilization laser lying removes optics. The mirror's position is 0.0025 fringes per cm of mirror travel, which even at the maximum translation of 125 cm amounts to a variation of only 0.3 fringes over the length of the mirror's scan. 1 3 6 The interferogram not only requires that its points be sampled at precise intervals, but also that one of these points occurs at an origin that is exactly reproducible from scan to scan. DA3 spectrophotometer acheives this by The BOM EM triggering the commencement of each scan at the ZPD of an interferogram of white light. The beams from the white light source follow the same optical path as that of the radiation of interest, with the incoherent nature of the white light producing an interferogram characterized by an intense pulse at ZPD (the WLZPD), and low intensity amplitudes at non-zero mirror translations. The occurrence of the pulse is precise to well within one laser fringe, so the actual W L Z P D trigger is marked as the laser fringe immediately following the pulse. The result is a synchronization signal which references the points in the 119 IR interferogram to a constant position along the scanning mirror's path.136,137 A Fourier transform infrared experiment is therefore the process of obtaining the infrared interferogram in conjunction with the white light reference interferogram and the time base generated by the He-Ne laser. from the IR These data are processed by Fourier transformation interferogram frequency domain. time domain to an IR spectrum in the The integrals of the Fourier transformation can be understood in terms of the phase differences between beams split frequency by the co reflects beamsplitter. off a When mirror moving a wave with with the IR angular velocity v, the frequency is Doppler shifted by an amount 38 1 Aco = 4JIV/X Expressed as a function of the (5.1) speed of light and the incident frequency, using the relation X= 2TIC/CO, the phase shift b e c o m e s 3 8 1 Aco = (v/c)2co (5.2) The magnitude of Aco is on the order of 1 kHz to 100 kHz, a frequency that can be processed easily as compared to the 1 0 frequencies of IR radiation 1 3 to 1 0 Hz 1 5 itself. The time-averaged beat intensity, I, produced by the combination of two waves out of phase by Aco i s 3 8 1 I = l (1 0 where l 0 + cosAcot)cos [(co + co')t/2] = (l /2)(1 + cosAcot) (5.3) 2 0 is the signal intensity when Aco = 0. Represented in terms of amplitude or electric field strength [E (co)], phase difference [5(co) 0 = Acot], and the reflectivity beamsplitter, equation (5.3) (R) and transmittance (T) becomes 38; 1 l(co,8) = cec-RT |E (co)2| [1 + cos8(co)] 0 (5.4) of the 120 where c is the speed of light and e equal to 8.85 x 1 0 - 1 2 C J- rrr . 2 1 is the vacuum permittivity , 139 0 Integrating over all frequencies of 1 the spectral components, l(8) = J l(co,5)dco = ceoRT[J |E (co)| dco + j [E (co)| cos8dco] 2 At zero path difference, (5.5) 2 0 0 or 8 = 0, the two terms in brackets in equation (5.5) are equal, so the ZPD intensity is given by: l The time-averaged difference, = 2ce RTJ |E (co)| dco 0 signal 0 intensity l(8), is the quantity interferogram (5.6) 2 0 as a function of phase measured at the detector. The points themselves are taken to be the oscillations of these intensities about l /2: 1 3 8 0 |l(8) - l /2| = 0 |E (co)| cos8dco (5.7) 2 C £ 0 R T J 0 The cosine Fourier transform of an interferogram of the form of equation (5.7) yields a spectral intensity distribution function l(co) in which intensity is a function of discrete frequencies: l(co) = However, equivalent since (1/KRT) J [l(8) - l /2]cos8d8 imperfections reflectivities (5.8) 0 in manufacture in the fixed do and moving not produce mirrors, sine components as well as cosine are introduced into the interferogram. The actual Fourier transform therefore employs the complex form of the e x p r e s s i o n 1 3 8 - 1 4 0 ' 1 4 1 l(v) = C J [l(8) - l /2] e - * d 8 i 2 v 5 0 In general form, the Fourier transform of function f(x) 3{f(x)} = F(a) = Jf(x)e-'«xdx (5.9) is 1 4 2 (5.10) The inverse Fourier transform of F(a) is therefore 3" {Fa)} = f(x) = 1 (1/2TI)J F(a)e « da ! x (5.11) 121 Likewise, the spectrum expressed in equation (5.9) is one member of a Fourier pair, which consists of two non-periodic functions related by the Fourier integral t r a n s f o r m s 141 : g(v) = J f(5)e'2"v6ds (5.12) f(8) = 1 g(v)e-'2*v8dv (5.13) A Fourier pair is illustrated graphically in Fig. 5.1. Fourier transform spectroscopy is able to exploit the Fourier pair relationship between domain the time domain (phase, 5) and the frequency (co or v), because frequency can be obtained with greater accuracy, resolution and phase differences frequency. speed by measuring than by directly rather With the Michelson and transforming measuring interferometer the relative integration cannot be performed over all space (-°° to +°°) but is limited to the range 0 - L where distance travelled included in the L is the total by the mirror integration maximum therefore inversely difference between increases, the number increases, extending information available for extraction theoretical mirror displacement. spectral the fixed of terms the amount into the spectrum l ( v ) . resolution proportional As the to of an the and moving The 1 4 1 interferometer maximum optical mirrors. 1 4 2 of is path Defining resolution as the full width at half height, the maximum unapodized resolution is: 1 4 4 Avi/2 = 1/(2L) (5.14) Imposing the 0 to L limits on an interferogram "boxcar" truncation interferogram is (see Fig. 5 . 2 ) . Fourier 1 4 4 transformed, When the contains the sine function [sine z = (sin z ) / z ] : is known as a a boxcar-truncated spectral 1 4 6 « 1 4 7 line shape 122 Fig. 5.1. A polychromatic signal in the frequency domain (above) Fourier transformed into the time domain (below). 1 14 123 F{D(x)} = 2L(sincz) where z = 2ji(a-a )L. The half-width of the center spike of this form 0 is very narrow: (5.15) A a = 1.207/2L, theoretical resolution of 1/2L. or about 20% wider than the However, the sidelobes next to the central peak have about 21% of its intensity, and the amplitudes of subsequent lobes are slow to die a w a y . 1 4 5 In order to approximate more closely the true frequency domain spectrum, an apodization function is often included in the data processing. This process dampens the effects caused by truncating the interferogram definite mirror displacement of L. at a Though there are many forms of apodization functions, the effect is to give decreasing weight to the data points recorded of the simplest is at large the triangular all sidelobes are positive of the center spike; the boxcar c a s e . 1 4 6 mirror and the displacements. function in 1 4 5 Fig. 5.3, largest is only about - One 1 4 6 in which 4.5% that the linewidth is increased by almost 50% over - The apodization applied to the aminoborane 1 4 7 experiment in this work was a cosine function "Hamming" or "Happ-Genzel". It referred produces spectral to as lines with negative sidelobes of only 0.0071 the height of the maximum peak, and lines about apodization. 145 2% broader than those from the triangular 124 ^ V Fig. 5.2. A boxcar function D(x) (above). The Fourier transform of a boxcar truncated interferogram is a spectrum with the line shape function F{D(x)} = 2Lsin(27ivL)/27tvL (where L denotes the maximum mirror displacement.) The full width at half-height (Avi/2) is 1.207/2L, and the strongest sidelobe has 21% the intensity of the maximum. 1 4 5 125 Triangular Fig. 5.3. spectrum (below). strongest D(x){1 - |x|/L} The triangular apodization function D(x) (above) produces a with the line shape function F{D(x)} = 2Lsin(27tvL)/(27cvL) The full width at half-height (Avi/2) is 1.772/2L, with the sidelobe only 4.5% of the maximum intensity. 2 145 126 V.C. Experimental. The aminoborane was prepared by pyrolysis of borane ammonia (BH3NH3, Alfa Products) according to the procedure of Gerry and coworkers the 1 3 1 , except that in the present work the temperature of solid N H 3 B H 3 was raised to only 67 ° C - 68 ° C for the first several hours, then lowered to 63 ° C - 65 ° C for the remainder of the experiment. 131 was The 70 ° C pyrolysis temperature employed in reference found to be uncontrolled thermal approximately 71 ° C . unnecessarily close to the decomposition, which temperature initiates violently of at At the time the interferogram was measured, the temperature of the solid ammonia borane was 63.5 (±0.5) ° C . The sample absorption cell, maintained at a set pressure of to an optical path of 9.75 m, IOOJJ. during data acquisition. was The B O M E M DA3.002 interferometer was fitted with a potassium chloride beam splitter and a liquid nitrogen-cooled HgCdTe detector. 127 V.D. The Asymmetric Rotor. A vibrational fundamental is infrared active if the dipole moment JI changes as a result of motion along the normal coordinate Qk, or in other words if the derivative (3|i/3Qk)o in the Taylor series expansion of the dipole moment u = uo + I ( 3 u / 3 Q ) Q k is non-zero. 1 4 8 0 (5.16) k The linear character of the dipole operator means that its components transform as translations along the principal axes, and therefore so do the various (3^i/3Q )o Qk's. k Aminoborane is a prolate asymmetric top molecule belonging to the point group C 2 v , whose character table is given in Table 5.II. irreducible representations of the normal vibrations are: 2Bi + 2 B , for a total of twelve fundamental vibrations. 2 out-of-plane reflection wagging vibration in the yz plane, representation is antisymmetric 5Ai + A + 2 The B H 2 with respect to transforms as the Bi Thus the V7 vibration represents and therefore (see Fig. 5.4). The translation along the c inertial axis and generates a C-type infrared band. Accompanying any molecular vibration are the rotational transitions involving changes in the total angular momentum, J . In order to understand the rotational selection rules for an asymmetric top molecule, one must write down asymmetric top rotational wave functions which are eigenfunctions of the symmetry operations of the molecular point group, in this case C the effects of the C 2 v 2 v . We begin by examining symmetry operations on the symmetric top wave functions, YJK(6,<|>). From equation (1.23) we know that: YjK(e,<>) - NPjK(cos 6)e'K<|> (5.17) 128 Table 5.II. Character table for the C point group, and the correlation of the axes of translation to infrared band type. The molecule-fixed axes x, y, z given here are related to the inertial axes a, b, c by the l representation. 