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A Calibration system for the visible region with application to the NO₂ and BO₂ molecules Steunenberg, Dinie M. 1989

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A CALIBRATION SYSTEM FOR THE V I S I B L E REGION WITH APPLICATION TO THE NO AND B 0 a  2  MOLECULES  By  DINIE M . STEUNENBERG B.Sc.  (Hon.  Chemistry)  University  of  B r i t i s h Columbia,  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in  THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY  We a c c e p t t h i s to  the  thesis  required  as  conforming  standard  THE UNIVERSITY OF BRITISH COLUMBIA February 1989 ©  DINIE STEUNENBERG  1986  In  presenting  degree  at the  this  thesis  in  University of  partial  fulfilment  of  of  department  this thesis for or  by  his  or  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  representatives.  an advanced  Library shall make it  agree that permission for extensive  scholarly purposes may be her  for  It  is  granted  by the  understood  that  head of copying  my or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  1 W ) V  K f t < \ .  Abstract  A method f o r c a l i b r a t i n g h i g h r e s o l u t i o n the v i s i b l e ultra-high  r e g i o n has been d e v e l o p e d  calibration  system i s  whose c a v i t y laser;  the  frequency  length  is  absolute is  based  free  obtained  sufficient then the  to  identify  the  o r d e r number t o  Two K=0 s u b b a n d s o f 2  at  N"=10 it  the  5 9 3 . 5 nm and 5 8 5 . 1  hyperfine  transitions  were s t u d i e d ;  was p o s s i b l e  d e s c r i b i n g the rotation,  to  although the  2  excited  state. and  the  where t h e y  exist.  was  Using the  splittings  of  the  fringe  absolute  of  is  the  transition  for analysis.  of  The  l e v e l s up  to  highly perturbed,  of meaningful  constants  electron  d i p o l a r parameters  found w i t h l i t e r a t u r e c a l i b r a t i o n system the  hyperfine  is  o r d e r number.  Aj e l e c t r o n i c  system i s  is  frequency  ratio  Values f o r the  (I,S)  be  frequency  with r o t a t i o n a l  obtain a series  Fermi contact  to  -  associated  Good a g r e e m e n t  accurate  B2  t h e HeNe  range can  'lock point' 2  at  (whose a c c u r a c y  nm were c h o s e n  obtained.  frequency  the  The  2  a transmission  fringe);  and s u b -  B0 .  fringe  HeNe f r e q u e n c y m u l t i p l i e d by t h e  •unknown'  N0  the  f r o m a c o m m e r c i a l wavemeter  2  Fabry-Perot etalon  For absolute  o r d e r number o f  in  a s t a b i l i z e d HeNe  spectral  determined with high accuracy. the  to  o f N0 of  on an e v a c u a t e d  o r d e r number o f  known and t h e  measurements,  spectra  servo-locked  spectra  and t e s t e d b y t a k i n g  r e s o l u t i o n molecular-beam spectra  Doppler intermodulated fluorescence  laser  intervals  spin-  were values small  were m e a s u r e d  ± 1 MHz, a s d e t e r m i n e d b y c o m p a r i s o n w i t h  earlier  microwave work. rotational  Larger  levels)  10 MHz, t h e r e b y  were f o u n d t o  demonstrating  system over s m a l l frequency The  transitions  to  analysis the  be c o n s i s t e n t t o  the  agreed w e l l  resonance  of  (0,0)  be a n a l y z e d .  state exhibited  accuracy of  B0  the  larger  and l\  bands o f  The g r o u n d s t a t e  with previous  two v e r y  covered the  2  results  interesting  better  while  permitted the  the of  2  A / 3  2  the  relative  (~480 GHz)  level 2  lies  2  A  A ^ and X^llg 2  constants  while  010  entirely  the  upper  there  is  states.  This  electronic  transition.  is  and E * v i b r o n i c  These  Fj and F  features  not normally p o s s i b l e  in  in a  2  an u n e x p e c t e d  energies,  f o r such comparatively  a n d w o u l d h a v e gone u n n o t i c e d b u t  high p r e c i s i o n of proved i t s  effect  the  data.  Once a g a i n t h e  200  -  separated  cm" , b e i n g r e p r o d u c e d , on a v e r a g e , 1  iii  to  ^ are  levels;  low  f o r the  extremely  c a l i b r a t i o n system  power, w i t h r o t a t i o n a l c o m b i n a t i o n d i f f e r e n c e s  b a n d s w i t h a common u p p e r s t a t e ,  of both  B o t h t h e v =0 a n d v = l u p p e r s t a t e s 2  2  have  a l l the v i b r o n i c l e v e l s  randomly p e r t u r b e d by h i g h l y i n g ground s t a t e v i b r o n i c is  a K-  2  5 / 2  between t h e  2" v i b r o n i c l e v e l . energies of  the  near  010 v i b r a t i o n a l l e v e l s t o b e d e t e r m i n e d a c c u r a t e l y  electronic  this  than  calibration  region  features:  a v o i d e d c r o s s i n g between t h e  s p i n components  the  (between  ranges.  5400A w h i c h a l l o w e d  levels,  intervals  (a few MHz) and somewhat  rotational  obtained  frequency  b y more t h a n  ± 0 . 0 0 0 3 cm" . 1  in  Table o f Contents  Abstract  i i  L i s t of Tables  v i i  L i s t of Figures  ix  Acknowledgement  xiii  Chapter 1  A C a l i b r a t i o n System f o r t h e V i s i b l e Region  1  1.1  Background  1  1.2  Theory  3  1.3 Chapter 2  1.2.1  Interferometers  3  1.2.2  The C a l c u l a t i o n o f t h e Free S p e c t r a l Range  5  Experimental D e t a i l s  7  A p p l i c a t i o n o f t h e C a l i b r a t i o n System t o t h e H y p e r f i n e S t r u c t u r e o f N0  13  2  2.1  Introduction  13  2.2  Experimental D e t a i l s  14  2.2.1  The M o l e c u l a r Beam Apparatus  14  2.2.2  F l u o r e s c e n c e C o l l e c t i o n Device  16  2.2.3  Laser-Induced F l u o r e s c e n c e  18  2.2.4  The Doppler Width  19  2.2.5  Lamb Dips  20  2.2.6  The L a s e r Beam Arrangement  23  2.2.7  V e l o c i t y Measurement  25  2.2.8  R o t a t i o n a l Temperature  27  iv  2.2.9 2.3  31  Theory  2.4 Chapter  C a l c u l a t i o n o f Frequencies  32  2.3.1  Background  32  2.3.2  M o l e c u l a r Hamiltonian  36  2.3.3  M a t r i x Elements  40  2.3.4  S e l e c t i o n Rules and I n t e n s i t y C a l c u l a t i o n s 46  Results 3  49  Application to B0 Molecule  the  Rotational  Structure  of  the  55  3  3.1  Introduction  55  3.2  Experimental D e t a i l s  57  3.2.1  B0 P r o d u c t i o n  57  3.2.2  Intermodulated  3.2.3  C a l c u l a t i o n o f Frequencies  59  3.3.1  Background  61  3.3.2  Hund's C o u p l i n g Cases  69  3.3.3  R o t a t i o n a l L e v e l s and S e l e c t i o n Rules  74  3.3.4  Case  (a) Hamiltonian and M a t r i x Elements  78  3.3.5  The Hamiltonian and M a t r i x Elements f o r Hund's Case (b)  81  3.3.6  The Hamiltonian f o r A!!,,, v = l , and i t s M a t r i x Elements  85  3.3  3.4  2  Fluorescence  57  Theory  2  2  Results  98  3.4.1  Least Squares F i t t i n g  3.4.2  Numerical R e s u l t s and D i s c u s s i o n  100  3.4.3  Conclusions  114  v  98  References  Appendix  I  Transition  Appendix  II  Sample i n t e n s i t y  Appendix  III  Observed t r a n s i t i o n s  frequenices  o f N0  calculation ( i n cm" ) 1  f o r the A^-X^lig system  o f B0  C a l c u l a t e d ground s t a t e Appendix IV  levels  of  U  B0  2  2  2  energy  i n wave numbers  (cm" ) 1  The u p p e r s t a t e e n e r g y l e v e l s < Appendix V  B0  2  i n wave numbers  vi  (cm" ) 1  List  2.1  Comparison o f microwave hyperfine  2.2  splittings  Comparison of rotational  of  Tables  and o p t i c a l l y  50  i n N0 . 2  a microwave  interval  measured  i n N0  and o p t i c a l l y along with  2  measured  50  calculated  values. 2.3a  Molecular constants  2.3b  Upper s t a t e r o t a t i o n a l  2.4a  Molecular constants  2.4b  Upper s t a t e r o t a t i o n a l  constants  3.1  The H a m i l t o n i a n m a t r i x  f o r the  states of  B0 . 2  respectively. origin 3.2  of  the  B0 . 2  respectively. origin  of  The ± r e f e r  of  vibronic  B0 . 2  vibronic  2  band 99.  2  for N0  band 99.  2 /  N0 ,  b a n d 115.  2  f o r N0 ,  b a n d 115.  2  000C^) v i b r o n i c the  e and f  52 52 53 53 82  levels  T , represents 0  f o r the to  X^iig 010 ( A)  the  vibronic  2  the  The t e r m v a l u e , vibronic  N0 ,  state.  The ± r e f e r  e and f  82  levels  T , represents 0  the  state. f o r the  The t e r m v a l u e ,  X n 2  010 ( S)  vibronic  2  g  T , represents 0  85  the  origin.  The H a m i l t o n i a n m a t r i x ( S)  to  The t e r m v a l u e ,  The H a m i l t o n i a n m a t r i x state  3.4  the  constants  f o r upper s t a t e of  The H a m i l t o n i a n m a t r i x state of  3.3  f o r upper s t a t e o f  v i b r o n i c states of  and f p a r i t y  levels  f o r the B0 . 2  A ^ 010 2  The ± r e f e r  respectively.  vii  ( A) 2  to  and the  95 e  3.5  Molecular B0 ,  U  2  3.6  3.7  BO  B0 ,  BO , 2  3.9  B0 , 2  3.10  3.11  B0  constants  constants  2  102  f o r t h e A ^ 000 l e v e l  of  104  f o r t h e A ! ^ 000 l e v e l  of  104  2  2  i n cm" . 