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Optical and infrared spectra of some unstable molecules Barry, Judith Anne 1987-12-31

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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.  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