2 v r Rotation (R) and Translation (T) axes E c Ai 1 1 1 A 2 1 1 -1 -1 Ra (Rz) Bi 1 -1 1 -1 T o Rb (Ry) B 1 -1 -1 1 T . Rc (Rx) C 2v 2 2 CTv(XZ) oV(yz) 1 T b a 129 where N is a normalization Legendre factor, Pj (cos 6) K is an associated polynomial, and the spherical polar angles 9 and $ are shown in Fig. 5.4. A C rotation about the a inertia! axis (C2< )) adds an amount n a 2 to <|>, but does not change the 6 coordinate: C (a)Yj (8,(t)) = NPjK(cos e)e'KM>+«) (5.18a) = NPjK(cos e)eiK<J)eiK« (5.18b) 2 K = eiK*Yj(e,<|>) (5.18c) K where ( - 1 for even K e'K* \ I. = -1 for odd K Note that the operation of C 2 on YJK(9,<)>) (5.19) gives a multiple of the original spherical harmonic, YJK(9,<1>). C 2 rotations about the b and c inertial axes are not symmetry operations of the CZM point group. Unlike CzW, the a the angles 6 and (>. associated Rodrigues Legendre formula a c v and a a b v operators reverse the directions of Both reflections change e into -9, causing the polynomial to become K a c v changes <> | to -<|>. o a b v opoosite direction and changes <J> t o n -<J>. reflections are K C0S 0) By the (5.20) projects the c axis in the The overall effects of the therefore: a Yj (9,<)>) = (-1)J NPjK(cos 9)e-iK<|> ac v 9). 149 P j ( - c o s 9) = (-1)J+KpjK( The operation of o Pj (-cos (5.21) +K K and o- YjK(9,(|>) = (-1)J NPjK(cos 9)e' ^e-' t> ab v +K K K( (5.22) Clearly the spherical harmonics themselves are not eigenfunctions of the reflection operators, though the linear combinations obtained 130 a(z) >b(y) Fig. 5.4. Schematic drawing of the C N H 2 B H 2 molecule in the x, y, z principal axis system and the a, b, c inertial axis system, showing the C2 o reflection planes. 2 v v 131 by taking Wang sum and difference f u n c t i o n s are eigenfunctions 150 of these operators: *FJK± = ( 1 / V 2 ) ( Y In equation (5.23) respectively) the sums correspond to J K and the ± Yj,. ) (5.23) K differences upper and (JK+ and J K . , lower asymmetry components of a JK level. The effects of the C asymmetric 2 top rotational symmetry v operations, wavefunctions, follow performed from on equations (5.18c), (5.19), (5.21), (5.22) and (5.23): C ( ) ^JKt = (-1) ¥JK± a (5.24) K 2 Cv a c ^ J K t = (1/V2)(-1)J-K Y j , . - ± ( - 1 ) J + K(i/V2) ( Y J K (1/V2)(-1)J+K Y ± K ± K (5.25) M a b x v F j K ± = ("1) (-1) K J+K, K Yj,. ) =+(-1) fjK± o J PJK± -±(-1)J*jK± (5.26) For even and odd values of K, and the + and - asymmetry components, the result of each operation can be tabulated using (5.24) through (5.26), as given in the first irreducible representations two sections of Table 5.III. The in the third section of Table 5.Ill are obtained by substituting even and odd values for J into section 2. The quantum numbers K a and K c in section 3 denote the projections of the angular momentum components J lowest and highest inertia. and J a The values of K and K a along the axes of c c corresponding to each irreducible representation are derived from the rule that K = J c - K a and K = J - K respectively. c a + 1, for the + and - asymmetry components, For example, for even J , even K a and the - asymmetry component, K must be odd, giving K K = eo. The eo notation c a c 132 Table 5.III. Character sets for an asymmetric top rotational wavefunction in the C point group. 2 v Wang sum & difference J K a functions J Keven + J Keven J Kodd J Kodd + Irred. representations ( K K ) a c E a 2 a c v 1 1 1 1 -(-1)J (-1)J o a b v (-1)J -(-1)J 1 -1 -(-1)J H) 1 -1 (-1)J -(-1)J J c E±/0± Jeven Jodd notation A i (ee) A (eo) E+ A (eo) A i (ee) E - B (oe) Bi(oo) o + B i (oo) B (oe) o- 2 2 2 2 133 indicates that the rotational wavefunction is symmetric with respect to rotation about the a inertial axis and antisymmetric with respect to rotation about the c inertial a x i s . 1 5 1 The E /0 ± notation ± given in the last column of Table 5.Ill is explained in Section V . E . From Table 5.Ill, the selection rules for a C-type band are: Ai <=> B and A <=> Bi (5.27) ee <=> oe and eo <=* oo (5.28) 2 or in K K a c 2 notation: The restrictions on changes in K and K are therefore: a c A K = ± 1 , ±3, ±5,... and A K = 0, ±2, ±4,... a C (5.29) so that C-type bands consist of the following branches, in notation: Branch AJ. Intensitv AKa +1 +1 0,0 strong PP -1 -1 0,0 strong Q 0 +1 0,-2 intermediate PQ 0 -1 0,+2 intermediate rp -1 +1 -2,-2 weak PR +1 -1 +2,+2 weak r AKaAJ 134 V.E. The Rotational Hamiltonian. V.E.1. The Hamiltonian The rotational without Hamiltonian energy, T, of a freely rotating interaction. representing the purely + 2 y [B - (B + B )/2] J / 2 + 2 z x y z (Bx - B ) ( J 2 + J.2)/4 y A where A J+ 2 A + J. = (J 2 h/87i cl a are be identified 2 to (in cm- ) A + U ) A + ( J - i J ) , and the quantities 2 B 2 y are the 1 (5.20) + A x kinetic rigid asymmetric top molecule is: Hrigid = (B + B ) J / 2 x vibration x y rotational with the and A, respectively, for the l constants. rigid-rotor r rotational representation B 152 B X ) y a = and B z constants B, C which is appropriate for a near-prolate asymmetric top molecule. The third term of equation (5.20) (which vanishes in a symmetric A- top) produces a matrix representation . for H 9 r| ld that contains off- diagonal matrix elements with AK ± 2: < J , K ± 2 | J 2 | J K > = (f|2/4)[J(J + 1) - K(K ± 1 ) ] x [J(J + 1) - ( K ± 1 ) ( K ± 2)] '2 1/2 ± 1 A (5.21) . The matrix of H 9 r| can be factorized at once into blocks containing |d only odd or even values of K in the basis set (because no matrix elements of the type AK = ± 1 arise from (5.20). can be further original factorized symmetric similarity top by taking basis transformation 150 These submatrices sums and differences functions by means of of a the Wang : |J,0+> = |J,0> |J,K±> = (1Al2){|J,K> ± |J,-K>} , (K > 0) (5.22) The four submatrices constructed from the basis functions |J,K > are ± designated E and Q ± ± for even and odd K, respectively. 135 To obtain a more accurate description of the rotational structure of an asymmetric top, centrifugal distortion must be considered. Centrifugal forces cause expansion (or stretching) a rotating and distortion in molecule, which lead to deviations from the rigid rotor Hamiltonian that increase with increasing angular momentum. distortion Hamiltonian, The H'd, is therefore treated as a power series which adds higher degree angular momentum terms to the rigid rotor Hamiltonian: H'd = (f> /4) I 4 T „ 8 JaJpJrJs (5.23) P y afJyS where T p s is the centrifugal distortion constant and a , (3, y and 5 = x, a y or z . Y The number of terms 1 5 3 equation (5.23) is 81. in the general However, power series of symmetry constraints reduce the number to 6 for an orthorhombic molecule (i.e., one which possesses at least two perpendicular planes of symmetry), since all terms vanish which are antisymmetric with respect to one or more of the symmetry operations. All of the remaining terms have only even powers of J , since those with odd powers change sign under the operation Further two of Hermitian reduction routes: asymmetric conjugation and time of the orthorhombic the "asymmetric top reversal. Hamiltonian reduction" follows for 1 5 2 ' 1 5 4 one of the general top, or the "symmetric top reduction" for asymmetric tops that are nearly symmetric. In the A-reduction the J + term terms is eliminated, leaving only of the type was treated using Watson's + J. A K = 0, ± 2 , whereas the "S" reduced Hamiltonian retains AK = ± 4 , ± 6 , . . . Aminoborane 4 "A" terms. reduced 4 136 Hamiltonian. is: Written out completely 1 5 2 up to terms in J , this 8 1 5 4 I W A ) = B (A)J X + B (A)j 2 X Y - 25jJ (Jx 2 + Oj K J4j - J 2 + 0 2 z Z ) - 8 [J 2 y K + B (A)J 2 y 2 K j J 2 J 2 (Jx + 0 4 Z J K J 2 - 2 Z A K J 4 Z - J y ) + ( J x - J y ) J z ] + <X>J J 2 J K - AJJ4 -A j 2 2 2 2 + 2(j)jj4(J 6 Z 2 - 2 X 2 6 jy ) 2 + <!>JKJ tfz (Jx - J y ) + ( J x " J y ) J z ] + <MJz (Jx - J y ) 2 2 2 2 2 + (3x - J y ) J z ] + L j J 2 2 + L Jz 4 8 + LjJKJ Jz 6 + 2ljj6(j 2 . j 2 ) 8 K x + lKjJ [Jz (Jx 2 4 2 y | + J K 2 4 + L 2 J K J4J 4 + L z J4[j 2(j 2 . j 2) z 2 x y The fitting 2 x 2 2 2 4 aminoborane E K I 2 z ] 6 2 2 (5.24) employed in this work to analyze the vj band included all matrix elements through to the off-diagonal sextic terms octic ) j Z 6 program 2 y 6 " Jy ) + (Jx - Jy )Jz ] + I K [ J ( J X " J y ) 2 A jJ Jz 2 K K (j 2 . j + + ( J x - Jy )J z ] 2 2 (J ), 6 plus the diagonal elements from the terms: K = <J,K | H ( ) | J,K> A R O T = [ B ( A ) + B ( A ) ] J ( J + 1)/2 X - Y + 1) - Aj J(J + 2 AjJ (J 2 K + <*>JKJ (J 2 + + LjJKJ (J + 3 1) K + 2 2 1) K 3 2 + (B (A) + [B (A) + Z 1)K 2 - A K K K K 2 + 1) K 2 4 + 2 Y + OJJ3(J 4 0 j J ( J + 1)K4 + 0 K 6 + LJKJ (J B (A)]/2}K X + 1)3 + 1)4 + LJJ4(J L KJJ(J + K 1)K 8 + L K K 8 (5.25) EK±2,K = <J,K±2 | Hrot< > | J , K > A = {[B (A) - B (A)]/4 X + Y (J)jJ (J + 1) + 2 2 - 8jJ(J + 1) - 4 > J ( J + 1)[(K JK 8K[(K ± 2) + K ]/2 + ± 2) + K ]/2 + 2 2 2 <J) [(K K {[J(J + 1- K ( K ± 1)][J(J + 1) - ( K ± 1 ) ( K ± 2)]} / 1 2 2 ± 2)4 + K4]/2} (5.26) 137 V.E.2. Coriolis The interaction. only perturbation present in the \-? fundamental, up to the limit of this analysis at K ' =11, a affecting the all levels to an extent which increases quadratically with rotational quantum coupling of two simply, is a Coriolis interaction globally number K. vibrations by the certain combinations of A Coriolis interaction rotation vibrations of the is the molecule. generate an Put internal angular momentum which is part of the total angular momentum of the molecule. In other words, rotational 155 are not separable. motion This internal, or vibrational, angular momentum a vector, written n , is and vibrational components are Tlx, Ily whose and TIz- To obtain the rotational Hamiltonian, the vibrational angular momentum must be subtracted from the total angular momentum, P, to give the rotational angular momentum. H -f|2(j 2/| x the rotation-vibration H = fi2{ [(P - x ft )]2/l x x Instead of the simple form + J 2/| y y Hamiltonian + [(P x J 2/| )/2 + H + z z (in joules) b e c o m e s - n ) ] 2 / | y + [(P y (5.27) v i b y 2 - fh)]2/| 2 1 5 5 }/2 ' 1 5 6 + H ' 1 5 7 : v i b (5.28) H =fj2(P 2/| x x + p 2/| y y P 2/| )/ + +fi (n 2/iy+ n 2 / i 2 2 x The first angular y term y z 2 - f i 2 ( n x P x / l x + ftyPy/ly + n z P z / l z ) + n 2/i ) + H in equation momentum, is the z z (5.29) v i b (5.29), independent of the vibrational rigid rotor Hamiltonian, while the third term, independent of rotational angular momentum, affects only the vibrational energy. The second vibrational and total angular coupling. scalar the term, a function momenta, of represents both the Coriolis The Coriolis interaction can therefore be considered as the product of the rotational and vibrational angular momenta, 138 the magnitude of which increases the faster the molecule and the nearer the vibrations approach degeneracy. rotates According to Jahn's rule two normal coordinates Qk and Q| are coupled via an a axis Coriolis interaction only if the product of their representations is of the same symmetry as P - 1 5 7 a fundamental a-axis). cm" 1 Thus the V 7 (Bi) at 1005 c m - undergoes an a-axis Coriolis interaction 1 with the nearby gives the A 2 v n (B ) fundamental at 1122 c m - , since Bi x B 1 2 2 symmetry species (corresponding to rotation around the The v n (the B H 2 vibration in turn interacts with symmetric bending vibration) vs (A-|) at 1145 by a c-axis Coriolis interaction, while the direct product of the V 7 and vs produces irreducible symmetries B i symmetry for a b-axis Coriolis interaction. Each of these three vibrations is therefore affected by the other two. The vibrational angular momentum, in units of fi, is defined a s II« - I k.l Ck|(«)qkPl(co|/co ) k 1/2 1 5 5 : (5.30) where the normal coordinate Q and its momentum conjugate, P = -ih3/3Q, are expressed in the dimensionless forms, q and p: Qk = Y k 1 / 2 Qk (5.31) Pk = Pk/Yk h (5.32) Yk = 27iCG)k/h (5.33) 1/2 The Coriolis coupling constant, Ck|(°0, is a measure of the angular momentum about the a-axis induced by the interaction of two normal vibrational modes, Qk and Q|, having frequencies (in cm- ) of 1 cok and co|. 