1  constants  f o r t h e X ! ^ 010 l e v e l s  of  107  f o r t h e A ^ 010 l e v e l s  of  109  2  i n cm" . 1  constants  2  i n cm" . 1  2 /  Fundamental B0 ,  of  2  1  Molecular n  f o r t h e X ^ 000 l e v e l  i n cm" .  Molecular n  constants  i n cm" .  Molecular 10  102  1  2 /  2  3.8  of  2  1  Molecular n  f o r t h e X ! ^ 000 l e v e l  i n cm" .  Molecular 1 0  constants  constants  form f i t t e d p a r a m e t e r s i n  i n cm" . 1  viii  115  List  1.1  A confocal  of  Fabry-Perot  Figures  interferometer  i n c i d e n t beam p a r a l l e l t o  the  w i t h an  axis of  4  the  interferometer. 1.2  2.1  A schematic  diagram of  " A b l o c k diagram of  the  the  Illustration  2.3  T h e m o l e c u l a r beam c r o s s e d beam w i t h  nozzle  Two l e v e l  2.5  The arrangement t o  2.6  A Gaussian v e l o c i t y  2.7  The G a u s s i a n v e l o c i t y a)  the  at  17  r i g h t a n g l e s by the collected  19 collect  Lamb d i p s i g n a l s .  b)  Bennet h o l e s as  with molecules the  convergence  laser  c)  separating  past  the  transition  Ri(2) a n d S ( 2 ) R  12  crossing Lamb d i p s  (500  is  22  the  symmetrically of  the  two  tuned i n frequency as t h e  laser  is  and tuned  frequency.  transitions  L I F (1.5  21  population distribution  Bennet h o l e s as t h e Bennet h o l e s  20  population distribution.  formation of  Doppler s h i f t e d ,  2.8  14  system.  beam i n t e r a c t s  the  15  both.  2.4  laser  8  and skimmer c o m b i n a t i o n .  fluorescence  perpendicularly to  showing  c a l i b r a t i o n system.  m o l e c u l a r beam a p p a r a t u s .  2.2  laser  of  the  GHz s c a n )  MHz s c a n ) .  ix  f r o m a) a n d b)  single the  corresponding  24  2.9  Experimental set  up f o r v e l o c i t y  circle  the  the 2.10  2.12  a r e a where b o t h l a s e r  m o l e c u l a r beam and f l u o r e s c e n c e  The m o l e c u l a r the  2.11  indicates  fluorescence  velocity  measurement.  The g r a p h o f  ln[ (IH/IQJ/AH]  o b t a i n the  rotational  molecular  beam.  N0  2  measurement.  with  axes  inertial  signal  versus  beams  cross  obtain  F„/k used to of  and p o i n t  26  observed.  used t o  temperature  (a,b,c)  is  The  the  N0  group  28  30  2  (x,y,z)  33  shown.  2.13  Gjv c h a r a c t e r t a b l e .  33  2.14  Walsh diagram a p p r o p r i a t e t o  N0  2  [after  reference  34  31]. 2.15  Existing rotational state of  levels  i n the  ground e l e c t r o n i c  N0 . 2  2.16  V e c t o r diagram of Hund's c o u p l i n g case  2.17  Shown a r e t h e  observed  (solid  and t h e  lines)  (dotted 3.1  B l o c k diagram showing  A sample  allowed  R(4)  forbidden  (b^).  41  transitions  47  transitions  lines).  fluorescence 3.2  37  intermodulated  58  The 150 MHz m a r k e r s  60  experiment.  3 0 GHz B0  scan.  2  are not  shown due t o  assigned  transitions  vibronic  symmetry,  boron isotope,  the  t h e i r high density.  The  have been l a b e l l e d w i t h  a superscript  indicating  and a b r a n c h l a b e l .  x  the the  The b r a n c h  label  is  R or P  (AJ=+1 o r A J = - 1 )  as  a s u b s c r i p t when r e q u i r e d a n d t h e  o r N" v a l u e  in  T h e DL^ p o i n t  3.4  Rough c o n t o u r d i a g r a m s o f  3.5  The p o t e n t i a l  3.6  energy  n  g  curves  and n  a d d i t i o n causes t h i s  u  62 orbitals. as  64  a linear  65  v i b r o n i c energy  bending v i b r a t i o n of  A 90°  2  to  look  3.8  Energy l e v e l  3.9  V e c t o r diagram of b)  levels  a linear  in a ^ g electronic  molecule  B0 .  67  like  a n g u l a r momentum.  The p a t t e r n o f  G=0,  J"  bends.  vibrational  a)  associated  separating  The d e g e n e r a t e b e n d i n g m o t i o n o f  the  component  group c h a r a c t e r t a b l e .  out-of-phase  3.7  spin  parentheses.  3.3  molecule  with the  state  diagram showing  the  f o r m e d when  symmetric is  triatomic  excited  transitions  Hund's c o u p l i n g case  (a)  68  [36]. studied.  with  70 71  G*0.  3.10  V e c t o r diagram of  Hund's c o u p l i n g case  (b).  74  3.11  V e c t o r diagram of  Hund's c o u p l i n g case  (c).  75  3.12  Rotational  l e v e l s which e x i s t  2 vibronic  levels.  Rotational  l e v e l s which e x i s t  3.13  n and A v i b r o n i c l e v e l s . the 3.14  missing  3.15  i n Table  A A F"(J') 2  ground s t a t e  77  i n the  ground s t a t e  79  The d o t t e d  lines  represent  levels.  D e f i n i t i o n of used  i n the  the  v i b r o n i c e n e r g i e s T ^ and A T  z  97  3.4.  combination d i f f e r e n c e .  xi  101  3.16  R e s i d u a l s from t h e upper s t a t e rovibronic a)  U  B0  2  levels plotted against  a n d b)  associated  3.17  10  BO .  The c r o s s e s r e p r e s e n t  the  diamonds r e p r e s e n t t h e ^  and b)  2  A  3 / 2  i n d i c a t e the  A graph of the  2~ a n d A  2  2  of  3 / 2  3.19  1  the  2  A and S  l o w e r  f o r a)  2  A  by c r o s s e s ) Circled  110  2  5 / 2  by and  data  t r a n s i t i o n s have  been  energies  ensure t h a t  levels  of the A n 2  u  112  The c r o s s e s r e p r e s e n t t h e A 2  circles  represent the  2  S  levels.  have been s c a l e d by a while the V  and A 2  -0.31J(J+1).  the d e t a i l s  of  the  5 / 2  have  This  scaling  energy  c a n be s e e n c l e a r l y .  The " r o t a t i o n l e s s " levels  while  (represented  2  been s c a l e d by a f a c t o r o f  levels  final  fit.  -0.3115J(J+1)  done t o  of J  (represented  vibronic states.  factor  is  fit  r o t a t i o n a l energy  l e v e l s while the The  and S  associated  e x c l u d e d from f i n a l  2  indicate  residuals.  ( r e p r e s e n t e d b y diamonds) .  v =l  105  residuals  levels plotted against  ( r e p r e s e n t e d by c r o s s e s )  points  n  J* for  the  R e s i d u a l s from t h e u p p e r s t a t e  ^upper  2  C i r c l e d data points  2  fit.  diamonds)  of the  t r a n s i t i o n s have been e x c l u d e d from  rovibronic  3.18  fit  energies  of  a l l the v i b r o n i c  studied with respect  to  a zero at  000(^3/2)  J=3/2  level.  xii  the X !^, 2  113  Acknowledgement  I wish to  t h a n k D r . A . J . M e r e r f o r p r o v i d i n g me w i t h  o p p o r t u n i t y t o work i n t h e resolution  electronic  challenging  spectroscopy  field  of  the  group a t UBC. as h i s  encouragement  I a l s o want t o  Dr.  throughout t h i s  work.  A . Adam w i t h whom I w o r k e d c l o s e l y  project.  Finally,  t h a n k s go t o  students,  postdoctoral  worked w i t h o v e r t h e  fellows  past  few  of  and t e c h n i c i a n s years.  xiii  and  continuing  on a l l  the v a r i e t y  high  His help  s u p p o r t was much a p p r e c i a t e d a s w e l l  the  thank  aspects of  this  graduate I h a v e met and  1  Chapter A C a l i b r a t i o n System f o r 1.1  1 the  Visible  Region  Background  One  problem  facing high resolution l a s e r spectroscopists  i s t o know, w i t h the h i g h e s t p o s s i b l e degree o f accuracy, the wavelength system.  of the l i g h t w i t h which they are p r o b i n g t h e i r  V a r i o u s methods are c u r r e n t l y used.  depend on the simultaneous  S e v e r a l methods  r e c o r d i n g of a secondary  standard w h i l e o t h e r s make use of the coherency and i n t e r f e r o m e t r i c t e c h n i q u e s .  wavelength  of l a s e r  Systems t h a t use  light  secondary  standards o f t e n use i o d i n e f l u o r e s c e n c e or a b s o r p t i o n l i n e s [1] because t h e r e i s an i o d i n e a t l a s c o n t a i n s over 22 000  [2] a v a i l a b l e which  l i n e s between 14800 cm"  1  measured t o an accuracy of ±0.002 cm" . 1  and 20000  Other  cm"  1  secondary  standards b e i n g used are iron-neon hollow cathode lamps [3] and uranium hollow cathode  lamps [4] whose s p e c t r a are  recorded using optogalvanic spectroscopy.  When secondary  standards are used f o r l a s e r work, one s i m u l t a n e o u s l y r e c o r d s the secondary  standard, the system b e i n g s t u d i e d and  i n t e r f e r o m e t r i c markers f o r i n t e r p o l a t i o n between the l i n e s of the secondary  standard.  A method o f c a l i b r a t i o n which does not i n v o l v e r e c o r d i n g a secondary  standard uses a 'lambda-  1  or 'wave-  1  meter.  There  are s e v e r a l d e s i g n s c u r r e n t l y i n use although they are a l l similar in principle  [5-7].  They are based on a t r a v e l l i n g  2  M i c h e l s o n i n t e r f e r o m e t e r and a comparison o f t h e unknown l a s e r frequency w i t h t h e known frequency o f a s t a b i l i z e d  laser,  o f t e n a HeNe l a s e r s t a b i l i z e d on an I h y p e r f i n e l i n e [ 8 ] . 