139 V.F. Band Analysis and Discussion. Aminoborane's is BH2-wag forms a C-type band whose appearance characterized by a central spike, due to the asymmetry of the molecule causing low-K Q branches to pile up about the band origin (see Fig. 5 . 5 ) . At high resolution (Fig. 5.6), it can be seen that the 158 spike is composed largely of the two lowest Q branches, P Q i and Q o r (using the notation AK aAJKa")- The | i n e s of the 1 1 B form of N H 2 B H 2 were assigned by a process of successive refinement of the upper state constants. The ground state constants were held fixed at the best values available so f a r , 1 3 3 and the structure of the band was calculated using a prediction program. As the upper state constants were improved the prediction became more accurate so that more lines could be assigned. The assignments were maximum of K upper state Boltzmann distribution value a limited to a equal to 11, as a result of the at room temperature. Lines of ammonia, present as an impurity in the spectrum, were used as an internal standard for absolute frequency calibration. The N H 3 frequencies were taken from the diode laser study by Job et a l . 1 5 9 A complete set of molecular constants cannot be given at this time because the V 5 fundamental has not yet been observed directly since its dipole derivative is very small. extremely difficult to analyze the Without lines from V 5 , it is V5-V7-V11 Coriolis interactions. However, it is hoped that a sufficient portion of the V 5 band can be assigned in the near future to allow a fit to be made. The data were fitted to the matrix elements in equation (5.25) and (5.26) by means of a least-squares program written by Dr. Wyn Lewis-Bevan. program the Hellmann-Feynman theorem is used In this to calculate the 140 Fig. 5.5. N H 2 B H 2 spectrum of the v 7 band and the vs and v n with which it undergoes Coriolis interactions. bands 141 Fig. 5.6. Center of the V 7 band of N H 2 B H 2 . 1 1 142 derivatives of the energy levels with respect (see Section IV.C). The computations to the parameters were performed on the University of British Columbia Computing Centre Amdahl 470 mainframe V/8 computer. Two sets of molecular constants appear in Table 5.IV. Both were obtained by ignoring the Coriolis perturbation, but one was produced from a reduced data set of 606 transitions With a maximum K ' of 6. In the the excited state, all constants were floated diagonal sextics, namely §j, above six reduces the standard 0.001 and < | > J K , <J>K- A except Eliminating all off- values K ' A deviation in the line positions from cnrr to 0.0003 c n r . This is expected from the K dependence Coriolis coupling. The standard errors of most constants 1 of the 1 improved when the data set was reduced, except for very small ones (= 10" 8 cm- ) 1 particular that and with matrix elements dependent on K . O K J ,L K K J and I_K, Note in which accompany the variables K , 6 J ( J + 1 ) K and K , are very poorly determined in the reduced data set. 6 This 8 reflects the importance of a wide range of K values in determining terms containing high powers of K . Without including constants in Table 5.IV internally the Coriolis terms are not true values. consistent sets which Coriolis interactions in have order particularly evident in that the A K variables - K 4 and - [ ( K ± 2 ) 2 sensitive to Coriolis interactions. Hamiltonian, fit the Rather, they comprise incorporated the to the data. and 8 K constants rather than positive as they should be. the in the are effects of This is negative, Since A K and 8 K accompany + K ] , these terms 2 are the most An estimate of -0.406 was made for the V7-V11 a-axis Coriolis coupling constant ( £ 7 , 1 1 ) from the V7 143 Table 5.IV. Molecular constants of the \j band of N H 2 B H 2 (in cm) , for both the full and reduced (K ' ^ 6) data sets. The numbers in parentheses denote one standard deviation in units of the last significant figures. Where a ground state constant is blank, it was fixed to zero. 1 1 1 a EXCITED STATE Reduced T Full 1004.68420(5) 0 GROUND STATE 1004.6831(2) A 4.51446(2) 4.51512(3) 4.610569(8) B 0.9060531(8) 0.90605(2) 0.916897(2) C 0.7646658(7) 0.76467(2) 0.763137(2) Aj 1.173(1) AjK A K <E>JK <E>KJ <DK -1.197(6) L-KKJ K x 10-5 6 . 7 ( 2 . 2 ) x 10-10 -4.3(7) x -6.7(1.6) LjK o 10-5 1.116(6) x 10-7 §J L 1.04(1) x 10-6 - 1 . 1 7 ( 3 ) x 10-4 K 8 x 1.161(3) x 1.15(2) x 10-6 10-5 - 0 . 6 8 ( 1 ) x 10-4 1.06(2) x 10-7 -1.89(2) x 10-5 -3.3(2) x 5.7(4) x x 10-7 3 . 7 ( 2 ) x 10-7 4.6(1.3) x 10-8 6.4(3) x 10-11 10-10 3 . 8 ( 2 . 3 ) x 10-9 0.0003 9.87(3) x x 8.692(8) 10-6 10-6 x 10-5 2.86(3) x 1 0 1.016(2) x 7 10-5 10-10 10-8 - 2 . 5 ( 7 ) x 10-11 1.542(2) 0.001 7.0(32) x 5.94(30) 10-11 x 10- 9 144 and v n data. estimate of This is in good agreement with a force field 1 6 0 -0.40. 1 6 1 Appendix I. NbN 3<D_3A Correlation B' B' A ' D X' D A' X' i D' h.i' h ' 0 h+i' b' e2qQ' 1 A ' D X ' D -0.462 0.105 A' X' i U Matrix h.i' h ' 0 h + 1 ' b' e2qQ' 0.025 0.013 0.028-0.107 0.034 0.055-0.002 0.039 0.022 1 -0.159-0.258-0.158 0.333 0.157-0.032 0.022 0.047-0.319-0.004 0.004 0 . 0 4 9 - 0 . 0 7 2 0.242-0.143 1 0.248-0.273 1 0 . 2 4 1 - 0 . 5 1 5 - 0 . 2 3 5 0.012 - 0 . 0 7 3 - 0 . 0 0 4 1 -0.573 -0.187 0.124-0.391 1 0.517 -0.1 17 0.121 1 0.002 0.066 I 0.151 0.133-0.005 0.671 0.049 0.392 0.011 -0.011-0.706-0.013 0.151-0.119-0.016 -0.027 -0.004 -0.002 -0.068 1 -0.027 -0.075 -0.002 1 0.101 -0.018 1 0.041 I Ul T B' B" 0 A " D A" X" D -0.038 0 . 9 9 5 - 0 . 4 6 2 - 0 . 0 1 7 X" - 0 . 2 7 8 - 0 . 4 6 4 0.974-0.181 -0.275 -0.100 0.272 X' -0.108 D 0.091-0.206 0.884 h.r h M 0 h + 1 " 0.026-0.113 0.011 -0.131 -0.025 0.032 0.021 A ' D D" f 0.232 0.013 0.030 b.i/o" 0.027 b 0 / + r 0.023 0.027-0.318-0.318 0 . 2 5 0 - 0 . 2 5 5 - 0 . 1 6 8 - 0 . 2 5 7 - 0 . 0 6 8 0 . 1 9 5 - 0 . 1 0 6 0.127 0.146 A' 0.152 0.026 -0.355 0.396 0.999 0 . 2 3 5 - 0 . 8 8 2 - 0 . 4 5 5 - 0 . 0 2 6 - 0 . 0 7 8 0.041 X' 0.592-0.091-0.060-0.114 0.242 0 . 9 6 5 - 0 . 2 6 8 - 0 . 0 6 7 0 . 0 8 4 - 0 . 3 4 5 0.670 0.672 0.150 0.398 0.377 - 0 . 9 0 2 - 0 . 0 0 4 0.227-0.267 - 0 . 5 2 0 - 0 . 3 8 2 0.680 0 . 2 8 4 - 0 . 0 6 3 0 . 1 2 3 - 0 . 0 5 0 - 0 . 7 0 1 -0.701 i D' - 0 . 5 1 2 - 0 . 0 2 2 0.089-0.145 - 0 . 2 4 7 - 0 . 0 5 3 0.246 0.518 h-i' 0.111 h ' -0.116 0 0.008-0.017-0.031 0.036 0.005 0.221 0.021 - 0 . 0 7 3 - 0 . 4 1 5 0.072 0.105-0.011 0.034 0.034 0.073-0.110 -0.020 0.151 b' 0.408 0.025-0.350 0.312 T 1 0 B" A " D X\f A" 0.007-0.097 1 0.147 0.081 0.670 0 . 3 3 0 - 0 . 8 7 7 - 0 . 1 9 1 - 0 . 0 5 0 - 0 . 0 7 8 0.153 0.996 0.994 0.047 0 . 0 1 8 - 0 . 0 5 8 - 0 . 0 0 8 - 0 . 0 4 7 0 . 0 0 1 - 0 . 0 1 2 0.043 0.036 0.159 0 . 3 6 8 - 0 . 3 0 1 - 0 . 1 1 0 0 . 0 7 4 - 0 . 1 1 2 0.050 0.401 0.400 0.027 0.022 y" 0.012 - 0 . 1 3 0 - 0 . 0 1 7 0.028 0.031 0.346 0 . 2 3 4 - 0 . 8 8 3 - 0 . 4 6 0 - 0 . 0 2 1 - 0 . 0 7 8 0.028 0.669 1 - 0 . 2 2 1 - 0 . 0 4 8 0 . 0 6 9 - 0 . 3 6 5 0.148 1 D" 0.269 0.028 0.014 0 . 0 4 5 - 0 . 3 5 0 0 . 3 9 7 - 0 . 1 7 6 - 0 . 3 5 1 - 0 . 3 0 8 - 0 . 0 4 2 0 . 1 8 7 - 0 . 0 6 9 0.307 I X" 0.442-0.005 I h_i" h " 0.910-0.012-0.082-0.062 0.096 1 -0.197-0.364-0.045 0.056 0.149 0.121 1 h " + 1 b.1/0" bo/ i" + 0.339 -0.352 0.324 0.671 0.316 0.078-0.008-0.877-0.880 0.318-0.186-0.200 -0.010-0.004 1 0 0.010 0.002 0.356 0 . 0 0 0 - 0 . 0 0 6 0.933 - 0 . 4 5 3 - 0 . 0 2 4 0.025-0.108 1 0.134-0.116-0.012 0 . 1 0 6 - 0 . 0 6 0 0.152 0 . 8 4 8 - 0 . 0 1 6 0.003 h+i' e2qQ'-0.016 0 . 0 1 5 - 0 . 0 1 7 - 0 . 0 0 3 0.017 0.077 -0.051-0.051 -0.01 1 -0.079 -0.078 1 0.153 1 0.150 0-993 APPENDIX II. Appendix II.A. 3 Transitions of the 02- Ai. 3 E J" F" 1 1 4 .5 4 .5 4 .5 1 1 1 5 .5 qR rR pR qR rR pR 2 2 2 .5 . 5 .5 .5 .5 2 .5 3 . 5 3 .5 2 2 2 2 2 2 2 2 2 3 .5 3 .5 4 .5 4 5 4 5 5 5 5 5 6 5 6 5 CO rR 3 3 3 3 3 3 1 1 1 1 2 2 3 3 3 3 3 2 2 3 1 2 2 2 5 5 2 2 2 Q J" pR qR CO rR PR qR 2 5 2 3 6 5 5 .5 16146 .7154 16146 .8310 16146 .9692 16147 .3250 3 3 6 .5 7 .5 4 4 4 5 .5 6 .5 8 .5 5 5 4 .5 5 .5 6 .5 7 .5 8 5 16147 16147 16147 16147 .3517 .3705 .3890 . 3833 qR rR 5 5 5 5 5 5 pR qR CO rR 16148 16148 16148 16148 3506 3604* 3684* 3763 PR qR 16148 16148 3712* 3873 CO rR pR 3 3 5 5 5 5 5 16148 16148 16148 16148 16 1 4 8 3985 4098 4032 4257 4399 3 3 3 4 5 5 16148 16148 4541 4469 3 3 3 .3 3 3 3 3 4 4 4 16148 16148 16148 16148 16148 16148 16148 16148 16148 16148 16149 16149 16149. 16149. 16149. 16149 16149. 4755 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5. 5 6. 5 6. 5 7 .5 7 .5 O. 5 1 .5 qR rR qR CO rR pR qR CO rR qR CO rR qR rR qR rR rR qR 1 .5 1 .5 2 .5 2 .5 2 .5 CO rR 3. 5 3. 5 3. 5 4 .5 qR CO rR qR CO rR 4 4 4 .5 4 .5 5. 5 qR CO rR qR 4 5. 5 CO 5 5 5 5 5 9 5 5 6 7 6 6 6 6 6 7 7 9 10 2 3 5 5 5 5 5 5 5 7 7 7 7 7 4 5 6 7 8 5 5 5 5 5 7 7 7 8 8 8 8 8 8 9 5 10 5 1 1 5 3 4 5 5 5 6 7 8 9 10 1 1 12 4 5 6 6 6 7 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4514 9 9 9 7 8 8 9 9 9 10 10. 10. 1 1 1 1 1 1 4670 16149. 4807 9 9 12. 5 12. 5 16149. 16149. 16149. 16149. 16149. 16149. 16149. 4930 5103 5375 5582 5782 6 1 18 6583 6996 7512 3518 3553 3607 3661 3725 3801 3875 3969 4064 4159 4282 4397 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 £ F ft 16146 .4540 16146 .5474 16146 .6629 16147 .4205 16147 .4442 16147 .4682 16147 .4643 16147 .5121 16147 .5698 16147 6272 16147 6948 16147 7669 16147 8436 pR qR rR 3<D_3A 147 System of NbN.a 5 5 5 5 5 5 5 .5 .5 .5 qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qo qO qO qO qO J" 16144 7074 16144 16144 9179 5975 16144 6982 16144 .8162 16144 5255 16144 .5854 16144 .7337 16144 .4281 16144 .4622* 16144 .5010 16144 .5462 16144 5975 16144 6555 16144 3 9 6 1 * 16144 4 2 3 2 * 16144 4 5 5 6 * 16144 5314 16144 5764 16144 16144 2829 2937 16144 16144 16144 16144 16144 3069* 3241 16144 16144 16144 16144 16144 16144 16 144 qQ 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 qO 16144 16144 CO qO CO CO CO CO qO CO CO qO CO CO qO CO CO qO 16144 16144 16144 16144 16144 16144 16144. 16144. 16144. 3445* 3674 3938 4232 4556 4915 2216 2325 2450 2 GOO 2781* 2976 3198 3445 3719 4024 1476* 1575* 1647* 1695* 1750* 1830 1888* 1989 2049* 2092* 2157 2226* 2279* 2350 2431* 2485* 2562 16144. 2645* 16144 2710* 16144. 2794 F" 3 3 6. 5 7 .5 PP pP 4 3. 5 4 .5 5. 5 pP 4 .5 PP PP 4 4 5 5 5. 5 6. 5 7 .5 5 5 5 8 .5 9. 5 3. 5 5 6 6 6 6 6 4 .5 5 .5 6 7 5 5 6 6 8 9 5 5 6 7 10 2 3 4 5 6 7 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 5 5 5 5 5 5 5 5 9 5 10 5 1 1 5 3 .5 4 .5 5 .5 6 .5 7 .5 8 .5 8 9 .5 8 8 8 9 9 9 9 1 0 .5 1 1 .5 12 .5 4 .5 5 .5 6 .5 7 .5 9 9 8 .5 9 .5 1 0 .5 1 1 .5 12 .5 13 .5 9 9 9 9 PP PP PP pP pP PP PP pP PP pP PP pP pP 1 6 1 4 1 . 71 14 16141. 8930 16140. 4155 16140. 4653 16140. 5293 16139. 4474 16139. 4858 16139. 5321 16139. 5875 16139. 6506 16139. 7216 16138. 3884* 16138. 4083* 16138. 4339 16138 16138 4648 5015 16138 16138 5435 PP pP 16138 16137 PP pP PP PP pP pP 16137 16137 16137 16137 16137 16137 PP pP PP pP 16137 16137 16137 16136 16136 16136 16136 PP pP PP PP PP PP 5909 6441 3347 3449 3592 3773 3992 4251 4548 4886 5263 5678 .2870 .2976 .3108 . 3270 1 6 1 3 6 .3461 16136 . 3682 PP 16136 16136 16136 16136 16135 16135 16135 16135 16135 16135 16135 16135 PP pP 1 6 1 3 5 .3681 1 6 1 3 5 .3961 PP PP pP pP PP pP PP pP PP pP .3931 .421 1 .4520 .4858 .2249 .2350 .2472 .2616 .2784 .2974 .3187 . 3422 148 Appendix II.A, continued. 3® &i. H J" 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 e 6 6 6 6 6 6 6 6 6 6 6 6 6 S 6 6 6 € € 6 6 6 e 7 7 7 7 7 7 7 7 Fn 5 .5 6 .5 6 .5 6 5 7 5 7 5 8 5 8 5 1 5 1 5 2 5 2 5 2 5 3 5 3 5 3 5 4 5 4 5 4 5 5 5 5 5 5 5 6 5 6 5 6 5 7 5 7 5 7 5 8 5 8 5 8 5 9 5 1 5 2 5 2 5 3 5 3 5 3 5 4 5 4 5 4 5 5 5 5 5 5 5 e 5 6. 5 6 5 7 5 7. 5 7. 5 8 5 8 5 8 .5 9. 5 9. 5 9. 5 10. 5 2. 5 3. 5 3. 5 4 .5 4 .5 5. 5 5. 5 5. b rR qR CO rR qR rR qR rR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR rR rR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR rR rR CO rR CO rR qR CO rR m3 2 P Q 16149 4943 16149 5134 16149 5292 16149 5446 16149 5678 16149 6026 16149 6298 16149 6681 16150 3267 16150 3306 16150 3348 16150 3401 16150 3454 16150 3513 16150 3581 16150 3649 16150 3726 16150 3808 16150 3891 16150 3989 16150 4086 16150 4 184 16150 4302 16150 4414 16150 4528 16150 4669 16150 4795 16150 4921 16150 5088 16150 5228 16150 5364 16150 5862 16151 2772 16151 2838 16151 2878 16151 2917 16151 2968 16151 3019 16151 3071 16151 3133 16151 3195 16151 3260 16151 3332 16151 3407 16151 3485 16151 3569 16151 3652 16151 3748 16151. 3843 16151 3937 16151 4046 16151 4153 16151. 4256 16151. 4385 16151 .4501 16151 4613 16151. 5009 16152. 2180 16152. 2246 16152 2287 16152. 2370 16152. 2419 16152. 2461 16152. 2521 16152. 2578 J" F It 9 13 5 9 13 5 10 5 5 10 6 5 10 6 5 10 7 5 10 8 5 10 8 5 10 8 5 10 9 5 10 9 5 10 9 5 10 10 5 10 10 5 10 10 5 10 1 15 10 11 5 10 12 5 10 12 5 10 13 5 10 14 5 1 17 5 11 8 5 11 8 5 11 8 5 11 9 5 11 9 5 11 9 5 11 10 5 11 10 5 1 110 5 11 11 5 11 11 5 11 11 5 11 12 5 11 12 5 11 12 5 11 13 5 11 13 5 11 14 5 11 15 5 12 7 5 12 8 5 12 8 5 12 9 5 12 9 5 12 9 5 12 10 5 12 10 5 12 10 5 12 1 15 12 1 15 12 1 15 12 12 5 12 12 5 12 12 5 12 13 5 12 13 5 12 13 5 12 14 5 12 14 5 12 14 5 12 15 5 12 15 5 12 15 5 CO qO qO q CO O q O CO q CO O CO qO CO CO qO CO CO q O CO qO qO q O CO CO q O CO CO qO CO CO q o CO CO q O CO CO q CO O CO qO qO qO C O CO CO CO qO CO CO qO CO CO qO CO CO q O CO CO q CO O CO qO CO CO qO CO 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16143 16143 16143 16143 16143 16143 16143 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16144 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 16143 2947* 3047* 0616* 0707* 0749* 0815* 0888* 0937 0989* 1018* 1075 1 132* 1 167* 1226 1289* 1328* 1393 1507* 1575 1775 1989 9767 9783 9824 9869 9889 9933 9982 0004 0053 0106 0134 0188 0246 0278 0334 0396 0431 0494* 0666* 0852 8591 8609 8671 8684 8722 8763 8778 8819 8864 8882 8928 8977 8997 9046 9097 9121 9175 9230 9257 9315 9374 9405 9464 9525 149 Appendix II.A, continued. 3<D2- Ai. 3 fl H J" Fn 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 6 5 6 5 6 5 7 5 7 5 7 5 8 5 8 5 8 5 9 5 9 5 9 5 10 5 10 5 10 5 11 5 11 5 3 5 4 5 4 5 5 5 5 5 6 5 6 5 7 5 7 5 8 5 8 5 9 5 9 5 10 5 10 5 10 5 1 15 qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR CO rR rR CO rR CO rR CO rR CO rR CO rR CO rR qR CO rR CO 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16152 16153 16153 16153 16153 16153 16153 16153 16 153 16153 16153 16153 16153 16153 16153 16 153 16153 16153 2632 2697 2762 2827 2900 2972 3047 3130 3210 3298 3388 3477 3578 3676 3772 3995 4096 1443 1507 1546 1621 1668 1758 1810 1914 1973 2091 2157 2290 2362 2432 251 1 2589 2754 0" 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 14 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 F 16 5 16 5 9 5 10 5 10 5 1 15 11 .5 12 .5 12 . 5 13 . 5 13 . 5 13 .5 14 . 5 14 5 14 5 15 5 15 5 15 5 16 5 16 5 17 5 10 5 1 15 12 5 13 5 14 5 15 5 16 5 17 5 18 5 10 5 1 15 12 5 13 5 14 5 15 5 16 5 17 5 18 5 19 5 12 5 13 5 14 5 15 5 16 5 17 5 18 5 19 5 20 5 12 5 13 5 14 5 15 5 16 5 17 5 18 5 19 5 20. 5 21 .5 13. 5 14 . 5 15. 5 16 . 5 17. 5 18. 5 CO qO qO qO CO CO qO qO CO CO qO CO CO qO CO CO qO CO CO qO qO qO qO qo qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qo qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO £ 16143 9559 16143 .9626» 16143 7435 16143 7524 16143 7553 16143 7562 16143 .7602 16143 .7698 16143 .7744 16143 . 7756 16143 .7801 16143 .7851 16143 . 7870 16143 7916 16143 7966 16143 .7986 16143 8036 16143 .8099 16143 8113 16143 .8168 16143 .8307 16143 6131 16143 6199 16143 6279 16143 6365 16143 6459 16143 6557 16143 6662 16143 6778 16143 6901 16143 4662» 16143 4721» 16143 4787 16143 4854 16 143 4931 16143 501 1 16143 5100 16143 5193 16143 5293 16143 5401 16143 321 1* 16143 3267 16143 3331 16143 3393 16143 3465 16143 3544 16143 3625 16143 3713 16143 3805 16143 1555 16143 1602 16143 1650 16143 1706 16143 1763 16143 1824 16143 1890 16143 1963 16143 2037 16143 2118 16142 9849 16142 9887 16142 9932 16142. 9978 16143. 0028 16143. 0081 Appendix II.A, continued. 3 02- Ai. 3 a R 0" 16 18 18 18 19 19 19 20 20 20 20 20 20 20 20 20 20 21 21 21 21 21 21 21 21 21 21 31 31 31 31 31 31 31 32 32 32 32 32 32 32 33 33 33 33 33 33 33 33 33 33 34 34 34 34 34 34 34 34 34 34 35 35 35 35 FM 19 5 20 5 21 5 22 5 15 5 22 5 23 5 15 5 16 5 17 5 18 5 19 5 20 5 21 5 22 5 23 5 24 5 16 5 17 5 18 5 19 5 20 5 21 5 22 5 23 5 24 5 25 5 27 5 28 5 29 5 30 5 31 5 32 5 33 5 28 5 29 5 31 5 32 5 33 5 34 5 35 5 28 5 29 5 30 5 31 5 32 5 33 5 34 5 35 5 36 5 37 5 29 5 30 5 31 5 32 5 33 5 34 5 35 5 36 5 37 5 38 5 32 5 34 5 35 5 36 5 qO QO qO qO qQ qO qO qQ qO qO qQ qQ qQ qQ qQ qQ qO qQ qo qQ qQ qO qQ qO qQ qO qQ qQ qQ qO qQ qO qO qQ qQ qQ qQ qQ qQ qO qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qO qQ qQ qQ qQ qO qO qQ 16143 16143 16143 16143 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16142 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16139 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 0140 0200 0266 0336 8074 8404 8462 6139 6166 6194 6230 6262 6301 6341 6387 6433 6486 4132 4152 4182 4207 4233 4261 4299 4337 4378 4417 8508 8488 8467 8423 8408 8382 8366 5394 5368 5321 5293 5268 5242 5217 2199 2171 2143 2113 2083 2053 2023 1994 1966 1938 8863 8832 8798 8764 8730 8696 8662 8628 8596 8562 5376 5299 5261 5222 Appendix II.A, continued. 3 C>2- Ai. 3 R Q. J " 35 35 35 36 36 36 36 36 36 36 36 36 36 37 37 37 37 37 37 37 37 37 37 38 38 38 38 38 38 38 38 38 38 39 39 39 39 39 39 39 39 39 39 40 40 40 40 40 40 40 40 40 40 41 41 41 41 41 41 41 41 41 41 42 F It 37 .5 38 5 39 .5 31 .5 32 5 33 5 34 5 35 5 36 5 37 5 38 5 39 5 40 5 32 5 33 5 34 5 35 5 36 5 37 5 38 5 39 5 40 5 41 5 33 5 34 5 35 5 36 5 37 5 38 5 39 5 40 5 41 5 42 5 34 5 35 5 36 5 37 5 38 5 39 5 40 5 4 15 42 5 43 5 35 5 36 5 37 5 38 5 39 5 40 5 41 5 42 5 43 5 44 5 36 5 37 5 38 5 39 5 40 5 41 5 42 5 43 5 44 5 45 5 37 5 qo qO qo qO qo qO qO qO qO qO qo qO qO qO qO qO qO qO qO qo qO qO qO qO qO qO qO qo qO qO qO qo qO qO qo qO qO qo qO qo qO qO qO qo qo qO qO qo qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO qO 16 138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16138 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16 137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16137 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16136 16135 5184 5147 51 10 1934 1893 1853 1812 1773 1729 1686 1642 1601 1559 8315 8270 8226 8181 8136 8087 8044 7995 7950 7904 4595 4548* 4505 4449 4403 4352 4305 4251* 4199 4150 077 1 0720 0669 0617 0563 05 10 0455 0403 0346 0293 6841 6787 6732 6677 6619 6562 6505 6444 6387 6329 2810 2749 2689 2631 2569 2510 2448 2387 2324 2261 8672 Appendix II.A, continued. 3<j>2- Ai. 3 E F 42 42 3 8 .5 3 9 .5 42 42 40 41 42 42 42 42 42 42 43 43 43 43 43 43 J" F 2 2 2 2 2 3 3 4 43 43 44 44 44 46 47 2 2 4 4 2 2 2 3 3 3 3 3 3 3 5 5 6 2 3 4 4 5 5 6 3 3 3 4 4 6 5 7 5 7 5 2 5 2 5 3 5 3 5 4 .5 4 .5 4 4 4 4 4 4 5 5 6 6 7 7 4 4 4 4 5 5 5 5 5 rR PR rR pR qR rR qR rR rR qR qR PR qR PR qR qR rR qR rR qR rR 16545 16545 16545 16545 16545 16545 16545 16545 16545 16546 16546 16546 16546 16546 16546 16546 9680* 8810 9621 8495 8954 9 5 1 1* 8684 9350* 9133* 8608 8555 8233* 8492* 8087* 5 qO 16135 .8485 5 5 qO qO qO 16135 16135 8421 8356 16135 16135 16135 8293 8227 8158 16135 16135 16135 16135 8091 4427 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 40 41 5 5 42 5 43 5 44 5 45 5 46. 5 47 .5 48 5 qO qO qO qO qO qO qO qO qO qO qO 16135 qO qO 16135 16135 qO qO qO qO 16135 16135 16134 16134 16134 16134 16134 16134 16134 16134 qO qO qO qO qO qO 16135 16135 16135 16135 4362 4298 4230 4164 4093 4024 3961 3889 3816 0081 0014 9947 9875 9804 9733 9664 9588 9515 9442 16546 16546 16546 16547 8422* . 8339 8792* 8246* 8763* 7981* 16547 16547 8109* 7965* 16547 16547 8139* 7948* 16547 16547 8158* 5 5 6 4 5 8 5 1 5 qO CO 5 6 6 6 6 6 2 2 2 3 4 CO qR rR 7910* 8197* .5 .5 qR rR 16547 .7888* 16547 .8209* 7 7 7931* 8178* F 3 3 3 3 3 4 4 4 4 4 4 4 4 16547 16547 16547 rR 8612 8549 2 .5 .5 .5 .5 qR rR 16135 3 6 6 6 6 6 7 7 7 qR rR qR 39 16135 d>3- A . J" 5 5 5 5 5 5 5 5 5 5 5 42 43 44 45 44 44 3 39 40 41 43 43 44 44 44 44 44 Appendix II.B. 43 44 45 46 38 qO qO 5 5 5 6 7 5 5 1 5 3 5 4 5 5 6 6 7 7 5 6 7 8 9 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5. 5 6 .5 7 .5 8. 5 10 4 qO rO qO rO qO qO qO qO qO qO rO qo rO qO CO qO qO qO qO qO qO qO qO qO qO qO qO qO 16542 16542 16542 16542 16542 16542 16542 16542 16542 16542 16542 16542 16542 16542 16542 8992 9654 9180 9944 9403 8182 8289 8379 16542 16542 8465 8574 9034 8707 9224 8858 7690 7723 6976 7028 7009 7054 7107 16542 16542 16542 16542 16542 16542 16542 16542 16542 16542 7165 7227 7298 7382 7473 6228 6265 6312 6354 6408 16542 16542 16542 16542 Appendix II.B, continued. 