2  In t h e i r p r e v i o u s sub-Doppler  work, t h e h i g h r e s o l u t i o n  e l e c t r o n i c spectroscopy group a t UBC c a l i b r a t e d t h e i r data against iodine fluorescence spectra. has r e c e n t l y developed  However, as t h e group  a m o l e c u l a r beam system which produces  s p e c t r a w i t h much narrower l i n e w i d t h s , a new c a l i b r a t i o n system w i t h g r e a t e r accuracy was r e q u i r e d .  There were two  p o s s i b l e ways o f upgrading t h e o l d system.  One was t o buy an  "Autoscan"  wavemeter upgrade system f o r t h e i r e x i s t i n g  l a s e r from Coherent  ring  Inc., and t h e o t h e r was t o b u i l d a new  c a l i b r a t i o n system w i t h i n t h e l a b which would perform a similar function.  The "Autoscan"  be a b l e t o a c q u i r e data  system [9] i s s p e c i f i e d t o  'seamlessly* over a 10 THz frequency  range w i t h ±2 5 MHz r e p r o d u c i b i l i t y , but i t cannot enough, w i t h a s u f f i c i e n t l y l a r g e number o f data  scan s l o w l y acquisition  p o i n t s , t o r e c o r d t h e v e r y h i g h r e s o l u t i o n data t h a t t h e m o l e c u l a r beam system was capable o f producing.  Spectral  l i n e s w i t h f u l l - w i d t h a t half-maximum (FWHM) o f 2 MHz c o u l d be d i s t o r t e d o r even missed a t t h e slowest data a c q u i s i t i o n r a t e available.  F o r t h i s reason, i t was decided t o develop a new  c a l i b r a t i o n system. The  system i s c e n t r e d around a 750 MHz f r e e s p e c t r a l  range e t a l o n whose c a v i t y l e n g t h i s s e r v o - l o c k e d t o a m u l t i p l e of t h e HeNe l a s e r wavelength; more d e t a i l s w i l l be g i v e n i n  3  Section  1.3.  controlled written three  S p e c t r a were r e c o r d e d u s i n g a c o m p u t e r -  laser  a t UBC.  channels  s c a n n i n g and d a t a a c q u i s i t i o n programme As t h e  of  data,  c a l i b r a t i o n markers,  computer v a r i e d t h e the  laser  molecular spectra,  frequency,  the  750 MHz  and 150 MHz i n t e r p o l a t i o n m a r k e r s ,  were  recorded.  Theory  1.2  Interferometers  1.2.1  Before d e s c r i b i n g the of  the  theory  of  the  c a l l e d an e t a l o n ) type  consists  their  focal  of  light.  transmitted difference different  to  (as  spherical mirrors  is  (called  the  maxima w i l l  there  fringes)  reflections light.  summary  If  of  the is  of  any  incident  the  w i l l b e maxima i n  when t h e  beam t h a t  optical  the  path  undergo  an i n t e g r a l m u l t i p l e  interferometer,  6v = c / 4 d  by  reflecting,  t u n a b l e monochromatic  be  (also  this  separated  100%  sent through  i n Figure 1.1),  passes through the  intensity  are not  t r a n s m i t a few p e r c e n t  numbers o f of  An i n t e r f e r o m e t e r  The m i r r o r s  light  a short  Fabry-Perot interferometer  t r a v e l l e d by p a r t s o f  wavelength  the  d.  intensity  from a l a s e r of  two c o n c a v e  When a beam o f  interferometer  the  confocal  w i l l be g i v e n .  length,  but are designed  c a l i b r a t i o n system,  the  of  light  separation  F i g u r e 1.1:  A confocal Fabry-Perot interferometer with an i n c i d e n t beam p a r a l l e l t o t h e a x i s o f t h e interferometer.  5  where 4d i s t h e path d i f f e r e n c e between two r a y s , c i s t h e speed o f l i g h t and Su i s c a l l e d t h e f r e e s p e c t r a l range (FSR) o f t h e i n t e r f e r o m e t e r quoted i n frequency  units.  F o r an  e t a l o n t o have a FSR o f 750 MHz, t h e m i r r o r s must be 10 cm apart. The C a l c u l a t i o n  1.2.2  of  To measure t h e frequency necessary  the  Free Spectral  Range  o f a dye l a s e r a c c u r a t e l y , i t i s  t o know t h e FSR o f t h e s t a b i l i z e d 750 MHz e t a l o n  with great p r e c i s i o n .  T h i s r e q u i r e s t h a t t h e wavelength o f  the HeNe l a s e r be known, along with t h e a b s o l u t e o r d e r number of t h e f r i n g e t o which the system i s l o c k e d .  In t h e p r e s e n t  case, these were determined by r e c o r d i n g t h e e t a l o n f r i n g e s t o g e t h e r with t h e I  2  f l u o r e s c e n c e spectrum over as wide a range  as p r a c t i c a l , which was about 3 000 t o r e c o r d the f u l l  cm" . 1  I t was not necessary  3000 cm" as t h e FSR c o u l d be determined  w i t h enough accuracy  1  from a 10 cm" p o r t i o n o f t h e spectrum t o 1  o b t a i n t h e r e l a t i v e order number o f t h e f r i n g e s i n another 10 cm" p o r t i o n some 100 cm" away. 1  The FSR c o u l d then be r e f i n e d ,  1  i n o r d e r t o l i n k a t h i r d 10 cm" p o r t i o n some 400 cm" away. 1  all,  eleven 1  1  In  10 cm" i n t e r v a l s were c o l l e c t e d over t h e r e g i o n s 1  16480 cm" t o 17700 cm" 15900 cm"  1  1  (using R6G l a s e r dye) and 14920 cm" t o 1  ( u s i n g DCM l a s e r dye) .  T h i s l a r g e range was  r e q u i r e d so t h a t t h e order number o f t h e f r i n g e a t t h e HeNe l a s e r frequency  t o which the system was l o c k e d c o u l d be  determined t o an accuracy  o f approximately  ±0.1. A l i n e a r  l e a s t - s q u a r e s programme was used t o f i t t h e wave numbers o f  6  the  I  2  lines  fringe  [2]  against  numbers.  their positions  i n terms  The e q u a t i o n g o v e r n i n g t h i s  of  relative  relationship  is  "N = "int + N*FSR where  is  the  recorded  (near  with r e l a t i v e fit  the  the  last  i n the  frequency  of  14920 cm" )  the  digit.  determined to  the  frequency  fringe  of  the  o r d e r number N i n wave number u n i t s .  FSR was d e t e r m i n e d t o  range  longest wavelength  and «/„ i s  1  (1.1)  Since the  be  1  f  1  frequency  1  p o l a r i z a t i o n - s t a b i l i z e d HeNe l a s e r 1  f r e q u e n c y must  cm" , and g i v e n t h a t  1 4 9 2 3 . 1 9 8 1 5 cm" , t h e  1 5 7 9 8 . 0 0 3 6 4 cm" , w i t h t h e  the  be 0.025038797 cm" w i t h ± 3 on  e x a c t HeNe l a s e r  15798.00 ± 0.01  From  fringe  on t h a t  system l o c k e d t o  k t  of  was  our  d a y was fringe  lie  found to  be  number  630941. Therefore,  given the  c o m m e r c i a l wavemeter, therefore equation It to  obtain its  frequency  A f t e r the  r e l o c k i n g the  Section  the  FSR o f  from  s y s t e m was t o  fringe,  and n o t ,  ensure  it  was known,  physical  that  it  to  was  any o f  an a d j a c e n t  nine  with  locked to  arrangement  the  arrangement  The o n l y d i f f i c u l t y  present  c a n be r e l o c k e d t o  etalon  the  f o r example,  l o c k i n g system i n i t s 1.3)  above p r o c e d u r e o n l y had  the  l o n g as  e t a l o n was n o t d i s t u r b e d .  With the  o r d e r number and  frequency with higher accuracy  s h o u l d be m e n t i o n e d t h a t  correct  its  (1.1).  be done o n c e .  the  a f r i n g e measured by a  one c a n work o u t  s y s t e m c o u l d be s h u t down a s of  of  one. (see  different  the  7  fringes.  The method used t o check f o r the c o r r e c t f r i n g e was  t o compare some p r e v i o u s l y c o l l e c t e d data, o r more s p e c i f i c a l l y , the d i s t a n c e between the 750 s e v e r a l N0  hyperfine t r a n s i t i o n s .  2  markers  and  I f the system i s not  t o the c o r r e c t f r i n g e , the c a v i t y l e n g t h FSR)  MHz  (and t h e r e f o r e the  w i l l be s l i g h t l y d i f f e r e n t ; the e f f e c t i s t o s h i f t  markers w i t h r e s p e c t t o the f i x e d N0 the s h i f t may  transitions.  2  be c a l c u l a t e d by m u l t i p l y i n g the FSR  r a t i o of the N0  2  f r e q u e n c i e s t o the HeNe frequency  s u b t r a c t i n g the FSR.  For N0  corresponds t o a 50 MHz  the  The  s i z e of  by  the  and  then  t r a n s i t i o n s a t 16850 cm" ,  this  1  2  shift.  r e g i o n s d i f f e r e n t from the N0  2  locked  For f u t u r e work a t wavelength r e g i o n s s t u d i e d (see Chapter 2),  the system would have t o be l o c k e d f i r s t t o the c o r r e c t f r i n g e u s i n g N0  2  and then switched  t o the new  r e g i o n where data  c o l l e c t e d on a d i f f e r e n t molecule c o u l d be used as a 'reference lock point' f o r that region. 1.3  Experimental  The  Details  c a l i b r a t i o n system which was  F i g u r e 1.2.  The  system i s designed  o f an argon ion-pumped r i n g dye 699-21).  