3 <E>3-3A2. E F 0" 12 13 14 15 16 17 qR rR rR rR rR 5 5 5 5 5 rR rR rR rR rR 16557 16557 16557 16557 16557 16557 18 19 1 1 12 13 5 5 5 5 5 rR rR rR rR rR 16557 16557 16558 16558 16558 4176 4219 1965* 1986 2008 16 16 14 15 5 5 rR rR 16558 16558 2029* 2056 16 16 16 17 5 5 rR rR 16558 16558 2086 2 1 16 16 16 16 17 17 18 19 20 12 13 14 15 16 17 18 19 20 5 5 5 rR rR rR 16558 16558 16558 2150 2186 2222 5 5 rR rR 16558 16558 9853* 9869 5 5 5 16558 16558 16558 9892 9917 9943 5 5 5 5 rR rR rR rR rR rR rR 16558 16559 16559 16559 9972 0004 0036 007 1 21 13 14 15 5 5 5 5 5 5 5 rR rR rR rR rR rR rR 0109 7605* 7635* 7652* 7679 5 5 5 5 5 rR rR rR rR rR 16559 16559 16559 16559 16559 16559 16559 16559 16559 16559 16559 16560 16560 16560 5268 5291 5314 15 15 15 15 15 15 15 16 16 16 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 18 18 19 19 19 19 19 19 19 19 19 19 20 20 20 20 20 20 20 20 20 21 21 21 8 8 10 11 n 5 5 5 5 5 4 4 15 15 15 16 17 18 19 20 2 1 22 14 15 16 17 18 19 20 21 22 23 16 17 5 5 5 5 5 5 5 5 5 5 18 19 5 5 5 20 5 21 22 23 24 16 17 18 5 5 5 5 5 5 .5 rR rR rR rR rR rR rR rR rR rR rR rR rR rR rR rR rR rR rR rR rR 16547 16547 16557 16557 16560 16560 16560 16560 16560 16560 16560 16561 16561 16561 16561 16561 7864* 8221* 3963 3980 4003 4027 4053 4076 4 1 10 4146 7702 7732 7765 7798 7832 7872 5248 5344 5371 5404 5437 5474 5514 2782 2808 2834 2863 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 14 F II 5 qQ 10 5 1 1 5 4 5 qQ qQ qQ qQ 9 5 5 6 7 8 9 5 5 5 5 5 10 1 1 5 12 5 7 5 8 .5 10 5 1 1 5 12 13 8 9 10 1 1 12 13 14 8 9 10 11 12 13 14 qQ qQ qQ qQ qQ qQ qQ 5 5 qQ qQ qQ qQ 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 15 5 16 5 10 5 1 1 5 12 5 13 5 14 5 15 5 16 5 17 5 5 5 5 12 13 5 14. 5 qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ 5 5 5 5 12. 5 qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ 15 16 17. 18. 2891 2922 16561 .2957 16561 2991 16561 3032 16562 .0156 16562 .0181 15 15 15 15 15 13. 5 14. 5 15 5 16. 5 16562 .0203 15 17. 16561 qQ 5 15 9 10 1 1 12 13 14 10 1 1 qQ qQ qQ qQ 5 qQ Appendix II.B, continued. 3 <J> -3A2. 3 B d" F" 21 21 19 5 20. 5 rR rR 21 21 21 21 .5 22. 5 23. 5 rR rR rR 21 21 24 5 25. 5 rR rR 16562 16562 0389 0425 22 22 22 17. 5 18 5 19 5 rR rR rR 16562 16562 16562 7433 7456 7483 22 22 22 22 20 21 22 23 5 5 5 5 rR rR rR rR 16562 16562 16562 16562 7506 7536 7569 7599 16 16 16 16 17 17 17 17 22 22 22 24 5 7633 5 5 rR rR 16562 25 26 7669 7706 18 19 5 5 5 5 16562 16562 16563 16563 16563 16563 4585 4608 4635 4660 4689 4722 23 23 23 23 23 23 23 23 23 23 24 24 24 24 20 21 5 14 5 5 5 5 5 5 5 5 rR 5 5 5 5 5 5 5 5 16 17 17 18 5 17 17 19 20 5 5 17 18 18 18 21 14 5 5 15 16 17 18 19 5 5 16563 16563 4785 4821 18 18 4864 rR rR rR rR rR rR rR 16563 16564 16564 16564 16564 16564 16564 16564 16564 16564 16564 16564 18 19 19 19 19 19 19 19 19 19 5 5 rR rR 16564 16564 rR rR rR rR rR 8543 8568 8597 8627 16564 16564 16564 8660 16564 .8687 16564 8729 16564 .8765 16564 8803 16565 .5304 16565 .5326 16565 .5349 1 6 5 6 5 .5381 1 6 5 6 5 . 54 1 1 16565 . 544 1 16565 .5474 16565 .5510 16565 .5547 24 24 24 24 25 26 27 25 25 25 20 21 22 25 25 25 25 25 25 25 23 24 5 5 25 26 27 28 29 21 22 23 24 5 5 5 5 5 5 .5 .5 .5 25 26 27 28 29 .5 .5 .5 .5 .5 rR rR rR rR rR rR rR rR rR rR rR rR rR 1609 1637 1668 1690 1724 1752 1785 1818 1857 1897 8514 5 5 5 5 5 5 5 5 qO qO qO 22 15 16 17 18 19 20 21 20 20 20 20 20 21 21 21 21 21 20 21 22 23 24 17 18 19 21 21 21 21 22 20 21 22 23 24 25 18 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 22 22 19 20 5 5 22 22 21 22 23 24 25 26 19 5 5 5 5 5 5 16566 .2068 16566 .2103 27 27 2 8 .5 2 9 .5 rR rR 16566 .2138 16566 .2171 23 23 qO qO qO 5 5 rR rR qO qO qO qO qO qO 5 20 5 2 1 5 18 19 27 qO qO 20 20 22 22 22 22 qO qO qO qO qO qO qO qO qO qO qO qo qO 5 5 20 20 rR rR rR . 1975 . 2006 .2034 20 14 15 5 5 2 3 .5 24 5 2 5 .5 2 6 .5 27 5 .5590 . 1951 18 19 22 23 16 17 16565 16566 16566 16566 16566 27 27 27 3 0 .5 2 2 .5 rR rR rR rR 15 16 17 5 5 5 5 5 5 5 5 5 5 4748 5 5 23 24 26 26 26 26 26 26 27 27 13 16 16 16 18 18 18 24 24 26 26 26 26 16 16562. 0321 16562. 0355 rR rR 26 27 28 18 19 rR 5 5 20 21 22 F 15 15 16563 16563 16563 22 23 24 25 19 rR rR rR rR rR rR rR H d" 16562 0228 16562. 0256 16562 0287 20 5 5 5 5 qO qO qO qO qO qO qO qO qO qO qO qO qo qO qO qO qO qO qO qO qO qO qO qO qo qO qO qO qO qO qO qO qO qO qO qO qO qO 16541 16541 16541 16541 16541 16541 16541 16541 16541 16541 1654 1 1654 1 16541 1654 1 5616 5662 3521 3547 3577 3608 3644 3683 3725 3767 1503 1529 1558 1591 16541 16541 16541 1628 1664 16541 1748 9346 16540 16540 16540 16540 16540 16540 1707 9370 9397 9428 9460 9494 16540 16540 9530 9571 16540 16540 16540 16540 16540 16540 16540 16540 16540 16540 961 1 7098 7123 16540 16540 4728 4755 16540 16540 16540 16540 16540 16540 16540 4781 4810 4843 4877 4912 4953 4993 16540 16540 16540 16540 16540 2245 2269 2297 2327 2357 16540 16540 2391 2427 16540 16540 16539 2466 2507 9633 16539 16539 16539 16539 9660 9688 9 7 17 9751 9784 9822 9858 16539 16539 16539 16539 16539 16539 7 150 7 180 7214 7247 7283 7322 7365 9901 6903 6928 155 Appendix II.B, continued. 3 d>3- A2. 3 R J" 27 F" 30.5 rR 27 31.5 rR Q. 16566.2211 16566.2254 F" J" 23 21 .5 qQ 16539 6956 23 22 5 qQ 16539 6988 23 23 23 24 5 5 qQ qQ 16539 16539 7021 7054 23 25 5 7092 26 27 5 5 qQ qQ 16539 23 23 24 24 24 5 5 5 5 5 5 7130 7172 4074 24 24 24 21 22 23 24 25 26 16539 16539 16539 16539 16539 24 24 27 28 25 25 5 5 qQ qQ 16539 16539 4275 4317 21 22 23 24 25 5 5 5 5 5 qQ qQ qQ 16539 16539 1073 26 27 28 5 25 25 qQ qQ 25 26 29 22 5 5 26 26 23 24 5 5 26 26 26 26 26 26 27 27 27 27 27 27 27 27 25 26 27 28 29 5 5 5 5 5 30 23 24 25 26 27 28 5 5 29 30 31 5 5 5 27 5 pR 3 3 3 3 3 3 3 3 3 3 4 4 4 4 5 5 5 5 5 5 5 qR CO rR pR qR CO rR 3 3 5 5 5 5 pR 3 3 3 3 3 3 3 5 5 6 6 6 6 7 5 5 5 5 5 5 5 16864 16864 16864 5123 16864 16864 16864 16864 16864 5240 5312 5385 4016 4167 4255 4348 16864 16664 2657 284 1 16864 16864 2950 3058 1039 1256 1383 rR 16864 16864 16864 16864 qR 16863 qR co rR pR qR CO 5 5 qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ 5 5 5 5 5 qQ qQ qQ qQ qQ qQ 16539 16539 16539 1099 1 129 1 158 1 191 16539 16539 16539 1227 1264 1307 16539 16538 1344 7968 16538 16538 8014 8043 8077 8107 8146 8183 8225 8264 4771 16538 16538 16538 16538 16538 16538 16538 16538 16538 16538 16538 16538 16538 16538 16538 4802 4829 4859 4895 4931 4969 5010 5050 3 d" 3 qQ qQ 4200 4236 <X>4- A3. 3 F » 3 4100 4131 4165 16539 16539 16539 25 d" qQ qQ qQ qQ 25 25 25 Appendix II.C. qQ qQ qQ 1509 9396 F M 7 7 7 2 3 4 7 7 7 7 5 6 8 9 9 9 9 9 9 10 10 10 8 10 5 4 6 8 9 1 1 12 6 7 8 <J" 5 5 5 5 qQ 5 5 5 1995* 1811* qQ qQ qQ 16860 16860 16860 16860 16860 16860 16859 1613* 1358* 1061* 0329* 9436* 5 5 5 qQ qQ qQ 16860 16859 16859 0412* 9476* 9134* 5 5 qQ qQ 16859 16859 5 5 5 5 5 qQ qQ 16859 16859 16859 16859 16859 8685* 8420* 7819* 7482* qQ qQ qQ qQ qQ qQ F" 7 8 9 6 6 6 6 7 7 7 7 10 3 4 5 6 7 7 7 7 8 9 7 7 5 5 5 5 5 5 5 7938* 8 8 10 1 1 3 4 7770* 7572 8 8 5 6 5 5 5 5 5 5 5 5 pP pP pP 5 5 pP pP pP pP pP pP PP pP PP PP pP pP pP PP 16854 16854 16854 16854 16853 16853 16853 16853 1702 1221 0689 0117 2286 2077 1823 1531 16853 16853 1202 0836 16853 16852 16852 16852 16852 16852 16852 0431 9993 9518* 1317 1151 0958 0726* 156 Appendix II.C, continued. 3 <I>4- A3. 3 E d" 3 3 4 4 4 4 4 F" 7 .5 7 .5 0 .5 2 .5 2 .5 2 .5 3 .5 4 4 4 3 .5 3 .5 CO rR rR PR 16863 16863 16865 16865 16865 16865 16865 qR CO rR 16865 .4154 1 6 8 6 5 .4201 16865 .4250* pR 16865 .3407 qR CO rR 16865 .3509 16865 .3568 16865 .3630 qR CO rR .9543 .9686 .5243* .4658* .4690* .4729* .4072 4 4 4 3 .5 4 .5 4 .5 4 .5 4 .5 4 4 5 .5 5 .5 pR qR 16865 .2600 16865 .2726 4 4 4 5 .5 5 .5 CO rR 6 .5 6 .5 6 5 6 5 7 5 pR 16865 .2796 16865 2870 16865 1656 16865 1802 16865 1886 16865 1970 16865 0577 16865 0743 16865 084 1 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 7 5 7 7 5 8 8 8 3 3 3 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 qR CO rR PR qR CO rR qR CO rR qR CO rR PR qR CO rR 5 5 PR 5 5 5 6 6 6 6 7 7 7 7 5 qR CO rR 5 5 5 5 5 PR qR CO rR pR 5 5 5 8. 5 qR CO rR qR CO rR 5 5 8.5 8 5 9 5 9 5 1 .5 2. 5 4 .5 4 .5 4 .5 5. 5 5. 5 5.5 6. 5 6. 5 6. 5 7 .5 16865 16864 16864 16864 0936 9557 9664 9773 16866 3218 3247 3283 2717 16866 16866 16866 16866 16866 16866 16866 16866 16866 16866 16866 16866 16866 2792 2833 2876 2 183 2274 2326 2378 1563 1667 1726 1785 0852 0975 1043 J" F n 10 10 10 10 10 9 J- 10 11 12 5 5 5 5 10 11 13 14 7 5 5 5 11 11 9 5 5 11 11 10 11 5 11 11 11 12 13 14 5 5 11 15 5 5 12 8 5 8 5 12 12 12 9 5 10 5 1 1 5 12 12 12 12 12 13 14 15 5 5 5 5 12 13 16 8 5 5 13 9 13 13 10 11 5 5 5 13 13 13 13 13 12 13 14 15 16 17 13 14 14 14 14 14 14 14 14 14 14 qR CO rR qR CO rR 16866 16866 16866 16866 1 6 8 6 6 1 109 16866. 0198 16866. 0273 16866. 0350 16865. 9338 16865. 9508 16867. 2702 16867. 2546 16867. 1976 16867. 2007 16867. 2037 1 6 8 6 7 . 1611 1 6 8 6 7 . 1647 1 6 8 6 7 . 1687 qR CO 1 6 8 6 7 . 1 182 16867. 1225 16 16 16 16 16 16 16 rR qR 16867. 1270 16867. 0689 16 17 qR rR rR rR p Q 5 5 5 5 5 pP pP pP 5 5 PP pP 5 5 PP PP 8 9 5 5 9 9 10 11 9 9 12 5 5 5 PP PP pP pP 10 11 12 4 5 6 7 qO qO qO qO 16859 16859 16859 qO qO qO qO 16859 16859 16859 16859 16859 6301 5991 6461 6301* 6124 9 9 9 9 16859 16859 5931 5719 9 9 qO qO qO 16859 16859 16859 5494 5251 4999 qO qO 16859 16859 4725* 4874 qO qO qQ qO 16859 16859 4725* 4567 16859 4389 16859 16859 4202 3998 3781 3552 qO qO qO qQ qQ qQ qO 16859 16859 16859 7360 7124* 6872 6598 3310 3310* 3183 qQ 16859 qQ 16859 qQ qQ qQ 0808 0642 0468 0283 0092 5 5 5 5 5 5 9 5 10 5 1 1 5 12 5 13 5 14 5 15 5 16 5 17 5 18 5 qQ qQ qQ 16859 16859 16859 16859 16859 16859 16859 16859 16859 16859 16859 16859 16859 16B59 16859 16859 16859 16859 qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ 3046 2896 2736 2564 2383 2187 1983 1771 1494 1377 1248 1112 0964 15 15 15 15 10 1 1 12 13 5 5 5 5 qQ qQ qQ qQ 16858 16858 16858 16858 9562 9451 9332 9205 15 14 15 16 17 18 19 1 1 12 5 5 5 5 5 5 5 5 qQ qQ qQ qQ qQ qQ qQ 16858 15 15 15 15 15 16 16 16858 16858 16B58 16858 16858 16858 16858 9070 8928 8777 8617 8453 8279 7512 7407 13 14 5 5 16858 16858 729B 7182 15 16 17 5 5 qQ qQ 18 19 5 5 5 7057 6926 6787 6644 6499 20 12 5 5 qQ qQ qQ qQ qQ 16858 16858 16858 16858 16858 16858 16858 6338 5344 qQ qQ qQ F" 8 8 8 8 8 16859 8 9 10 13 5 10 10 10 6 7 8 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 5 5 5 5 5 9 5 10 5 1 1 5 12 5 13 5 14 5 6 5 7 5 8 5 9 5 10 5 1 1 5 12 5 13 5 14 .5 15 .5 PP pP 16852 16851 0184 16851 16851 16851 9520 9147 8748 16851 16850 16850 16850 0102 9946 9764 9558 16850 16850 16850 9325 9072 9865 8793 8494 pP 16850 16850 8171 pP pP 16850 16849 8802 PP pP pP 16849 16849 16849 8656 8488 8298 PP pP 16849 16849 16849 16849 16849 16849 8089 7861 7615 7352 PP PP pP PP PP pP pP pP PP PP pP PP PP pP 16848 7827 7070 6771 7402 16848 7261 16848 7106 16848 6934 16848 6745 16848 6534 16848 6316 16848 .6086 16848 .5835 16848 .5575 Appendix II.C, continued. 3 04- A3. B. J" , Fm 6 6 6 6 6 6 6 e 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 7 5 7 5 8 5 8 5 8 5 9 5 9 5 9 5 io 5 10 5 10 5 2 5 3 5 5 5 5 5 5 5 6 5 6 5 6 5 7 5 7 5 7 5 8 5 8 5 8 5 9 5 9 5 9 5 10 5 10 5 10 5 1 15 1 15 1 15 3 5 4 5 6 5 6 5 6 5 7 5 7 5 7 5 8 5 8 5 8 5 9 5 9 5 9 5 10 5 10 5 10 5 1 15 11 5 1 15 12 5 12 5 12 5 4 5 5 5 6.5 6. 5 6. 5 7 .5 7 .5 7. 5 CO rR qR CO rR qR CO rR qR CO rR rR rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR rR rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR qR CO rR rR rR qR CO rR qR CO rR 3 Q. 16867 16867 16867 16867 16867 16866 16866 16866 16866 16866 16866 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16867 16867 16867 16867 16867 16867 16B67 16867 16867 16867 16867 16867 16869 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16868 16869 16869 16869 16869 16869 16869 16869 16869 0742 0792 0141 0197 0254 9532 9597 9659 8867 8940 9008 1433 1266 0739 0768 0795 0420 0456 0488 O057 O097 0136 9650 9695 9738 9200 9247 9297 8706 8760 8815 8172 8234 8292 0122 9957 9467* 9496* 9521 9187* 9219* 9250 8873 8909 8943 8526 8566 8604 8144 8188 8231 7731 7779 7826 7289 7341 7390 8738 8579 8346 8369 8391 8123 8152 8177 J" F N 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 18 18 19 19 19 19 19 19 19 19 13 .5 14 .5 15 .5 16 .5 17 5 18 .5 19 5 20 5 21 5 13 5 14 5 15 5 16 5 17 5 18 5 19 5 20 5 21 5 22 5 15 5 16 5 17 5 18 5 19 5 20 5 2 15 22 5 qO qO qO qO qQ qO qO qo qO qO qO qO qO qO qQ qQ qQ qQ qO qQ qO qO qO qQ qQ qO qQ 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16858 16857 16857 5253 5158 5037 4922 4801 4676 4546 44 12 4269 3054 2967 2868 2771 2665 2556 2439 2325 2197 2072 0564 0475 0383 0284 0186 0081 9973 9862 Appendix II.C, continued. 3 <J>4- A3. 3 R J" F n F n 9 9 9 8 .5 8 .5 8 .5 9 9 9 9 9 9 .5 9 .5 9 .5 1 0 .5 1 0 .5 9 9 9 1 0 .5 1 1 .5 1 1 .5 9 9 1 1 5 12 5 CO rR qR 9 12 12 9 9 9 9 22 22 22 22 22 22 22 13 13 13 17 18 19 20 16869 16869 16869 16869 16869 20 20 20 20 20 .5 .5 .5 qO qQ qQ 16857 9745 16857 8116 16857 .8037 .5 18 .5 1 9 .5 qQ qQ 20 21 5 5 16857 .7960 16857 .7872 16857 7782 16857 7689 16857 7593 16857 7492 qQ qQ qQ qQ 16857 7390 7286 1 6 8 6 9 .7051 16869 6621 21 21 16 17 5 5 qQ qQ 16857 16857 5463 5393 CO rR 16869 6663 6704 21 18 5 qQ 16869 21 19 5 qQ 16857 16857 5318 5236 qR CO rR 16869 16869 16869 6246* 6292 6334 21 20 21 5 5 qQ qQ 16857 16857 5155 5 5 5 5 rR rR rR rR 16880 16880 0768 0699 22 23 24 0628 0557 qQ qQ qQ qQ 5 16880 16880 5 5 5 5 rR rR rR rR rR rR rR 16880 16880 16880 0476 0398 0313 5 5 qQ qQ 16857 16857 16857 16857 16857 16857 5 5 qQ qQ 16857 16857 2550 2480 16880 16880 0228 0151 22 22 5 5 16857 16857 2410 2330 16880 16881 0063 4627 16857 16857 2250 2167 16881 16881 16881 16881 16881 5 5 5 5 5 24 24 5 5 5 5 5 24 24 24 24 25 26 27 28 5 5 5 5 qR rR rR rR rR rR rR rR rR rR 25 25 25 5 5 20. 5 21 .5 22 . 5 rR rR rR 25 25 23 5 24 . 5 rR rR 25 25 25 25 25 26 26 26 26 26 25. 5 26 .5 27. 5 28. 5 29. 5 21 . 5 rR rR rR rR rR rR rR rR rR rR 26 26 26 26 26 .7600 .7635 .7668 .7299 .7335 20 20 23 15 16 17 20 20 21 22 23 24 24 24 24 qR CO rR qR CO rR 19 qQ qQ 5 5 5 5 22 24 16869 .7877 16869 .7907 16869 .7936 22 5 2 3 .5 24 5 21 22 23 24 25 26 19 22 22 qR CO rR 22. 5 23. 5 24. 5 25 .5 26. 5 27 5 28 5 29 5 30 5 rR rR rR rR rR 16869 .7369 16869 .6975 16869 7013 20 20 16881 16881 16881 16881 16882 16882 16882. 21 21 21 21 21 22 22 25 17 18 16857 5070 4983 4894 4800 4706 2691 2621 22 22 19 20 21 22 23 24 5 5 qQ qQ qQ qQ 4569 4512 22 22 25 26 5 5 qQ qQ 16857 16857 2084 1997 4450 4387 23 23 18 19 qQ qQ 4320 4251 4 181 4113 4043 23 23 23 23 23 23 23 23 24 24 24 20 21 22 23 24 25 26 27 19 20 21 5 5 5 5 5 5 5 5 5 5 5 5 5 9789 9727 9664 qQ qQ qQ qQ qQ qQ 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 24 24 24 24 24 22 23 24 25 26 5 5 5 5 qQ qQ qQ qQ 16856 16856 16856 16856 5 24 24 27 28 20 21 qQ qQ qQ qQ 16856 16856 16856 25 25 5 5 5 5 25 25 25 25 25 25 25 25 26 22 23 24 25 26 27 28 29 21 5 5 5 5 5 5 5 5 5 26 26 26 22 23 24 5 5 1360 1305 1254 16882 1 193 1 6 8 8 2 1 138 16882. 1085 16882. 1020 16882. 0958 16882. 0897 16882. 0832 16882. 7961 16882. 7919 16882. 7862 16882. 7815 16882. 7766 16882 .7 7 1 2 16882 7660 16882 7603 16882 7548 16882 7491 22 22 5 qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 16856 9598 9528 9457 9386 9309 9233 9157 6761 6706 6645 6585 6522 6461 6391 6324 6256 6185 361 1 3560 3509 3454 3396 3342 3277 3217 3154 3093 0330 0284 0234 0188 Appendix II.C, continued. 3 3>4- 3. 3 a R Q. d" 26 26 26 26 26 26 27 27 27 27 27 27 27 27 27 28 28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29 a Transitions in units of crrr . asterisk. 1 £ F 25 26 27 28 29 30 22 24 25 26 27 28 29 30 31 23 24 25 26 27 28 30 31 32 24 25 26 27 28 29 30 31 32 33 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 qQ qO qQ qQ qQ qQ PQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qQ qO qQ qQ qQ qQ qQ qQ qQ qQ qQ qo qQ qO 16856.0135 16856.0079 16856.0028 16855.9973 16855.9916 16855.9862 16855.6942 16855.6846 16855.6800 16855.6754 16855.6705 16855.6663 16855 .6610 16855 .6560 16855.6509 16855.3426* 16855.3366 16855.3329 16855.3288 16855.3251 16855.3199 16855.3117 16855.3064 16855.3030 16854.9763 16854.9724 16854.9690 16854.9656 16854.9620 16854.9575 16854.9542 16854.9505 16854.9466 16854.9423 Blended lines are denoted by 160 APPENDIX III. Transitions of the V 7 F u n d a m e n t a l NH 1 1 2 BH . a 2 a E Branch J " rRO 0 1 2 1010. 1049 1011 . 9 0 7 7 1013 .7738 3 4 5 6 7 1015. 7095 1017 . 7 2 4 4 * 1019. 8287* 1022 .0 4 0 8 * 1024. 3696 8 9 1026. 8280 1029 .4 2 1 4 * 10 1 1 1032. 1525* 1 0 3 5 . 0 1 15 1037 . 9 8 5 4 1041 .0 5 3 8 1044 . 1942 10 1 1 12 13 14 15 1047. 3830 1050. 5983* 16 17 15 16 17 18 19 20 21 22 23 rR1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 1 1 1 1 12 12 13 13 14 14 15 15 16 16 17 17 1053. 8206* 1057 . 0 3 4 9 1008 .2841* 1008 .1281 1007 .9039 4 1007 .6227 5 1007 . 2 9 8 3 * 1006 .9468 1006 .5875* 1006 . 2366 1005 . 9094 1005 .6170 9 r01 3 2 2 3 4 4 1069 . 6 8 4 6 * 1072 . 7 8 8 5 1019 . 0 4 0 7 * 1018 . 8 9 1 2 1 0 2 0 .. 8 4 6 0 * 1 0 2 0 .. 4 0 5 8 * 1 0 2 1 .. 8 5 7 6 * 5 5 6 6 7 7 8 1 0 2 2 .. 7 1 8 9 1 0 2 4 .. 6 5 8 6 * 1 0 2 3 .. 2 6 4 5 1 0 2 6 .. 6 6 8 2 1 0 2 4 .. 6 4 5 9 * 1028 .7450* 1026 . 0 2 5 3 1 0 3 0 .8941 1027 .4255 1033 . 1 127 1028 .8737* 8 9 9 10 10 1 1 1 1 12 12 13 13 14 15 16 17 18 19 20 21 22 1035 .4025 1030 . 3893 1037 .7617 1048 1039 1050 104 1 1053 1044 2 3 6 7 8 1060. 2328* 1063 . 4078 1066 . 5 5 8 2 1031 1040 1033 1042 1035 1045 1037 1 rOO . 9923 .1900* .7003 .6999* .5309* .2437 .4928 .8616* .6025 .5364* .8683 .2621 . 2964* 23 24 25 26 27 .6386* 1061 . 7 9 8 1 * 28 29 3 3 4 4 19 20 1052 .5335* 1064 .7856* 5 5 20 1055 .5552* 6 18 18 19 1058 1046 1058 1049 .0367* .8873* .8497* r02 E Branch J" Branch 12 13 14 of 1005 1005 1004 1004 1004 rPO 1000 .7299* 999 . 4985 9 9 8 ,, 4 0 1 6 * 9 9 7 ,. 4 5 6 7 * 9 9 6 .6731 9 9 6 .0551 995 .6010 9 9 5 ,. 3 0 0 8 15 16 17 18 19 1016 .0602 1015 . 3731 1015 .8385 rP 1 1016 .8055 1014 .3608 1017 . 3 6 5 7 1013 .8658 1018 .0803 1 0 1 3 .. 2 9 7 8 1 0 1 8 ., 9 7 0 8 * 1 0 1 2 ., 6 6 4 3 * 20 21 4 5 5 6 7 7 8 9 9 10 10 1 1 1 1 1020. 0551 1011. 9726 1 0 2 1 .3 4 9 6 * 101 1 .2 3 2 7 1022 . 861 1 12 13 13 14 14 1010. 4563 1024 . 5 9 6 3 1022 . 4 5 8 3 1022 . 6 2 2 5 * 1022 . 2841 5 6 7 12 13 14 .9840* .8502 . 7468* 1004 . 6 7 1 1 * 1004 .6182* 1003 .6 7 8 6 1003 . 4 1 1 5 1003 . 1891 1003. 0034 1002 . 8 4 7 8 * 1002. 7176 1002 . 6 0 6 2 * 1002 . 5 3 4 5 * 1022 . 6 4 5 7 * 1022 . 6 7 2 4 * 1022 . 5 7 4 6 1022 .6 4 5 7 * 1005 .0696 1003 .5316* 1002 . 0 7 9 7 * 10 1 1 . 3645 1009. 6590 1008 . 8 5 5 6 1026 . 5538 1008 .0 6 4 9 1007 . 2 9 8 3 * 1006. 5875* 1005. 9323 1005 . 3 4 3 3 1004 . 8 2 5 3 * 1004 . 3 7 9 1 * 1 0 0 4 . 0 0 1 1* 2 3 4 8 9 . 1540 1015 . 1 179* 1016 .3770* 1014 . 7 8 0 1 * J" rP2 9 9 5 ,. 1 3 9 9 * 9 9 5 ,. 0 9 5 9 9 9 5 ,. 1 4 7 1 * 9 9 5 ,, 2 6 6 9 9 9 5 .4321 995 .6203 995 .8127 995 . 9953 996 .1585* 1008 . 1 2 8 1 * 1008 .3194* 1006 .0794* 1 0 0 7 ,. 0 7 9 3 * 1 0 0 5 ,, 9 2 0 5 * 1 0 0 1 .9 1 7 4 * 999 .8510* 1003 . 8 4 5 6 997 . 8300 1002 .9319 9 9 5 .8851 1002 .1016* 994 .0435* 992 .3300* . 6831 . 7678 .0897* 1000 990 1001 989 . 3753 15 15 16 17 18 19 9 9 9 ,. 5 6 7 8 * 988 . 1679 9 8 7 ,. 1 5 8 2 * 9 8 6 ., 3 4 9 2 9 8 5 ,. 7 3 7 9 * 9 8 5 ., 3 2 4 1 * 20 21 4 5 6 7 9 8 5 .. 0 7 8 7 * 9 8 4 .. 9 9 9 0 1 0 1 5 ,, 9 5 0 9 * 1 0 1 4 ., 2 5 6 6 * 1 0 1 2 .,564 1* 1 0 1 0 ., 8 8 4 9 * 8 9 9 10 10 1 1 1009. 2266 1007 . 5 9 7 1 * 1 0 0 6 .. 2 0 4 7 * 1 0 0 6 .. 0 0 3 7 * 1 0 0 4 .. 0 0 1 1 * 1001 . 7 1 4 2 1 0 0 2 ., 9 5 8 0 999 , 3473* 1001 . 5 2 0 8 996 9305 1000. 1527* 12 12 13 13 14 14 15 15 994 .4831* 998 . 8561* 992 .0 4 2 0 Appendix III, 161 continued. a Branch J " rR1 21 22 rR2 23 2 2 3 3 4 4 5 5 1031 1030 1032 1032 1034 1034 1035 .4699 1 0 3 7 .71 1 0 8 9 9 1036 1039 1038 1041 1039 .8455 . 4445 . 1416 .2140 .3617 1043 1040 1044 104 1 .0269 .5153 .8897* 18 18 19 19 1 0 4 8 ., 7 2 1 8 * 1 0 5 9 .. 6 6 7 4 1050. 2396* 20 20 21 21 22 23 23 24 24 1062 . 0 5 1 0 1051 . 91 10 1064 . 5001 1053. 7518 1067 .0 1 2 0 * 1069. 5835 1057 . 9821 1072 . 2 1 6 5 * 1060. 3822 1074 . 8 8 3 6 1062 . 9 7 0 8 1077 . 7 0 3 0 * 1065. 7372 1080. 3614* 1036 . 2744* 1036 . 2744* 1037 . 9 0 6 0 * 25 25 26 26 27 3 3 4 4 5 5 6 6 7 7 8 8 7 8 8 9 1037 . 9 0 6 0 * 1039 . 5 2 6 2 * 1039. 5262* 1041 . 1 3 8 3 * 1 0 4 1 .1 3 0 4 * 1042 . 7 3 6 3 * 1042 . 7 1 8 4 * 1044 . 3 2 3 3 1044 . 2834 9 10 10 1 1 1020 .7012* 1022 . 8 9 1 9 1020 .0366 1 1 1 0 2 3 .1151 1019 .2616 12 13 13 14 14 1023 1018 1023 1017 .4372* . 3825 15 15 16 16 1025 .2282 1015 .2320 1026 . 1794 17 17 1014 .0559 1 0 2 7 ,. 3 4 1 9 18 18 19 19 20 20 21 22 23 24 1 0 1 2 .. 8 4 3 9 * 1 0 2 8 ., 7 3 2 4 * 101 1 .,6 1 1 1 1 0 3 0 .. 3 5 9 4 * 1 0 1 0 .. 3 8 0 0 * 1 0 3 2 .. 2 2 6 2 * 1 0 0 9 .. 1 6 8 3 * 1 0 0 7 .. 9 9 5 4 1 0 0 6 .. 8 8 1 7 * 28 29 4 4 5 5 7 7 8 8 9 9 10 10 1 1 1 1 12 12 13 13 1 0 0 5 .. 8 4 2 0 * 1 0 0 4 ,. 8 8 8 6 * 1 0 0 4 .. 0 5 5 0 * 1 0 0 3 ., 2 6 7 4 * 1 0 0 2 ., 5 9 9 5 * 1 0 0 2 ., 0 2 0 7 * 1029 . 5 4 5 5 * 1029. 5455* 1 0 2 9 ., 4 9 1 2 * 1 0 2 9 ., 4 9 1 2 * 1 0 2 9 ., 3 2 9 0 * 1 0 2 9 .. 3 4 7 5 1 0 2 9 ., 2 1 3 7 1 0 2 9 ., 2 5 5 1 1 0 2 9 ., 0 6 8 5 1 0 2 9 ., 1 5 1 0 * 1 0 2 8 .. 8 8 4 6 1 0 2 9 .. 0 3 6 2 * 1 0 2 8 .. 6 5 1 2 1 0 2 8 ., 9 1 7 4 1 0 2 8 ., 3 5 6 8 1 0 2 8 ., 7 9 6 1 1 0 2 7 .. 9 8 6 9 1 0 2 8 ., 6 7 9 6 14 14 15 1 0 2 7 .. 5 2 6 5 1 0 2 8 ., 5 7 7 2 1 0 2 6 .. 9 5 9 7 * 15 16 16 17 17 1 0 2 6 ., 9 5 9 7 * 1 0 2 6 ., 2 7 7 8 * 1 0 2 8 ., 4 6 0 8 1 0 2 5 .. 4 6 7 0 1 0 2 8 .. 4 7 5 3 1024 .5247* 18 rP2 d" 16 pP1 9 8 9 ., 6 3 6 7 17 17 18 9 9 6 ., 5 0 0 3 987 . 3044 9 9 5 ., 4 4 7 5 * 985 .0787* 20 20 21 21 23 24 2 3 4 5 6 7 8 9 pP2 997 . 6372 16 18 19 .8812 .4096* 1024 .4698 1016 . 3552 25 26 27 r03 Branch 1022 .6132* 1022 .0408* 1022 .6225* 1021 .6947 1022 .6619* 1021 .2534 1022 .7473* 12 .6205 1046 .8063 1042 .6999* 1048 .7838* 1 0 4 3 .. 7 8 2 4 1050 .8236 1 0 4 4 .. 8 9 7 5 * 1 0 5 2 .. 9 2 3 9 * 1 0 4 6 .. 0 7 4 4 1 0 5 5 ,, 1 0 6 6 1 0 4 7 ., 3 4 0 8 1057 . 3 5 2 3 14 15 15 16 16 17 17 6 7 .6636 . 5071 7 8 13 13 14 U' r02 .0100 .9418 .3278* .0212* 1036 .0076 12 12 Branch 1065 .1156* 1027 .7066* 1027 .7066* 1029 . 3 5 9 9 1029 .3377* 6 6 7 10 10 1 1 1 1 rR3 1 0 5 8 .68 1 3 * 1061 . 8 7 0 9 * 982 . 9912 993. 5984* 981 . 0 7 1 6 992 . 8029* 979 . 3436* 976 .5387* 975. 4816* 997 . 3140 995. 3837 993 . 3620 991 . 2 4 2 3 989 .0151 986 .6704 984 . 1997 9 8 1 ., 5 9 5 3 10 1 1 12 13 14 15 16 17 18 2 978 975 972 969 966 963 3 3 3 4 4 989 988 988 986 986 985 984 5 5 6 6 7 7 8 960 956 953 989 ., 8 5 4 4 ,, 9 8 0 2 ,. 9 8 3 2 .. 8 7 9 8 .6919 .4440 . 1606 .8643 . 5738 .9796 .8431* .4041 .0031 . 8626 .0845 . 3387 .0876 9 8 3 .8095 982 .0125 9 8 2 ,. 2 5 1 5 10 1 1 979 . 8592 9 8 0 ,. 