A 750 MHz  e t a l o n was  developed i s shown i n  t o measure the  laser  frequency  (Coherent Inc. model  chosen over o t h e r  commercially  a v a i l a b l e e t a l o n s because the accuracy  of our  B u r l e i g h WA-20VIS wavemeter (1 i n 10 )  i s such t h a t i t can  6  d i s t i n g u i s h between two FSR  f r i n g e s 750  MHz  apart.  existing  With a s m a l l e r  e t a l o n , the order numbers would be g r e a t e r than 10 , 6  so  fen. generator wavemeter lock-in  automatic  -BZ3-  6 O-  Q  HV amp.  level  -0 A  control  J  stabilized  _  j#  H*&Hff PD2  J  T  \  \  P  BS6  o 1 1  A  —&—V  ) V I 0»  /  t  BS5  PDl  4H  t o computer  iris2  >  1  BS4 70R/30T HeNe l a s e r  pol2  to computer  passive 150  ^n\2  )  ramped 750 to  oscilloscope  (  )  BS2 50T/50R BS3 85T/15R  Ar* i o n l a s e r  ^BSl  85T/15R  I  t o experiment  Figure 1.2: A schematic diagram of the c a l i b r a t i o n system.  09  9  that  t h e wavemeter w o u l d n o t h a v e t h e  them,  while  l a r g e r FSR e t a l o n s  accuracy to  lead to  too  measure  few c a l i b r a t i o n  markers. A small  f r a c t i o n of the  a beamsplitter to  the  (BS1)  experiment.  subdivided at  while the  and a p a s s i v e  one o f  its  etalon,  dye-laser  as  length repeatedly. detected  one t o  is  'ramped'  A 'ramped'  which 750 MHz  etalon  translator  changes  this  is  d i s p l a y e d on a n  if  the dye l a s e r  l o n g i t u d i n a l mode.  f u r t h e r i n C h a p t e r 2,  Section 2.2.8  A further p o r t i o n of the  by b e a m s p l i t t e r BS4, and s e n t t o t h e  l o c a t e d approximately 1 meter i n f r o n t o f  the  angle  of  to  d i r e c t the  entry is  the  The purpose o f  same f r o m d a y t o  day.  This  on t h i s  angle.  E r r o r s o f up t o  0.04  the  the  l a s e r beam r e p r o d u c i b l y s u c h  i m p o r t a n t a s t h e wave number r e a d i n g o b t a i n e d i s dependent  and  Burleigh  iris  is  they  l a s e r beam  o b t a i n a wave number r e a d i n g .  wavemeter,  is  The o u t p u t  wavemeter t o (irisl)  has  translator  p a r t of the c a l i b r a t i o n system;  Section 3.2.3.  picked off  sent  150 MHz e t a l o n p r o v i d e s i n t e r p o l a t i o n m a r k e r s  which form a n e c e s s a r y be d i s c u s s e d  see  is  by  further  The t r a n s m i s s i o n o f  by a p h o t o d i o d e ,  which allows  passive  C h a p t e r 3,  a  a p p l i e d to the  s c a n n i n g c o r r e c t l y on a s i n g l e  will  is  output  end m i r r o r s mounted on a p i e z o e l e c t r i c  oscilloscope  from t h e  the  picked off  (BS2 and B S 3 ) ,  l i g h t to  150 MHz e t a l o n .  a ramped v o l t a g e  cavity  remainder of  two o t h e r b e a m s p l i t t e r s  etalon  the  output i s  The f r a c t i o n p i c k e d o f f  s e n d s m a l l amounts o f  such t h a t  dye l a s e r  that  is  highly  cm" c a n o c c u r 1  because o f t h i s The  effect.  remainder o f t h e l a s e r beam i s sent t o t h e c e n t r a l  component o f t h e c a l i b r a t i o n system, t h e p r e s s u r e - and temperature-stabilized  750 MHz e t a l o n .  This s t a b i l i z a t i o n i s  done by means o f a vacuum housing which i s kept a t a p r e s s u r e below 10 microns and a temperature o f approximately 48°C ( s t a b i l i t y ±0.004°C [10]) by a d i f f u s i o n / m e c h a n i c a l  pump  combination and a c i r c u l a t i n g o i l bath r e s p e c t i v e l y . o f t h i s l a s e r beam i s a l s o d e f i n e d by i r i s e s iris3),  as t h e shape o f t h e t r a n s m i s s i o n  ( i r i s 2 and  f r i n g e s obtained  the e t a l o n depends d i r e c t l y on t h e i n p u t angle [11]. s i t u a t e d i n f r o n t o f t h e e t a l o n matches t h e e l e c t r i c d i s t r i b u t i o n of the l a s e r with the f i e l d d i s t r i b u t i o n for  a TEMQO mode i n t h e e t a l o n .  of l i g h t through t o t h e d e t e c t o r  The path  The l e n s field required  T h i s a l l o w s t h e maximum amount [11].  The p o l a r i z e r ( p o l l )  ensures t h a t t h e dye l a s e r beam i s v e r t i c a l l y p o l a r i z e d . reason f o r u s i n g orthogonal  from  The  p o l a r i z a t i o n s f o r t h e two beams  (the HeNe l a s e r beam i s h o r i z o n t a l l y p o l a r i z e d ) i s so t h a t they can be separated  by a prism p o l a r i z e r a f t e r p a s s i n g  through t h e e t a l o n , and sent t o d i f f e r e n t d e t e c t o r s . final dye  beamsplitter  The  (BS5) combines t h e HeNe l a s e r beam w i t h the  l a s e r beam such t h a t they f o l l o w the same path w i t h i n t h e  etalon.  The HeNe l a s e r (Spectra P h y s i c s model 117A) i s a  p o l a r i z a t i o n - s t a b i l i z e d helium-neon l a s e r where s t a b i l i z a t i o n i s achieved  by a l t e r i n g t h e c a v i t y l e n g t h t o m a i n t a i n t h e  r e l a t i v e i n t e n s i t i e s o f t h e two o r t h o g o n a l l y - p o l a r i z e d modes  11  which a r e allowed t o operate i n t h e c a v i t y ; o n l y one o f these modes i s allowed t o emerge from t h e l a s e r . beam passes through a p o l a r i z e r (A/4)  The HeNe l a s e r  (pol2) and a quarterwave p l a t e  combination which p r e v e n t s feedback, t h a t  i s , reflected  l i g h t returning  t o t h e l a s e r , which would upset i t s  stabilization.  The f i n a l p o l a r i z e r  (pol3) ensures t h a t t h e  HeNe l a s e r beam i s p u r e l y h o r i z o n t a l l y p o l a r i z e d the  two l a s e r beams a r e monitored by two photodiodes (PD1  PD2).  The HeNe l a s e r s i g n a l  p o s i t i v e feedback loop c o n s i s t i n g  (from PD2) i s sent t o a of a l o c k - i n amplifier  P r i n c e t o n A p p l i e d Research model 5101), a f u n c t i o n ( C i r c u i t m a t e FG2), a h i g h v o l t a g e a m p l i f i e r  locks  a transmission fringe of the etalon  frequency, u s i n g t h e f i r s t d e r i v a t i v e d r i f t away from t h e peak p o s i t i o n .  i s then a p p l i e d end  Spectrum  T h i s loop  t o t h e HeNe l a s e r  o f t h e s i g n a l t o monitor A correction  voltage  t o the p i e z o e l e c t r i c t r a n s l a t o r on which the  m i r r o r i s mounted.  often  (EG&G  generator  (Burleigh  A n a l y z e r RC-46) and an automatic l e v e l c o n t r o l .  any  entering  etalon. The  and  on  I n i t i a l l y i t was found t h a t t h e system  l o s t lock a f t e r several  hours because o f d r i f t i n g i n the  HeNe l a s e r frequency and c i r c u i t i n s t a b i l i t i e s , which caused the  c o r r e c t i o n v o l t a g e t o become t o o l a r g e .  level control  The automatic  c i r c u i t was then added; t h i s a d j u s t s an o f f s e t  v o l t a g e t o t h e p i e z o e l e c t r i c t r a n s l a t o r such t h a t t h e c o r r e c t i o n v o l t a g e from t h e l o c k - i n a m p l i f i e r remains v e r y near zero.  The system can now be l o c k e d f o r s e v e r a l  weeks.  12  The dye l a s e r s i g n a l a data channel  (from PD1) i s a c q u i r e d by t h e computer as  o f 750 MHz c a l i b r a t i o n markers.  In summary, w i t h t h e a c c u r a t e d e t e r m i n a t i o n o f t h e FSR o f the s t a b i l i z e d 750 MHz e t a l o n , i t i s now p o s s i b l e t o c a l c u l a t e the o r d e r number o f a t r a n s m i s s i o n f r i n g e g i v e n t h e wave number r e a d i n g o b t a i n e d from t h e B u r l e i g h wavemeter.  This  a l l o w s c a l c u l a t i o n o f t h e p r e c i s e wave numbers o f t h e t r a n s m i s s i o n f r i n g e s , thereby c a l i b r a t i n g t h e s p e c t r a l obtained.  data  The accuracy o f t h e system i s d i s c u s s e d i n S e c t i o n  2.4 o f Chapter 2.  Chapter 2 A p p l i c a t i o n o f the C a l i b r a t i o n System t o the H y p e r f i n e S t r u c t u r e o f 2.l  N0  2  Introduction The spectrum  of N0  was  2  f i r s t seen by Brewster  in  1834  [12] and s i n c e then many s c i e n t i s t s have put i n y e a r s of work attempting t o u n r a v e l i t s m y s t e r i e s . f i r s t t e s t of the new  N0  2  was  chosen  m o l e c u l a r beam system a t UBC  f o r the f o r several  reasons: N0  2  absorbs throughout  the v i s i b l e r e g i o n , which  meant t h a t the most convenient l a s e r dye, 6G, -  c o u l d be  rhodamine-  used,  m o l e c u l a r beams of N0  2  are simple t o make; one o n l y  needs a 10% mixture of N0 p r e s s u r e of l e s s than one  2  i n argon a t a b a c k i n g atmosphere,  s e v e r a l papers on m o l e c u l a r beam s p e c t r a of  N0  [8,13-15] were a v a i l a b l e , so t h a t comparisons  2  with  the e a r l i e r data c o u l d e a s i l y be made, r o t a t i o n a l a n a l y s e s had a l r e a d y been done [16-18] f o r v a r i o u s bands i n the rhodamine-6G r e g i o n , the UBC  system  should have s u f f i c i e n t r e s o l u t i o n t o  see the n i t r o g e n magnetic The a c t u a l band systems chosen of the B - A 2  2  2  1  hyperfine structure.  