6 4 0 6 * 977 .6288 978 . 9530 975 . 3221* 9 7 7 ,. 1 7 0 7 * 972 .9398 9 7 5 ,. 2 7 4 6 1 1 12 12 13 13 14 9 7 0 ,. 4 8 5 1 9 7 3 ,. 2 5 5 1 * 967 .9602 9 7 1 ,. 1 0 0 9 965 . 3686 968 . 7443* 14 15 962 960 963 957 8 9 9 10 16 16 17 17 18 19 19 20 .7151 .0028 .7555 .2391 9 6 0 .9989 954 .4276 951 .5756 955 .0372 948 .6880 951 .8544 162 Appendix III, continued. R Branch J " rR3 9 9 10 10 11 11 12 1050 . 1895 1052 . 1491 13 14 14 1051 .5302 1053 .7169 1052 .7941 1055 .2980 1053 .9720* 16 17 17 18 18 19 19 20 20 21 21 22 22 24 25 26 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 1061 1057 1063 1058 1028. 5612* 19 20 20 21 1023 . 4 5 0 6 * 1022 . 2521 1029. 0362* 1020. 9383 21 22 22 23 5 1029. 4670 1030. 0639 1019 .5236 1018 .0 0 5 6 * 1036. 2669* 5 6 1036. 2669* 1036 .2034* 6 7 1036. 2034* 1036. 1289* 1 0 3 6 ., 1 2 8 9 * 1 0 3 6 .. 0 3 9 5 * 1036 . 0 3 9 5 * rQ4 7 8 8 9 9 10 10 1 1 1 1 12 .6198* 1065 .4484 1 0 5 9 ,. 4 0 0 5 1 0 6 7 .. 3 1 3 6 * 1 0 6 0 .. 1 8 2 9 * 1 0 6 1 ., 9 6 3 0 * 1062 . 9 9 9 0 * 1064 . 1769* 1044 . 6741 1044 . 6741 1046 . 2965 1046 . 2965 1047 . 9076 1047 . 9 0 7 6 1049. 5070* 1049 . 5 0 7 0 * 1051 .0 9 4 1 * 1051 .0 9 4 1 * 12 13 13 14 14 15 15 16 16 17 18 18 19 19 20 20 21 22 23 23 24 24 25 25 26 26 27 27 1052 . 6 6 7 6 * 1052 .6 6 7 6 * 1054. 2260* 1054 . 2 2 6 0 * 13 14 14 15 1058 .7858 1060. 3155 1060. 2586* 1061 .7 9 8 1 * 15 16 16 17 1061 .7 0 0 8 * 1063. 2645 1063 . 1060 1064 . 7 190* 1064. 4676 1066 . 1620 1065 . 7771 20 20 18 .8923* .8196 .6434 12 12 13 18 19 19 r03 1056 .8992 1055 .0612 1058 .5274 1056 .0591 1060 .1880* 1056 .9737 1055. 7739* 1055. 7658* 1057. 3044* 1057. 2868* 1058. 8177* 17 18 J" 1048 .7838* 1050 .5882* 12 13 15 15 16 rR4 1045 .8996 1045 .8216 1047 .4676* 1047 . 3248 1049 .0294 1 0 6 7 .. 5 9 7 6 1 0 6 7 ., 0 2 4 4 * 1 0 6 9 ,. 0 3 0 3 * 1 0 6 8 .. 1 9 5 9 * E Q Branch r05 6 6 7 7 9 9 10 10 1 1 1 1 12 12 13 Branch pP2 pP3 iP 20 21 945. 7720 948. 5616 21 22 22 23 23 24 942 . 8300* 945. 1839* 939 .8683 9 4 1 .7 4 7 5 936 .8755* 25 4 4 5 5 6 6 7 7 8 938 . 2798* 934 . 8032 978 . 7082 978. 6884 9 7 7 . 0 1 16 976 .9555* 975. 3303* 9 7 5 ., 1 9 6 1 9 7 3 ., 6 7 2 0 9 7 3 ., 4 0 9 9 9 7 2 ,, 0 4 6 5 * 1 0 3 5 ., 9 3 5 8 * 1 0 3 5 ., 9 3 5 8 * 1 0 3 5 ,. 8 1 5 9 * 1 0 3 5 ,, 8 1 5 9 * 1035 .6753* 1035 .6843* 1035 .5144 8 9 9 10 10 1 1 9 7 1 ,. 5 8 9 2 * 9 7 0 .4631 1035 .5309* 1035 . 3291 1035 . 3631 1035 .1149* 1035 . 1769 1034 .8680 1034 .9734 1 1 12 12 13 13 14 14 1034 .5800 1034 . 7537 1034 .5206 1033 .8482* 1034 . 2 7 8 3 1033 . 3823 1034 .0309* 1032 .8330 1033 .7838* 1032 . 1 8 5 3 * 1031 .4235 1030 .5349 15 15 16 965 965 963 964 961 963 959 961 957 959 955 958 952 956 950 954 947 1033 1029 1033 1028 1032 1027 1032 1025 1033 1042 1042 1042 1042 1042 1042 1042 16 17 17 18 18 19 19 20 20 21 21 22 .1511 .5080 .0169 . 3354* .9486 .0079 .9670 .5505 .0915* .8357* .8357* .7645 .7645 .5844 .5844 .4754 1042 .4754 1042 .3520* 1042 .3520* 1042 .2139 1042 .2139 1042 .0604 22 23 23 24 24 pP4 25 4 4 5 5 6 6 7 7 8 8 9 9 10 9 6 9 ,, 7 2 8 2 968 . 9224 9 6 7 .8217 967 .4219* 952 945 950 942 947 939 945 937 942 934 .8621 .9497 .8465 . 4893 .7748* .0167 .6248* .5062 .4145* . 9296 . 1299 . 2623 .7749 . 4808 . 3476 .5650* .8489 . 5032 . 2803 .2831 .6462* .8973 .9468 . 3408 . 1897* .6114 .3056* 939 .7111* 970 .7243* 970 .7243* 969 .0060* 969 .0060* 967 .2789* 967 .2789* 965 .5417* 965 . 5467* 963 .8095 963 .7943* 962 .0712 9 6 2 .0391 9 6 0 .3347 163 Appendix III, continued. E Branch J " 21 21 22 22 23 23 5 e 7 8 9 10 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 24 24 25 6 6 7 7 8 8 9 9 10 10 1 1 1 1 12 12 13 13 14 14 15 15 16 16 17 17 18 18 1070 .4622* 1069 .2947* 1071 .9036 1070 1073 1071 1052 1054 J " r05 13 14 14 15 .3068* .3239* 15 16 .1956* .9239* .5354* 1056 .1367* 16 17 17 18 18 1057 .7263 1059 . 3 0 4 9 1060 .8713 19 19 1062 .4240 1063 .9636 1063 .9636 1065 .4879 1065 .4879 1067 .0112* 1067 .0112* 1068 .4896* 1068 .4896* 1 0 6 9 .. 9 6 7 7 * 20 20 21 21 22 23 23 r06 1069 .9616* 107 1 . 4 2 6 0 * 1071 .4088* 7 7 8 8 9 9 10 10 1 1 1 1 12 12 13 13 14 1072 1072 1074 1074 .. 8 6 5 0 ., 8 4 5 7 ., 2 8 6 3 ., 2 5 1 6 1075 . 6 8 6 1 * 1075 . 6301 1077 .0691 1076. 9773 1078. 4 3 1 8 * 1078 . 2 8 8 8 1079 . 5577 14 1081 . 1 1 0 1 * 1080. 7772 1 0 8 1 .9 3 6 3 * 15 15 16 1061 .0 3 8 0 1 0 6 1 .0 3 8 0 1062 . 6 3 9 6 1062. 6396 1064 . 2 3 0 0 * 1064 . 2 3 0 0 * 1065 . 8 0 9 8 1065. 8098 16 17 17 18 1067. 3781 1067. 3781 1068 . 9 3 4 8 1068. 9 3 4 8 1 0 7 0 ., 4 7 9 4 1 0 7 0 ., 4 7 9 4 1 0 7 2 .. 0 1 0 5 1072. 0 1 0 5 1 0 7 3 ., 5 2 9 6 1 0 7 3 .. 5 2 9 6 1 0 7 5 ,. 0 3 3 6 * 1 0 7 5 .. 0 3 3 6 * 1 0 7 6 .. 5 2 3 8 * 1076 .5238* 1078 .9823* 1078 .9823* 1079 .4562* 1079 .4562* E Q. Branch 18 19 19 20 20 21 21 22 22 23 r07 23 24 24 8 9 9 1042 .0604 1041 .8900* 104 1 . 8 9 0 0 * 104 1 . 7 0 1 3 * 1041 .7013* 1 0 4 1 :,4 9 1 3 * J" pP4 10 1 1 1 1 12 12 13 13 14 14 1041 .4988* 104 1 ., 2 5 9 9 1041 .2739 104 1 . 0 0 5 7 1 0 4 1 ,. 0 2 8 8 1040 .7239 1040 .7620 1040 .4126 1040 .4768 1040 .0675 1 0 4 0 .17 14 1 0 3 9 ,. 8 4 4 8 * 1 0 3 9 ,. 2 5 0 2 * 1039 .4988* 1049 . 2726 1 0 4 9 ,, 2 7 2 6 1 0 4 9 ,, 1 9 0 7 1 0 4 9 ,. 1 9 0 7 1 0 4 9 ,, 0 9 7 4 1 0 4 9 ,0 9 7 4 1 0 4 8 ,. 9 9 3 0 1 0 4 8 .. 9 9 3 0 1048 .8757 1 0 4 8 ., 8 7 5 7 1048 1048 1048 1048 1048 1048 1048 1048 1048 1048 1047 1047 1047 .7456 ,. 7 4 5 6 .6023 . 6023 ,. 4 4 4 4 ,. 4 4 4 4 1047 1047 1047 1047 1047 .6606* .4220 .4220 . 1643 . 1643 1046 .8878* 1046 .8878* 1046 .5863* 1046 .5950* 1046 .2623* 1046 . 2 7 7 3 * 1045 .9127* 1045 .9401* 1055 .5844* .5005 . 5005 .3983 .3983 1 1 12 13 1055 .0186 .2833 .2833 .1515* 17 17 948 . 7994 9 4 7 .. 3 4 2 1 18 18 19 9 4 7 ., 3 1 3 4 9 4 5 ,. 3 6 1 3 9 4 5 ,, 8 6 6 4 19 9 4 3 ,, 3 3 1 3 944 .4455 941 . 2457 22 22 23 23 24 5 5 6 6 7 7 8 8 9 .6606* 943 939 941 936 .0298* . 1012 .5931 .8755* 9 4 0 . 1075 934 .6179 938 .5446* 960 . 8783 960 959 959 957 957 955 .8783 .1505* .1505* .4145* .4 1 4 5 * .6703* 955 .6703* 953 .9166* 9 10 10 1 1 1 1 12 12 13 13 14 14 15 15 16 16 17 953 .9166* 952 .1558* 17 939 .6135 9 3 7 .9768 937 .7859 18 18 19 19 pP6 956. 6959 9 5 5 . 1994 954 .8808 15 15 16 16 21 21 pP5 9 6 0 . 2 7 18 958 .6069 958 .4922* 956 .8920 953 . 5372 9 5 3 ., 0 4 3 1 951 .9 1 3 4 * 951 . 1774 9 5 0 ., 3 3 2 1 9 4 9 ., 2 7 9 1 20 20 .2726 . 2726 .0851 .0851 .8814* .8814* 1055 1055 1055 1055 1055 1055 1055 10 10 1 1 Branch 20 23 6 6 7 7 8 952 950 950 948 948 946 946 945 945 943 943 941 .1558* .3875* .3875* .6166* .6099* .8395* .8262* .0593 .0298 . 2608 .2365 .5041* 941 .4301 939 .7341* 936 935 934 928 950 950 949 949 947 .2363* .9466* .5220* .3993* .8855 .8855 . 1497 . 1497 . 4049 Appendix III, 164 continued. E Branch rR6 d" 19 19 20 20 21 21 22 23 23 24 rR7 1087. 8 3 2 1 * 1087 .8 1 7 0 * 25 25 1089 . 1610* 1089. 1333 26 26 7 1090. 4248* 1090. 3575* 1069. 0363* 1069 .0 3 6 3 * 1073 .7749 12 12 13 14 15 15 16 16 17 17 18 1076 .8 7 8 5 * 1076 . 8 7 8 5 * 1078 . 4 1 2 5 * 1079. 9289* 1081 . 4 4 3 8 20 21 21 23 23 24 25 25 8 8 9 9 10 1 1 1 1 12 12 13 14 15 15 16 16 17 17 18 19 d" r07 13 14 14 r08 .0824* .6390 .6390 . 1847 . 1847 .7134* .2408* .7500 .7500 1089 . 247 1 1089 .2471 1090 .7314 1090 .7314 1092 .2029* 1093 .6596 1054 .3329 1054 . 1 2 5 9 * 19 1053 .9050* 1053 .6692 1053 .6692 1053 .4223* 1053 .4223* 9 10 1 1 12 12 13 13 14 14 15 16 17 17 18 19 1081 . 4 4 3 8 1082. 9402 1082 . 9 4 0 2 1084 .4 2 3 2 1084 .4 2 3 2 1085. 8867* 1087. 3487 1 0 8 7 .. 3 4 8 7 1080 1081 1081 1083 1083 1084 1086 1087 1087 17 18 22 22 1075. 3329 1075 . 3329 1 0 8 8 ,, 7 9 4 0 * 1 0 9 0 .. 2 2 0 0 * 1 0 9 0 .. 2 2 0 0 * 1 0 9 3 .. 0 1 0 2 * 1093 .0102* 1094 .3885* 1095 .7295* 1095 .7295* 1076 .9357 1076 .9357 1078 .5143 1078 .5143 15 15 16 17 1055 .0186 1054 .8669 1054 .8669 1054 . 7 0 2 3 1054 .7023 1054 .5236* 1054 .3329 20 20 21 21 1070. 6266 1070. 6266 1072 . 2067 1072 . 2 0 6 7 1073 . 7 7 4 9 10 1 1 1 1 19 19 Branch 1083. 7281* 1083 . 7 2 8 1 * 1085. 1 118* 1086 .4 8 5 8 * 1086 .4 7 0 8 * 24 7 8 8 9 9 10 rR8 1080. 8983* 1080. 8983* 1082 . 3 2 2 7 * 1082. 3227* E a 20 20 21 2 1 22 PQ1 23 23 24 25 26 26 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 Branch d" pP6 8 9 9 10 10 1 1 1 1 12 12 13 13 14 14 15 1053 . 1481* 1053 . 1481* 1061 . 8 1 6 1 * 1061 . 7 1 2 7 * 1061 . 5 9 8 9 1061 .4747 1061 .4747 15 18 1061 . 3371 1061 .3371 1061 . 1 8 8 0 106 1 . 1 8 8 0 1061 .0259* 1060 .8523* 1060 .6641 1060 . 664 1 pP7 1060 .4628* 1060 .2462* 1060 .0175 1060 .0175 1059 .7736 1059 .7736 1059 .5139* 1 0 5 9 .. 2 3 9 1 1059 . 2391 1058 .9472* 1 0 5 8 .. 6 4 0 2 * 1058 3147 1 0 5 8 .. 3 1 4 7 1003. 2853 1003. 5143 1003 . 7086 1003 . 8702 1 0 0 4 . O O I 1* 1004 . 1081 1004. 1938 1004 . 2634 1004 . 3 2 0 5 1004. 3688 1004 . 4104 1004 . 4 4 8 0 * 1004. 4 8 3 0 1004 . 5 1 6 9 9 4 2 . 1 175 9 4 2 . 1 175 9 4 0 .3385 940 . 3385 938 .5518 938 .5518 936 .7573* 936 934 934 929 .7573* .9561* 18 19 19 927 .6847* 20 20 21 21 925 .8776* 925 .8515* 924 .0542* 924 .0132* 22 22 7 7 8 8 9 922 .2312* 922 .1676* 12 12 13 13 14 14 15 15 16 16 17 17 pP8 945 .6509 945 .6509 943 . 8883 943 .8883 .9561* .5201* 929 .5118* 927 .6994* 10 10 1 1 1 1 1 0 0 0 .. 9 8 1 3 * 1 0 0 1 .. 1 1 2 4 1001 . 3 0 1 5 * lOOl . 5405 1001 . 8 1 7 4 1002 . 1 190* 1002 . 4 2 8 7 1002 . 7 3 5 3 1003. 0 2 3 7 947 .4049 18 18 20 20 21 21 8 8 9 9 10 10 1 1 1 1 12 12 13 13 14 940 940 939 939 937 . 7662 .7662 .0210 .0210 .2674* 9 3 5 .5052 935 . 5052 933 . 7342 933 . 7342 931 .9544 9 3 1 .. 9 5 4 4 9 3 0 .. 1 6 5 8 9 3 0 .. 1 6 5 8 928 . 3691 928 . 3691 926 . 5645 926 . 5645 9 2 4 .. 7 5 2 0 924 . 7520 922 . 9322* 922 .9322* 921 . 1042* 9 2 1 .1 0 4 2 * 917 .4283* 917 .4283* 915 .5818* 915 .5818* 930. 5386 930. 5386 928 . 7860 928 . 7860 927 .0244 927 .0244 925. 2535 925 .2535 923. 4737 923 .4737 921 .6 8 5 6 921 .6856 919 .8884 Appendix III, continued. Q. Branch JJ * rR8 19 20 20 21 22 22 23 24 24 25 27 rR9 28 9 9 1 1 1 1 13 13 14 14 15 15 16 17 17 18 19 19 pR1 20 20 21 21 22 1 2 3 4 5 6 8 9 10 1 1 12 13 14 15 16 pR2 2 3 4 5 6 7 7 8 8 9 9 10 10 1 1 1 1 12 12 1093 1095 1095 1096 1097 .6596 .1036* .1036* .5326* Branch J" pQ1 24 25 26 27 .9470 1097 .9470 1 0 9 9 .34 1 4 * 1100 .7295 1 100.7295 1 102 . 0 9 7 8 * 1 104 . 7 8 0 8 * 1106 .0987* 1084 . 7 4 6 8 1084 .7468 1087 .8486 p02 1090 1090 1092 1092 1004 .7359* 1004 . 7 7 9 0 * 993 .4676 993 .0378 6 7 7 8 8 9 9 .4124 1 0 9 3 .. 9 0 9 7 1093 .9097 1 0 9 5 .. 3 9 2 9 * 1096. .8673* 1096. , 8 6 7 3 * 1098. .3274* 1 0 9 9 ,. 7 7 3 8 1099. . 7738 1 101 .2 0 7 9 1101. 2 0 7 9 1 1 0 2 .6 2 8 4 * 1 1 0 2 .6 2 8 4 * 1 1 0 4 .0 2 2 3 * 1 0 0 4 .. 1 6 5 1 1005 .6467 1 0 0 7 ,. 0 2 1 6 1008. . 2 7 6 6 * 1 0 0 9 .. 3 9 8 4 1010. 3 7 4 5 * 101 1 .. 8 4 2 2 1 0 1 2 ., 3 2 5 9 * 1 0 1 2 .. 6 4 5 1 1012 . 8 1 2 6 1 0 1 2 .. 8 4 3 8 * 1 0 1 0 . 01 10 1004 . 6 4 9 5 * 101 1 .7 0 2 3 1005. 4748 1013. 2673 1006. 2 1 6 0 1014 . 6 8 4 9 1006. 8 7 5 0 * 1015. 9388 1007. 4514 29 30 2 2 5 5 6 .9030 .9030 .4124 1 0 1 2 .. 7 6 8 7 * 1 0 1 2 .. 6 0 5 1 1 0 1 2 ., 3 8 3 6 1012 . 1293* 9 9 7 ., 8 4 8 8 * 999.,1911* 1000. 4591 1001 . 6 3 7 1 * 1006. ,3518* 1 0 0 8 ., 2 1 8 5 * 1003. 7 3 3 0 28 3 3 4 4 1087 .8486 992 993 992 993 991 994 991 .5503* .5828* .6182* .6495* .6954* .8895* .9070 . 1894 . 2583 994 .5445 9 9 0 .4366 994 .9527 989 . 4200 9 9 5 .4 1 0 7 988 .1900* 995 .9129 986 .7319 10 1 1 1 1 12 12 13 13 14 15 16 17 18 19 996 985 997 983 997 23 24 25 26 27 28 29 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 1 1 1 1 pP8 .7770* .6490 .4059 10 20 21 22 P03 Branch 1004 1004 1004 1004 1004 pP9 .4521* .0383 .0207* . 1079 .6071 .9447 . 1994 . 7843* . 3473* .8737 .3537* .7778 .14 14 1 0 0 1 ,. 4 4 4 3 * 1001 .6879 1 0 0 1 ,. 8 7 8 0 1 0 0 2 ,. 0 2 0 7 1 0 0 2 ., 1 1 9 0 * 1 0 0 2 ., 1 9 3 4 * 1 0 0 2 ., 2 3 6 1 * 1002, ,2564* 1 0 0 2 ., 2 8 5 1 * 1 0 0 2 ., 3 0 3 7 * 985 .4164 985 . 4382* 980 998 998 999 999 1000 1000 1001 985.,4283* 985. 3671* 985. 4382* 985. 2945 985. 4813* 985. 1924* 985 .5695 985. 0520* 985. 7 177* 984. 8605 985 .9377 984 .6054 986 .2380 984 .2701 986 .6 2 2 0 983 .8 3 4 7 * rPO 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 23 9 9 10 10 1 1 1 1 12 12 13 13 14 14 15 15 16 16 17 17 18 18 10 10 1 1 12 12 13 13 14 14 15 15 16 16 17 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 919 .8884 9 1 8 .0 8 3 0 9 1 8 .0 8 3 0 916. 2684 916. 2684 914 .4465 914 . 4465 9 1 2 .6 1 6 0 912 .6 1 6 0 910. 7782 910. 7782 908 .9318 908. 9318 907 .0794 9 0 7 .0 7 9 4 905. 2186* 9 0 3 .. 3 5 1 7 * 9 2 0 .. 2 2 8 4 9 2 0 .. 2 2 8 4 9 1 8 .. 4 6 6 8 * 9 1 8 .. 4 6 6 8 * 9 1 6 .. 6 9 7 0 9 1 6 .. 6 9 7 0 9 1 4 .. 9 1 7 7 9 1 4 .. 9 1 7 7 9 1 3 .. 1 2 9 9 913 . 1299 91 1 . 3 3 3 0 911 . 3 3 3 0 9 0 9 .5262 909 .5262 907 .7126 907 .7126 905 .8895 9 0 5 .8895 904 .0587' 904 .0587' 9 0 9 .8565' 9 0 9 .8565' 9 0 8 .0866' 9 0 6 .3071 906 . 3071 904 .5200 9 0 4 .5200' 902 . 7215' 902 .7215 9 0 0 .9151 900 .9151 899 . 1004 899 . 1004 897 . 2758 1005 .0696 1003 . 5316 1002 .0797 1000 .7299 999 .4985 998 .4016 997 .4567 9 9 6 .6731 9 9 6 .0551 995 .6010 9 9 5 .3008 9 9 5 . 1399 995 .0959 9 9 5 .1471 166 Appendix III, continued. a E Branch J " pR2 13 13 14 14 15 15 17 17 pR4 1008 .7512* 1019 .3825* 18 1009 .3150* 1019 .5032 18 19 1 0 0 9 .5251 1 0 0 9 .. 6 9 7 6 * 21 1018 .9620* 1018 .5698 995 .5791* 997 . 3516 22 pR3 J" 1017 .0141 1007 .9534 1017 .9005 1008 .3822* 1018 .5919 5 6 6 7 7 1001 . 0 6 7 0 * 999 .8603* 9 10 1 1 1001 .2852 1002 .6512 1007 .0793* 1 1 12 1003 .9510 1009 . 1494 12 13 14 14 15 16 17 1005 1006 1013 1007 1015 1017 1018 19 7 8 9 10 10 1 1 1 1 12 12 13 13 14 14 15 16 17 17 18 19 20 21 . 1782 .3285* . 2431 . 3972 .2062 .0540* .8015 102 1 . 7 5 2 1 * 9 9 0 .6643 992 .2798* 994 .0085 9 9 5 .. 6 9 2 7 9 9 5 .. 4 8 2 5 997 .4076* 9 9 7 .. 0 6 4 5 9 9 9 .. 1 6 2 7 * 9 9 8 .. 6 2 7 0 1 0 0 0 .. 9 6 9 2 * 1 0 0 0 ., 1 6 5 9 * 1 0 0 2 .. 8 3 5 7 1 0 0 1 .. 6 7 3 9 1 0 0 3 ., 1 4 5 4 1006. 7 6 9 6 * 1008 . 8 3 5 7 * 1006 . 9 4 9 1 * 1007 . 1670* 1008 . 5 1 8 1 * 1009. 6 9 7 6 * 1010. 8 0 0 8 * 987 .0907* 983 .2784* 987 .6387* 13 14 982 .5775 988 .2622 14 15 981 .7067 938 .9500 980 .6406* 9 8 9 .6941 979 .3608* 15 16 16 17 17 18 19 20 21 22 23 24 996 .8616* 999 .1801* 998 .3831* 8 8 12 12 13 P04 25 26 28 4 4 5 5 6 7 7 8 8 9 9 10 10 1 1 1 1 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 25 26 27 E Branch J" rPO 16 17 18 19 995.2669 20 21 4 995.9953 996.1585* rPI 5 9 9 0 .4827 977 .8422* 991 .3031 992 . 1436 992 .9882 993 . 8224 994 .6295* 995 . 3947 9 9 6 . 1022 996 .7844* 997 .4076* 998 .4732* 977 .4141* 977 977 977 977 .4141* .3718* .3718* .3208* 977 . 2778 977 . 2625 977 .2310* 977 977 977 977 977 977 976 977 . 1963* .. 1 8 9 1 * .. 1 1 9 9 * .. 1 6 0 9 * .. 0 3 1 9 .. 1 5 2 6 * ,. 9 2 8 6 . 1707* 976 . 8040 977 . 2397* 976. 6517 977 . 3603* 976. 4627 977 .5502 976 .2262 977 .8212* 975. 9274 978 . 1846 9 7 5 . 551 1 978. 6466 975. 0783 979. 2104 974 .4872 979. 8744 973 . 7537 980. 6335* 972 .8532 981 .4 7 6 4 * 971 .7609 982 . 3944* 970. 4513 983 .3726* 984 . 3966 985. 4507 986 .5186* 1008.1281* 1008.3194* 5 6 7 7 1006.0794* 8 9 999.8510* 1003.8456 1O07.0793*' 1005.9205* 1001.9174* 9 997.8300 10 10 11 1 1 1 0 0 2 . 9 3 19 995.8851 12 13 13 992.3300* 1000.6831 990.7678 14 14 1001.0897* 989.3753 999.5678* 9 8 8 . 1679 987. 1582* 15 15 16 17 18 19 rP2 995.4321 995.6203 995.8127 20 21 4 5 6 7 8 9 9 10 10 1 1 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 20 20 21 21 23 24 1002.1016* 994.0435* 986.3492 985.7379* 985.3241* 985.0787* 984.9990 1015.9509* 1014.2566* 1012.5641* 1010.8849* 1009.2266 1 0 O 7 . 5 9 7 1* 1006.2047* 1006.0037* 1004.0011* 1001.7142 1002.9580 999.3473* 1001.5208 996.9305 1000.1527* 994.4831* 998.8561* 992.0420 997.6372 989.6367 996.5003 987.3044 995.4475* 985.0787* 982.9912 993.5984* 981.0716 992.8029* 979.3436* 976.5387* 975.4816* 167 Appendix III, continued. R Branch P04 P05 p06 U" 28 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 6 6 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 17 17 18 18 19 19 20 20 21 21 22 23 987 . 5992* 969. 2386* 969. 2386* 969. 1859* 969 . 1859* 969 . 1260* 969 . 1260* 969 .0586* 969 .0586* 968 . 9853* 968 . 9853* 968.,9064* 968 ,9064* . 968 . 8264* 968. 8202* 968 . 7435* 968 .7285 968 .6588 968 .6318 968 .5784 968 .5291 968 .504 1 968 .4193 968 .4421* 968 . 3020 968 .3970* 968 . 1728 968 . 3793 968 .0280* 968 .3970* 967 .8656* 968 .4421* 967 . 6744* 968 .5614* 960 .9167 960 .9167 960 .7837 960 . 7837 960 .7057* 960 .7057* 960 .6200 960 .6200 960 .5279 960 .5279 960 .4291 960 .4291 960 .3237* 960 .3237* 960 .2128* 960 .2128* 960 .0969* 960 .0969* 959 .9748* 959 .8587* 959 .8487* 959 .7366* 959 .7183* 959 . 5839 959 .6133* 959 .4937 959 .4453* 959 .3002* 959 .3758* 959 . 1627* 958 .9872* 168 Appendix III, continued. R Q. Branch P07 P08 p09 j " 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 1S 16 17 17 18 18 19 19 20 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 17 18 18 19 19 20 20 21 21 22 22 24 24 12 12 13 13 14 14 15 15 16 16 952 . 3931* 952 . 3931* 952 .3136* 952 .3136* 952 .2261 952 .2261 952 . 1301 952.. 1301 952,.0267 952 .0267 . 952 .9134* 952 .9134* 951 .7966 951 .7966 951 .6714 951 .6714 951 .5399 951 .5399 951 .4016* 951 .4016* 951 .2580* 951 .2580* 951 .1094* 951 .1094* 950,.9490* 943 .9068 943 .9068 943,.8256 943 .8256 943 . 7367 943 .7367 , 943,.6391 943 .6391 943 .5336 943 . 5336 943 .4193 943 .4193 943 . 2973 943,. 2973 943 . 1668 943 . 1668 943 .0298* 942 .8855* 942 .7328 942 . 7328 942,.5744 942 .5744 942,.4087 942 .4087 942,.2373* 942 .2373* , 942 .0601* , 942,.0601* 941 .6879* 941 .6879* , 934,.9675* 934 .9675* , 934 .8515* , 934 .8515* , 934 .7237* 934 .7237* , 934 .5948* , 934 .5948* 934 .4536* , 934 .4536* E Appendix III, continued. a a Branch PQ9 ^Transitions in units of cm asterisk. J " 17 17 18 18 21 21 934.3056* 934.3056* 934.1575* 934.1575* 933.6237* 933.6237* Blended lines are denoted by an 170 REFERENCES 1. J.T. Hougen, The Calculation of Rotational Energy Levels and Line Intensities in Diatomic Molecules. (National Bureau of Standards Monograph115, 1970). 2. M.E. Rose, Elementary Theory of Angular Momentum. (John Wiley and Sons, Inc., New York, 1957), Ch. 1. 3. J.H. Van Vleck, Rev. Mod. Phvs. 23. 213 (1951). 4. A. Messiah, Quantum Mechanics, vol. 2, (North-Holland Publishing Co., Amsterdam, 1962), Appendix C. 5. A.R. Edmonds, Angular Momentum in Quantum Mechanics. (Princeton University Press, Princeton, 1960), C h . 3. 6. A. Messiah, Quantum Mechanics, vol. I, (North-Holland Co., Amsterdam, 1964), Appendix B. Publishing 7. M.E. Rose, ibjd_, p. 235. 8. A. Messiah, ibid, vol. 1, Ch. 13. 9. M.E. Rose, M i , Ch. 4. 10. B.L. Silver, Irreducible Tensor Methods. (Academic Press, New York, 1976), Ch. 5. 11. A.R. Edmonds, M i , Ch. 5. 12. B.L. 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Optical and infrared spectra of some unstable molecules Barry, Judith Anne 1987
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Title | Optical and infrared spectra of some unstable molecules |
Creator |
Barry, Judith Anne |
Publisher | University of British Columbia |
Date | 1987 |
Date Issued | 2010-09-22T02:33:50Z |
Description | Some unstable gaseous molecules, cobalt oxide (CoO), niobium nitride (NbN) and aminoborane (NH₂BH₂), were studied by high resolution optical spectroscopy. A portion of the "red" system of CoO, from 7000 Å to 5800 Å, was measured using laser induced fluorescence techniques. Three bands of the system, with origins at 6338 Å, 6411 Å and 6436 Å, were rotationally analyzed. The lower levels of these parallel bands are the Ω = 7/2 and 5/2 spin-orbit components of a ⁴∆i electronic state. Available evidence indicates that this is the ground state of the molecule; its bond length is 1.631 Å. This work completes the determination of the ground state symmetries for the entire series of first row diatomic transition metal oxides. The hyperfine structure in the ground state is very small, supporting a σ²δ³π² electron configuration. The upper state, assigned as σδ³π²σ*, has large positive hyperfine splittings that follow a case (aβ) pattern; it is heavily perturbed, both rotationally and vibrationally. The sub-Doppler spectrum of the ³Φ₋³∆ system of NbN was measured by intermodulated fluorescence techniques, and the hyperfine structure analyzed. Second order spin-orbit interactions have shifted the ³Φ₃₋³∆₂ subband 40 cm⁻¹ to the blue of its central first order position. The perturbations to the spin-orbit components were so extensive that five hyperfine constants, rather than three, were required to fit the data to the case (a) Hamiltonian. The ³∆₋³Φ system of NbN is the first instance where this has been observed. The magnetic hyperfine constants indicate that all components of the ³∆ and ³Φ spin orbit manifolds may be affected, though the ³∆ state interacts most strongly, presumably by the coupling of the ³∆₂ component with the ¹∆ state having the same configuration. The Fermi contact interactions in the ³∆ substates are large and positive, consistent with a σ¹δ¹ configuration. In the ³Φ state the (b + c) hyperfine constants are negative, as expected from a π¹δ¹ configuration. The ³∆ and ³Φ bond lengths are 1.6618 Å and 1.6712 Å, respectively, which are intermediate between those of ZrN and MoN. The Fourier transform infrared spectrum of the V7 BH₂ wagging fundamental of NH₂BH₂ was rotationally analyzed. A set of effective rotational and centrifugal distortion constants was determined, but the band shows extensive perturbations by Coriolis interactions with the nearby V5 and V11 fundamentals. A complete analysis could not be made without an analysis of the V5-V7-V11 Coriolis interactions, which is currently not possible because the very small dipole derivative of the V5 vibration has prevented its analysis. |
Subject |
Molecular spectra Infrared spectroscopy |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-09-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0046956 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
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