f o r study were two K>0  e l e c t r o n i c t r a n s i t i o n a t 593.5 nm  subbands  and 585.1  nm.  14  They were chosen as they were both noted t o be s t r o n g t r a n s i t i o n s by Smalley e t a l . [15]; f o l l o w i n g Smalley e t a l . ' s numbering scheme these w i l l be c a l l e d bands 99 and 115 respectively. 2.2  Experimental D e t a i l s 2.2.1  The M o l e c u l a r Beam  A b l o c k diagram up t o produce N0  2  Apparatus  o f the m o l e c u l a r beam apparatus, as s e t  f l u o r e s c e n c e s p e c t r a , i s shown i n F i g u r e 2.1.  A m o l e c u l a r beam o f N0 was formed by expanding 2  10% N0  2  a mixture o f  i n argon through a n o z z l e and skimmer combination as  shown i n F i g u r e 2.2.  The n o z z l e diameter used was 50 fim and  the skimmer diameter was 500 pm.  pressure 300 T o r r  nozzle  F i g u r e 2.2  pressure ~10  -3  Torr  pressure <10  -6  Torr  skimmer  I l l u s t r a t i o n o f the n o z z l e and skimmer combination.  to  F i g u r e 2.1  A b l o c k diagram  calibration  of the m o l e c u l a r beam apparatus.  16  Before p a s s i n g through t h e n o z z l e , t h e gas mixture obeys t h e normal Maxwell-Boltzmann v e l o c i t y d i s t r i b u t i o n f o r molecules i n thermal e q u i l i b r i u m a t room temperature.  The  expansion through t h e n o z z l e c o n v e r t s t h e random motions of thermal e q u i l i b r i u m i n t o a d i r e c t e d mass flow, which reduces the width o f t h e v e l o c i t y d i s t r i b u t i o n  [15]. The skimmer  p i c k s out t h e c e n t r a l p o r t i o n o f t h i s expansive beam t o produce a w e l l c o l l i m a t e d m o l e c u l a r beam, where t h e v e l o c i t y range and a n g u l a r divergence a r e both s m a l l .  W i t h i n t h i s beam  c o l l i s i o n s a r e i n f r e q u e n t s i n c e t h e molecules a r e a l l moving forward a t n e a r l y the same speed.  As a r e s u l t o f t h e  expansion t h e molecules a r e r o t a t i o n a l l y c o l d , as w i l l be d i s c u s s e d f u r t h e r i n S e c t i o n 2.2.8. 2.2.2  Fluorescence C o l l e c t i o n  Device  Two d i f f e r e n t experiments were done on N0 . 2  One i n v o l v e d  a s i n g l e l a s e r c r o s s i n g where t h e t o t a l f l u o r e s c e n c e was c o l l e c t e d and t h e l i n e w i d t h was determined by t h e a n g u l a r d i v e r g e n c e o f t h e beam (see S e c t i o n 2.2.3).  The second  experiment used Lamb d i p methods (see S e c t i o n 2.2.5) where the l i n e w i d t h was reduced by sampling t h e middle of t h e m o l e c u l a r beam and o n l y a f r a c t i o n o f the. t o t a l f l u o r e s c e n c e was collected.  In both cases, the m o l e c u l a r beam i s c r o s s e d a t  r i g h t a n g l e s by t h e l a s e r beam as shown i n F i g u r e 2.3.  The  a p e r t u r e s i n t h e i n p u t and output arms reduce t h e s c a t t e r e d room and l a s e r l i g h t r e a c h i n g t h e c o l l e c t i o n system.  The  F i g u r e 2.3  The molecular beam c r o s s e d a t r i g h t angles by the l a s e r beam w i t h f l u o r e s c e n c e c o l l e c t e d p e r p e n d i c u l a r l y t o both.  18  f l u o r e s c e n c e c o l l e c t i n g d e v i c e , a l s o shown i n F i g u r e 2.3, d e v i s e d t o i n c r e a s e the amount of molecular  was  fluorescence  c o l l e c t e d ; i t includes a s p a t i a l f i l t e r to eliminate scattered light.  The  l a s e r beam and the molecular  c e n t r e o f the c o l l e c t i n g system where N0 isotropically.  The  beam c r o s s i n the 2  molecules f l u o r e s c e  c o l l e c t i n g system uses a s e r i e s o f  lenses  (L1-L4) t o focus l i g h t upward t o the p h o t o m u l t i p l i e r tube (PMT) fed  (RCA  tube type C31034-02).  The  c u r r e n t from the PMT  is  t o a l o c k - i n a m p l i f i e r (EG&G P r i n c e t o n A p p l i e d Research  Model 128-A) which measures the s i g n a l by phase s e n s i t i v e detection.  The major reasons why  s c a t t e r e d l i g h t i s a problem  i s t h a t i t i s modulated, by the e x t e r n a l chopper, a t the same frequency  as the molecular  a m p l i f i e r cannot separate  f l u o r e s c e n c e , so t h a t the the two  signals.  I t i s also  p o s s i b l e f o r s c a t t e r e d l i g h t t o o v e r l o a d the PMT. filter  (F)  (Corning c o l o u r e d g l a s s f i l t e r  f r o n t o f the PMT  lock-in  2-58)  A cut-off  was  placed i n  t o reduce the background s i g n a l from the  l a s e r l i g h t w h i l e a l l o w i n g molecular  f l u o r e s c e n c e t o the r e d  of the c u t - o f f wavelength (37% t r a n s m i s s i o n between 637-648 nm)  to  pass.  2.2.3  Laser-Induced  Fluorescence  A molecule can absorb r a d i a t i o n o f the frequency state  appropriate  and be e x c i t e d from i t s ground s t a t e i n t o an upper  ( F i g u r e 2.4),  from which i t decays back t o lower s t a t e s ,  e m i t t i n g i t s e x t r a energy as spontaneous emission  or  19  Figure fluorescence. its  original  that the  is  s t a t e but  excitation.  it  total  of  laser  excitation  all  which r e s u l t s  MicroVAX  spectrum.  from  computer  i n frequency,  to  is  and  Laser-induced  with a linewidth of approximately  in this  manner.  This  linewidth  is  w h i c h c a n be a c h i e v e d w i t h an a b s o r p t i o n  or u n c o l l i m a t e d flow  divergence  the  decay back  spectroscopy  fluorescence  experiment,  (LIF) s i g n a l s  narrower than t h a t  excitation  scans the  and s t o r e s t h e  10 MHz were a c h i e v e d  cell,  the  system.  does not n e c e s s a r i l y  in laser  In t h i s  programmed s u c h t h a t  fluorescence  Two l e v e l  The m o l e c u l e  important i s  collects  2.4  system,  because  the  small,  t h e m o l e c u l a r beam r e d u c e s t h e D o p p l e r  width.  The D o p p l e r W i d t h  2.2.4  Molecules with v e l o c i t y frequency  w = u> (l±v/c) 0  where t h e  away f r o m o r t o w a r d s t h e transition  f r e q u e n c y as  assumed t h a t light  source.  light  absorb r a d i a t i o n at  ± r e f e r to molecules  source.  shown i n F i g u r e  the molecule I n a gas  v will  is  cell,  The w  0  2.4,  is  where  moving  the it  not moving w i t h r e s p e c t a Gaussian v e l o c i t y  a  is to  the  distribution  20  is  present  range of  as  shown  frequencies  absorbed.  in Figure than the  This gives  2.6,  which  results  true transition  a widened  spectral  i n a wider  frequency  line  which i s  be D o p p l e r b r o a d e n e d w i t h a c e r t a i n D o p p l e r w i d t h . m o l e c u l a r beam r e d u c e s velocity laser  the  d i s t r i b u t i o n of  molecules  s a i d to  The  D o p p l e r w i d t h by n a r r o w i n g  the  being  with respect  the  to  the  beam.  2.2.5  Lamb D i p s  Lamb d i p s  c a n be u s e d t o  the  transitions  not  equal  crossings  to  being detected.  the  transition  A a n d B (see  laser  reduce  f u r t h e r the  When t h e  frequency,  Figure  2.5)  the  laser  l i n e w i d t h of frequency  laser  interacts  beam  with  is  at  molecules  beam  10% NOa "in A r  Figure  2.5  The arrangement t o  collect  symmetrically Doppler-shifted i n opposite frequency, very  as  shown i n F i g u r e  intense l i g h t  source,  it  2.7a.  Since  distorts  the  Lamb d i p  signals,  s e n s e s from t h e the  laser  velocity  true  beam i s  a  21  F i g u r e 2.6  A Gaussian v e l o c i t y p o p u l a t i o n  distribution.  22  ——  B F i g u r e 2.7  velocity  A  The Gaussian v e l o c i t y p o p u l a t i o n d i s t r i b u t i o n showing a) the formation of Bennet h o l e s as the l a s e r beam i n t e r a c t s w i t h molecules s y m m e t r i c a l l y Doppler s h i f t e d , b) the convergence of the two Bennet h o l e s as the l a s e r i s tuned i n frequency and c) the Bennet h o l e s s e p a r a t i n g as the l a s e r i s tuned past the t r a n s i t i o n frequency.  23  d i s t r i b u t i o n i n t h e lower s t a t e by d e p l e t i n g t h e p o p u l a t i o n a t the two v e l o c i t i e s ±v, l e a d i n g t o what a r e known as "Bennet holes"  [19].  As t h e l a s e r frequency  c e n t r e frequency  o f t h e molecular  i s scanned toward t h e  l i n e , t h e two "Bennet h o l e s "  move towards each other, c o i n c i d e a t t h e t r u e t r a n s i t i o n frequency  ( F i g u r e 2.7b) and then separate  again  ( F i g u r e 2.7c).  A phase s e n s i t i v e d e t e c t i o n method i s used, w i t h t h e l a s e r beam b e i n g chopped p r i o r t o c r o s s i n g A w h i l e a l o c k - i n a m p l i f i e r i s used t o d e t e c t t h e modulated f l u o r e s c e n c e a t c r o s s i n g B. Linewidths  down t o 2 MHz were achieved u s i n g t h i s method.  A  comparison o f t h e Ri(2) and S (2) l i n e s o b t a i n e d by s i n g l e R  12  c r o s s i n g L I F and t h e corresponding  Lamb d i p s i s shown i n  F i g u r e 2.8. 2.2.6  The L a s e r Beam  Arrangement  A Coherent Inc. model 699-21 r i n g dye l a s e r , pumped by an argon i o n l a s e r , was used as t h e source t o e x c i t e the N0 molecules i n t h i s experiment.  The output  2  o f t h e dye l a s e r  (see F i g u r e 2.1) i s focussed by two l e n s e s , L I and L2, which t o g e t h e r produce a l a r g e r e g i o n where t h e l a s e r beam i s t i g h t l y focussed.  T h i s reduces t h e s c a t t e r e d l i g h t and allows  the l a s e r beam t o be focussed down t o t h e same spot s i z e a t both c r o s s i n g s A and B.  The path o f t h e dye l a s e r beam i s  c o n t r o l l e d by a s e r i e s o f m i r r o r s .  The f i r s t two m i r r o r s (ml  and m2) r a i s e t h e l a s e r beam t o t h e same h e i g h t as t h e molecular  beam, s i n c e t h e l a s e r i s on a separate  optical table  24  Rl(2) + *S (2) 12  LIF  1.5 GHz scan  F i g u r e 2.8  Ri(2) and S (2) t r a n s i t i o n s from a) s i n g l e c r o s s i n g L I F (1.5 GHz scan) and b) t h e c o r r e s p o n d i n g Lamb d i p s (500 MHz s c a n ) . The t r a n s i t i o n s a r e l a b e l l e d F' - F". R  12  25  i n another room, below t h e a x i s o f t h e molecular the l a s e r beam i s d i r e c t e d t o B, two m i r r o r s  beam.  When  (m3 and m.4) a r e  needed t o c o n t r o l i t s h o r i z o n t a l and v e r t i c a l placement through t h e molecular  beam apparatus.  In a s i n g l e c r o s s i n g  LIF experiment t h e l a s e r beam c r o s s e s a t B where t h e h i g h s e n s i t i v i t y PMT i s s i t u a t e d , although  i t i s possible to rotate  the t o p f l a n g e o f t h e beam chamber t o put t h e PMT a t A and, w i t h t h e use o f a b e a m s p l i t t e r through A.  (BS), send t h e l a s e r beam  M i r r o r s m5 and m6 have t h e same purpose as m i r r o r s  m3 and m4 when u s i n g c r o s s i n g A. To r e c o r d Lamb d i p s i g n a l s , t h e l a s e r i s f i r s t  sent  through c r o s s i n g B and then r e f l e c t e d o f f two m i r r o r s  (m7 and  m8, F i g u r e 2.6) t o b r i n g i t back through c r o s s i n g A, upstream of c r o s s i n g B.  In t h i s case, the mechanical chopper (CI)  is  moved from i t s p o s i t i o n i n f r o n t o f c r o s s i n g B t o between m i r r o r s m7 and m8. 2.2.7 The estimated  V e l o c i t y Measurement  speed o f t h e molecules i n t h e molecular by u s i n g Doppler s h i f t s .  A p o r t i o n of the l a s e r  beam was p i c k e d o f f by t h e b e a m s p l i t t e r the a x i s o f t h e molecular Doppler s h i f t  (Av)  beam c o u l d be  (BS) and sent  beam as shown i n F i g u r e 2.9.  along The  of the fluorescence s i g n a l i s r e l a t e d t o  26  1  laser  F i g u r e 2.9  beam  Experimental s e t up f o r v e l o c i t y measurement. The c i r c l e i n d i c a t e s t h e area where both l a s e r beams c r o s s t h e m o l e c u l a r beam and f l u o r e s c e n c e i s observed.  27  the v e l o c i t y o f t h e molecules  V  =  (v) by  Ai/  c  v  sin*  (2.1)  The v a l u e o f e i s 90° f o r t h i s experiment. i s shown i n F i g u r e 2.10.  A typical  signal  I t i s i n t e r e s t i n g t o note t h a t t h e  h y p e r f i n e s t r u c t u r e , which i s e a s i l y r e s o l v e d when t h e l a s e r beam c r o s s e s t h e m o l e c u l a r beam p e r p e n d i c u l a r l y , i s completely unseen i n t h e v e r y much Doppler broadened l i n e o b t a i n e d  from  the c o a x i a l l a s e r beam because o f t h e spread o f v e l o c i t i e s i n the m o l e c u l a r beam.  A v a l u e o f 440 ± 30 m/s was found  f o r the  molecular v e l o c i t y .  T h i s i s a reasonable number as i t i s  somewhat l e s s than t h e t e r m i n a l v e l o c i t y o f argon which i s 558 m/s [ 2 0 ] . 2.2.8  R o t a t i o n a l Temperature  I t i s w e l l known t h a t c o o l i n g o c c u r s i n s u p e r s o n i c beams formed by t h e f r e e expansion  o f a gas through a n o z z l e [20].  By means o f i n t e n s i t y measurements, i t i s p o s s i b l e t o estimate the r o t a t i o n a l temperature experiment.  o f t h e m o l e c u l a r beam i n t h i s  The i n t e n s i t y o f a g i v e n r o t a t i o n a l l i n e i s  p r o p o r t i o n a l t o [21] T  I  _ „ KJ  •» „  -F(KJ)/kT  = ci/Afugoe  ,1  (2.2)  F i g u r e 2.10  The m o l e c u l a r f l u o r e s c e n c e s i g n a l used t o o b t a i n the v e l o c i t y measurement.  29  where c = c o n s t a n t  d e p e n d i n g on t h e  electronic  and v i b r a t i o n a l  transition, v = t r a n s i t i o n frequency , A  KJ  = Honl-London l i n e  g  KJ  = statistical  F(KJ)  weight  = term v a l u e ,  k = Boltzmann  strength of  i.e.  lower  energy  the  this  N0  2  experiment,  K is  some r e a r r a n g e m e n t ,  summed t o ition,  give  I .  of the the  I  separate  total  The v a l u e s  N  of  f o r the  reflected  l a s e r power.  experiments.  = -F„/kT  f o r the  ± 1.6  have  rotational from  [21] (2.4)  i n the  H  i n these values,  of the  is  which  graph  The is  probably fluctuation  graph leads  been  trans-  = (N+l)/(2N+l) = N/(2N+1)  e r r o r b a r s shown,  20.2  also  (2.3)  F / k (shown i n F i g u r e 2 . 1 1 ) .  The s l o p e  a r e 1;  to  o f An h a v e b e e n c a l c u l a t e d  large scatter  i n the  temperature of  leads  l o g a r i t h m i c term have been used  l n [ (Ig/IoJ/Ag] v e r s u s  reason  the  the  g factors  h y p e r f i n e components  intensity  A«(R l i n e s ) A«(P l i n e s ) Averages of  state,  beam.  e q u a t i o n 2.2  0  intensities  lower  w r i t t e n as N r a t h e r t h a n J .  ln[(I«/I )/A ] The  of  z e r o and t h e  r o t a t i o n a l quantum number i s  After  state,  constant,  T = r o t a t i o n a l temperature of In  factor,  to  K f o r t h e NO2 beam u s e d  in  a rotational in  these  30  31  2.2.9  C a l c u l a t i o n of  Each N0 channels,  2  spectrum was  c o v e r i n g 1.5  r e c o r d e d ; the N0 the 150 MHz  Frequencies recorded as 4096 data p o i n t s , or  GHz.  Three separate s i g n a l s were  f l u o r e s c e n c e s i g n a l , the 750 MHz  2  markers, as was  shown i n F i g u r e 2.8a.  markers and The  markers have been d i g i t a l l y smoothed t o remove n o i s e  150  MHz  generated  by the j i t t e r of the l a s e r , and the peaks were then f i t t e d a Gaussian channel  to  f u n c t i o n t o o b t a i n the peak c e n t r e s i n terms of  numbers.  The p o s i t i o n s of the 750 MHz  markers were found  i n terms of channel  calibration  numbers u s i n g a  g r a p h i c s c u r s o r r o u t i n e on the MicroVAX computer. p o s i t i o n s were r e f i n e d w i t h a Gaussian  These  function f i t t i n g  program which obtained the b e s t peak p o s i t i o n i n f r a c t i o n a l channel  numbers.  A f i t t o the 750 MHz  because, r e g a r d l e s s of how w i t h the Fabry-Perot asymmetric.  markers was  w e l l the l a s e r beam was  required aligned  c a v i t y , the f r i n g e s were o f t e n s l i g h t l y  T h i s asymmetry made i t d i f f i c u l t t o judge the  c e n t r e p o s i t i o n of the marker a c c u r a t e l y by eye. were measured t o ±\ channel  The N0  f r a c t i o n a l 150  marker numbers (as were the p o s i t i o n s of the 750 MHz 2  peaks  u s i n g the same g r a p h i c s c u r s o r  r o u t i n e and then converted t o c o r r e s p o n d i n g  F i n a l l y , the N0  2  markers).  f r e q u e n c i e s were c a l c u l a t e d from the known  f r e q u e n c i e s o f the 750 MHz  markers.  MHz  32  2.3  Theory 2.3.1 N0  2  Background  i s a s t a b l e gaseous t r i a t o m i c f r e e r a d i c a l  t h a t i t has an odd number o f e l e c t r o n s ) . schematically  I t i s shown  i n F i g u r e 2.12 w i t h i t s i n e r t i a l  p o i n t group (x,y,z) axes l a b e l l e d .  (meaning  (a,b,c) and  I t i s c l a s s i f i e d as an  asymmetric t o p because i t s t h r e e p r i n c i p a l moments o f i n e r t i a , I , I and I , a r e a l l d i f f e r e n t . a  b  N0  c  belongs t o t h e  2  point  group, whose c h a r a c t e r t a b l e i s shown i n F i g u r e 2.13, and which has f o u r symmetry s p e c i e s , A  1#  A, B 2  and B .  lf  The  2  e l e c t r o n i c s t a t e s can be c l a s s i f i e d a c c o r d i n g  t o how t h e  e l e c t r o n i c wavefunction t r a n s f o r m s under t h e symmetry operations  of the  p o i n t group.  In t h e ground s t a t e o f N0 , 2  the 17 v a l e n c e e l e c t r o n s a r e i n t h e f o l l o w i n g o r b i t a l s [22] (3a ) (2b ) (4a ) (3b ) (lb ) (5a ) (la ) (4b ) (6a ) 2  2  1  2  2  1  2  2  2  1  2  2  1  2  2  2  which l e a d s t o an e l e c t r o n i c s t a t e w i t h symmetry A^ of t h e unpaired  1  1  The s p i n  e l e c t r o n (S=^) leads t o a s p i n m u l t i p l i c i t y  (2S+1) o f 2, so t h a t the ground s t a t e i s A . 2  t  The Walsh  diagram f o r XY -type molecules (Figure 2.14), which shows how 2  the e n e r g i e s  o f t h e o r b i t a l s change w i t h bond angle, i s  c o n s i s t e n t w i t h N0 being bent i n i t s ground s t a t e . 2  i s t h a t t h e unpaired  The reason  e l e c t r o n i n t h e ai=irn o r b i t a l has a much  lower energy i n t h e bent c o n f i g u r a t i o n , and outweighs t h e e f f e c t s of the four electrons i n the n  a  o r b i t a l which a r e more  33  z b  X c  F i g u r e 2.12  N0 w i t h i n e r t i a l axes shown.  (a,b,c) and p o i n t group  2  C  2v  A  l  A  2  B  l  B  2  E  C  2  o" (xz) v  o" ' (yz) v  1  1  1  1  1  1  -1  -1  1  -1  1  -1  1  -1  -1  1  F i g u r e 2.13  character table.  (x,y,z)  34  F i g u r e 2.14  Walsh diagram a p p r o p r i a t e t o N0 31].  2  [after  reference  35  stable i n the l i n e a r configuration.  In t h e v i s i b l e  region,  t h e r e a r e t h r e e p o s s i b l e e l e c t r o n i c s t a t e s which can be reached through a b s o r p t i o n 1st e x c i t e d s t a t e  . . . ( l a ) ( 4 b ) (6ai)  2nd  ... ( l a ) ( 4 b ) ( 6 a )  2  excited state  1  The i n N0  2  2  2  2  2  2  2  2  1  . . . (la ) (4b ) (2b )  f i r s t excited electronic state,  2  2  2  2  e l e c t r o n i c t r a n s i t i o n studied  2.14 by a d o t t e d  1  2  3rd e x c i t e d s t a t e The  of r a d i a t i o n :  2  1  A 2  1  2  Ba  2  Bi  i n t h i s work goes t o t h e  B , and i s shown i n F i g u r e 2  line.  zero n u c l e a r s p i n s o f t h e two e q u i v a l e n t  oxygen atoms  l e a d t o a l t e r n a t e r o t a t i o n a l l e v e l s b e i n g absent.  S p e c i f i c a l l y , t h e s i g n o f t h e t o t a l wavefunction must remain unchanged under r o t a t i o n by 180° around t h e symmetry  (z) a x i s  (z)  (C  2  ) as t h i s i s t h e 'equivalent  two oxygen n u c l e i .  r o t a t i o n ' t o exchanging t h e  T h i s means t h a t t h e product o f e l e c t r o n i c ,  v i b r a t i o n , r o t a t i o n and n u c l e a r s p i n wavefunctions must t r a n s f o r m as A  x  or A .  wavefUnction i s A  2  1;  S i n c e t h e ground s t a t e v i b r o n i c  t h e product o f t h e r o t a t i o n and n u c l e a r  s p i n wavefunctions must t r a n s f o r m as Aj o r A . 2  The n u c l e a r  s p i n wavefunctions a r e c l a s s i f i e d under a sub-group o f which i s C A B  and  E  2  l  l 1 - 1  t h e numbers o f A and B symmetry s p i n wavefunctions a r e  36  g i v e n by  [23] nAsy* =  (21+1) (1+1)  (2.5) n^ya = Since n  Bsym  exist.  (21+1)1  =0, o n l y r o t a t i o n a l l e v e l s w i t h Ai or A  T h i s i s shown i n F i g u r e 2.15  not e x i s t have been cross-hatched. a c c o r d i n g t o standard  symmetry  2  can  where the l e v e l s t h a t do The  l e v e l s are  asymmetric top n o t a t i o n ,  v  labelled , where N  i s the r o t a t i o n a l angular momentum quantum number and K  and \%.  a  are i t s p r o j e c t i o n s onto the i n e r t i a l a and c axes respectively. K  a  S i n c e N0  2  i s a near p r o l a t e asymmetric top,  and Kc n o t a t i o n w i l l be dropped i n favour of u s i n g K,  the  In the upper B 2  p r o j e c t i o n of N onto the a - ( p r o l a t e ) a x i s .  2  v i b r o n i c s t a t e , the same symmetry c o n d i t i o n s apply, the being that only B  x  2.3.2  and Rj r o t a t i o n a l l e v e l s  Molecular  result  exist.  Hamiltonian  To f i t s p e c t r o s c o p i c data, expressions  the  i t i s necessary  to obtain  f o r the energy l e v e l s of the system b e i n g  T h i s i s done by s o l v i n g the time-independent  studied.  Schrodinger  equation H* = E* u s i n g the a p p r o p r i a t e Hamiltonian functions  (*)  operator  (2.6) (H) and  t o s o l v e f o r the energy l e v e l s  (E).  basis For  the  37  3  3j2  3  U  3  13  //////  a  -  //////  2 a ////// 303  //////  '11  2 •02"  a  //////  l i o ////// In  l o i ••////// Ooo K=0  F i g u r e 2.15  K = 2  K = l  E x i s t i n g r o t a t i o n a l l e v e l s i n t h e ground e l e c t r o n i c s t a t e o f N0 . 2  38  systems s t u d i e d ,  t h e e f f e c t i v e Hamiltonian can be w r i t t e n H  where H  rot  eff -  H  rot +  + Hgj. + H  (2.7)  hfs  i s t h e r o t a t i o n a l Hamiltonian, H^ i s t h e r o t a t i o n a l  c e n t r i f u g a l d i s t o r t i o n term, H i n t e r a c t i o n and H  hfs  sr  i s the electron  i s t h e hyper f i n e  spin-rotation  interaction.  The r o t a t i o n a l Hamiltonian f o r an asymmetric t o p can be written  i n t h e form [24] H  rot  = AN  + BN + CN  2  X  where N , N and N a r e components X  Y  Z  (2.8)  Y  of the r o t a t i o n a l  angular  momentum o p e r a t o r (N) r e f e r r e d t o t h e m o l e c u l e - f i x e d a x i s system.  The r o t a t i o n a l c o n s t a n t s , A , B and C , a r e d e f i n e d i n  terms o f t h e p r i n c i p a l moments o f i n e r t i a as  A =  h 8 7 T  2  h  B = C l  A  8 7 T  2  C l  C = B  h  (cm ) -1  8 7 T  2  C l  c  The q u a r t i c c e n t r i f u g a l d i s t o r t i o n terms have been shown by Watson [25] t o be H  cdl  = - A ^ - A ^ ^ - A K N ^ - S H ^ ^  (2.9)  where t h e c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s a r e A„, A ^ , A , 5 K  and 6 . K  S e v e r a l h i g h e r order c e n t r i f u g a l d i s t o r t i o n terms  ( s e x t i c and o c t i c ) were a l s o i n c l u d e d , requirement  based on t h e i r  i n p r e v i o u s N0 work [26], and these a r e 2  H  39  ^  /V  H H, HKH,  fi  = HN  cd2  K  A 4A 9  + HKHN/N  Z  A 9<Vl  Afi  A o  + H N - LjN,  + H N K N . V  2  (2.10)  8  M  H N K , H ( s e x t i c ) and L x f o c t i c ) a r e t h e d i s t o r t i o n  K  K  constants.  The c e n t r i f u g a l d i s t o r t i o n terms a r e r e q u i r e d  s i n c e the r o t a t i o n a l Hamiltonian approximation  i s based on t h e r i g i d  rotor  which does not a l l o w t h e bonds o f a molecule t o  s t r e t c h while i t rotates. The  e l e c t r o n s p i n - r o t a t i o n Hamiltonian  may be w r i t t e n  [24] A  H  A A sr  A A  A A  A  A  A A  A  A  A  A  A  A  = -a (N S +N S +N S )-a(2N S -N S -N S )-b(N S -N S ) 0  x  x  y  z  y  z  2  x  z  x  y  y  x  x  y  (2.11)  y  where Van V l e c k ' s s p i n - r o t a t i o n parameters (a, a and b) [ 2 7 ] 0  are r e l a t e d t o t h e d i a g o n a l elements o f t h e s p i n - r o t a t i o n t e n s o r by 9o =  ( - 1 / 3 ) ( exx yy+ zz) +£  (-1/6) (2e  a = b  The  =  z z  -  £  e x x  -  £ y y  )  (2.12)  ( xx- yy) e  £  f i n a l term i s t h e h y p e r f i n e H a m i l t o n i a n  operator,  which has t h r e e terms a r i s i n g from i n t e r a c t i o n s o f t h e N 14  nuclear spin  (1=1),  [28]  xyz E TiiliSi + A  H  hfs  = a I«S + c  i  The  1 21(21-1)  xyz E xiili A  2  (2.13)  i  f i r s t term i s t h e Fermi c o n t a c t i n t e r a c t i o n , which  i n v o l v e s t h e o v e r l a p o f t h e n u c l e a r s p i n wavefunction that o f the electron spin.  with  The second term i s t h e e l e c t r o n  40  s p i n - n u c l e a r s p i n d i p o l a r i n t e r a c t i o n , which t r e a t s t h e two s p i n s as i n t e r a c t i n g b a r magnets.  The t h i r d term i s t h e  n u c l e a r quadrupole i n t e r a c t i o n which i s needed because t h e nucleus has a s p i n g r e a t e r than \.  T h i s term i s t h e  i n t e r a c t i o n o f t h e n u c l e a r quadrupole moment w i t h t h e e l e c t r i c f i e l d g r a d i e n t i n t h e molecule.  The c o n s t a n t s  involved i n the  hyperfine  i n t e r a c t i o n a r e a , t h e Fermi c o n t a c t i n t e r a c t i o n  constant,  T^, Tyy and T^, t h e magnetic d i p o l e - d i p o l e  c  i n t e r a c t i o n constants and  Xxx,  i n t h e p r i n c i p a l a x i s system; and X z z / X y y  t h e n u c l e a r quadrupole c o u p l i n g c o n s t a n t s ,  also i n the  p r i n c i p a l a x i s system. 2.3.3  Matrix  Elements  For N0 t h e most a p p r o p r i a t e 2  which t o c a l c u l a t e t h e matrix  set of basis functions i n  elements o f t h e Hamiltonian ( i n  o t h e r words, t h a t which g i v e s t h e s m a l l e s t o f f - d i a g o n a l elements) i s c a l l e d Hund's case ( b ) . w  T h i s corresponds t o t h e  p h y s i c a l s i t u a t i o n where t h e r e i s no f i r s t - o r d e r s p i n - o r b i t c o u p l i n g , and where t h e e l e c t r o n s p i n - r o t a t i o n i n t e r a c t i o n i s l a r g e r than t h e h y p e r f i n e e f f e c t s .  The v e c t o r c o u p l i n g scheme  is written N + S = J  J + I = F  (2.14)  f o r which a c o u p l i n g diagram i s g i v e n i n F i g u r e 2.16. The case ( b ) b a s i s f u n c t i o n s a r e w r i t t e n w  | NKSJIFM >, where K i s F  the p r o j e c t i o n o f t h e r o t a t i o n a l angular momentum N along t h e  41  F i g u r e 2.16  V e c t o r diagram o f Hund's c o u p l i n g case (b ) . BJ  42  near-symmetric t o p a x i s , J i s t h e v e c t o r sum o f N and S, and F i s t h e t o t a l angular momentum i n c l u d i n g n u c l e a r s p i n ; t h e s p a c e - f i x e d p r o j e c t i o n quantum number M i s not needed i n t h i s F  work s i n c e no e x t e r n a l f i e l d s a r e i n v o l v e d .  The quantum  numbers I and F a r e o n l y needed f o r t h e h y p e r f i n e  structure.  The m a t r i x elements o f t h e e f f e c t i v e H a m i l t o n i a n i n a case (b ) b a s i s a r e g i v e n below.  I n t h i s b a s i s t h e asymmetric  BJ  top r o t a t i o n a l H a m i l t o n i a n  (Equation 2.8) has m a t r i x elements  of t h e form AK=0,±2, AN=0.  The r o t a t i o n a l and e l e c t r o n  spin  f i n e s t r u c t u r e m a t r i x elements have been taken from r e f e r e n c e 26, w h i l e t h e h y p e r f i n e matrix elements have been taken from r e f e r e n c e 29.  In those cases where primes (') a r e used t o  i n d i c a t e a change i n quantum number, t h e s e l e c t i o n r u l e s from the 6-j and 9-j symbols a r e shown f o l l o w i n g t h e m a t r i x element. <NJSK | H +H +H | NJSK> = \ (B+C) N (N+l) + [A-^ (B+C) ] K rot  cd  2  sr  -  h[J(J+1)-N(N+1)-3/4] 3K  2  x  [ A  °  A  -  A K -A  +  H^K'N (N+1) + H  "  <NJSK±2 | Hrot+H^+Hsr | NJSK>=  4  K  LgK  ,,  K*  N(N+1)  {  N(N+1)  N(N+1)K -A N (N+1) +H K 2  H K  (2.15)  ]  2  2  K N (N+1) +H„N 2  N K  2  6  K  N  2  3  (N+1)  3  8  <*(B-C)  -*b[J(J+l)-N(N+1)  - 3/4]/[N(N+l) ]-6"„N(N+l)  (2.16)  - ^5 [K +(K±2) ] } [N(N+1)-K(K±1) ]* 2  2  K  X  [N(N+1)-(K±l)(K±2)]  43  <N-USK|H  <N-l  R O T  +H  C D  JSK±2|fi  f  <NKSJ'IF|H  F E R A L  +H  W T  S R  +H  |NJSK>  C D  +H  B P  =  (3a/2-^uI^) K C ^ - k ] */N  |NJSK>  = ±^b [ N ( N + l ) - K ( K ± l ) ] *  x  [ ( N T K - 1 ) (N=FK-2) ]*/N  |NKSJIF>  =  (-1)  x  N+S  * ' (-1) J  J + I + F + 1  (i  J  IJ  with A J = 0 , ± 1  [ ( 2 J • + 1 ) ( 2 J + l ) ]*  1  F !  i  (J  i j  s  J+I  x  [1(1+1)(2I+1)S(S+1)(2S+1)]*  E q  x x x  W i t h  quad  F  [(2J'+1)(2J+1)(2N'+1)(2N+1)]*  J  AJ=0,±1  *  X  fl J ' F7 (J I 1\  ,  l j  AF=0  dip  ,  (2.19)  \S J ' N I  <N'K'SJ'IF|H |NKSJIF> = - 7 3 0 gg„/x,/i«(-l)  <N'K SJ IF|H  (2.18)  [ S ( S + 1 ) ( 2 S + 1 ) I ( I + 1 ) (21+1) ]*  x  W i t h  (2.17)  2  AN=0,±1,±2  |NKSJIF> =  (N» N 2 - J S S 1 (J J 1 1  (-1) '" ' N  (2.20)  K  2 N\ l-K' q Kl  /N'  T  2 Q  (C)  AK=0,±2  (-1)  J+I+F  (-1) ' N  +s+J  [I (21-1) ]  [(1+1) [ (1+1) (21+1) (21+1)(21+3) (21+3)]*[( ]*[ (2J'+1) (2J+1) (2N'+1) (2N+1) ]* / \j  l J ' F\ I 2 J  J N J» S \ ( J N 2j  E (-1) '" ' q  /N' 2 N\ \-K' q Kj  B  AJ=0,±1  K  (2.21)  1  AN=0,±1,±2  T (VE) 2  q  AK=0,±2  In t h e above m a t r i x elements, s e v e r a l parameters have not been defined. correction  These a r e n, t h e l e a d i n g c e n t r i f u g a l to f  a  [26], and t h e d i p o l a r  distortion  and quadrupole c o u p l i n g  44  c o n s t a n t s which are r e l a t e d t o the p r e v i o u s c o n s t a n t s by gg M MNT (C) = *sT 2  N  c  0  B  gg Li MNT (C) = (1/724) 2  Nj  6  ±2  (2.22)  (T^-Tyy)  eQT (VE) = ^ x z z 2  0  eQT (VE) = (1/724) ( x x x " Xyy) 2  j2  S i n c e t h i s study o f N0 was undertaken as a t e s t of the 2  c a l i b r a t i o n and m o l e c u l a r beam systems, the amount o f data c o l l e c t e d was not e x t e n s i v e . the  following  manner.  T h e r e f o r e the data was f i t t e d i n  The r o t a t i o n a l and h y p e r f i n e c o n s t a n t s  determined by P e r r i n e t a l . [30] from a wide c o l l e c t i o n o f microwave and i n f r a r e d t r a n s i t i o n s , were used w i t h the above m a t r i x elements t o determine the ground s t a t e energy l e v e l s . These energy l e v e l s were then used w i t h the t r a n s i t i o n frequencies l i s t e d levels.  i n Appendix I t o o b t a i n upper s t a t e  energy  These energy l e v e l s were then f i t t e d u s i n g much  s i m p l i f i e d m a t r i x elements.  These m a t r i x elements c o n t a i n a  r o t a t i o n a l term, T , a Fermi c o n t a c t parameter, b,,, a d i p o l a r H  parameter, T^, and an e l e c t r o n i  = i  i ( xx e  + e  yy)=  2  (  a _ a  o)  •  T  n  e  spin-rotation  parameter,  m a t r i x elements a r e  <NKJF|H |NKJF> = T„ tenB  <NKJF | H  ferai  | NKJF> = [F ( F + l ) - J (J+l)-2 ] 4J(J+1) X [N(N+1)-J(J+1)-3/4]  (2.23)  (2.24)  45  <NK,J =J±l F|H ,  /  feimi  |NKJF> =  (2.25)  b^7(F+J+2)(J+l-F)(F+J-l)(F+2-J)(N+J+3/2)(J+^-N)(N+J-^)(N+3/2-J) 4j/4J -l 2  <NK, J=N+ ^ , F | H  dip  | NK, J=N+ \ , F> =  (2.26)  -^T [N +2N+ll/4-F(F+l) ] [SK^-NfN+l) ] 2  Z2  (N+l)(2N+1)2N+3) <NK, J=N- ^ , F | H  dip  | NK, J=N- \ , F> =  (2.27)  ^T [N +7/4-F(F+l) ] [3K -N(N+1) ] 2  2  22  N(2N-1)(2N+1) <NK, J=N- ^ , F | H  dip  | NK, J=N+ \ ,Y> =  (2.28)  -T [3K -N(N+1) ]7(N+F+5/2) (N-F+3/2) J(N+F-^) (F-N+3/2) 2  22  8N(N+1)(2N+1) <NK, J=N+^,F|H |NK, J=N+^,F> = -% N sr  (2.29)  7  H | NK, J=N- h ,F> = hi (N+l)  <NK, J=N-  (2.30)  sr  The r o t a t i o n a l term v a l u e s were then f u r t h e r  f i t t e d t o the  expression T„ = T + BN(N+1) - DN (N+1) 2  2  0  t o determine T and  0  (the band o r i g i n ) , B (the r o t a t i o n a l  (2.31) constant)  D (the c e n t r i f u g a l d i s t o r t i o n constant) f o r t h e upper  state.  46  2.3.4  S e l e c t i o n R u l e s and  Intensity  Calculations  Because more l i n e s appear i n the s p e c t r a than are expected from the s e l e c t i o n r u l e s f o r case (b ) w  was  necessary  coupling, i t  t o c a l c u l a t e the r e l a t i v e i n t e n s i t i e s of the  h y p e r f i n e components i n order t o a s s i g n the s p e c t r a . (b )  In case  c o u p l i n g the expected t r a n s i t i o n s f o l l o w the s e l e c t i o n  w  rules  [16] AN=0,±1, AJ=0,±1 (except AJ^O,  AN^O  f o r K=0);  AN=AJ;  AK=0,±1; AF=0,±1; AS=0 For t r a n s i t i o n s between K=0  s t a t e s these s e l e c t i o n r u l e s l e a d  t o R branches (AN=+1) and P branches (AN=-1). the  'allowed'  An example of  t r a n s i t i o n s i s shown i n F i g u r e 2.17,  s o l i d l i n e s i n d i c a t e the R(4) state N value).  The  F  and F  x  transitions 2  where the  (4 b e i n g the ground  n o t a t i o n r e f e r s t o the  two  e l e c t r o n s p i n components, J=N+^ and J=N-^, r e s p e c t i v e l y , of a r o t a t i o n a l l e v e l N.  The  t r a n s i t i o n s between F  t  ' e x t r a ' l i n e s are  and F  2  levels.  spin-forbidden  They break the AN=AJ  s e l e c t i o n r u l e s and are shown as d o t t e d l i n e s i n F i g u r e The  assignment of the r o t a t i o n a l quantum number N f o r  band 99 was  taken from p r e v i o u s work ( r e f e r e n c e s 17 and  w h i l e the assignment of band 115 99;  2.17.  18)  f o l l o w e d c l o s e l y t h a t of band  i n o r d e r t o a s s i g n the J and F quantum numbers, the  r e l a t i v e l i n e s t r e n g t h s of the h y p e r f i n e components had t o  be  47  F i g u r e 2.17  Shown are the observed allowed R(4) t r a n s i t i o n s ( s o l i d l i n e s ) and the f o r b i d d e n t r a n s i t i o n s (dotted l i n e s ) .  48  c a l c u l a t e d , as f o l l o w s .  The m a t r i x elements o f the e l e c t r i c  d i p o l e moment were c a l c u l a t e d u s i n g <N'J'SKIF |/i|NJSKIF> a (-1) " 1  J  +F  (-1)  [29]  r+J+3/2  (-1)"'~  K  x  [ (2F'+1) (2F+1) ]*[ (2J'+1) (2J+1) ]*  x  [(2N'+1)(2N+1)]  x  (j« F' [F J  l l l j  (2.32)  ( N ' J'  J  N  h I 1  1 0  / N'  I-K  1  N  1  K  These m a t r i x elements are e i t h e r zero o r v e r y s m a l l f o r AN=£AJ t r a n s i t i o n s but, because the e l e c t r o n s p i n and h y p e r f i n e s p l i t t i n g s i n t h e ground s t a t e are comparable  i n s i z e , the  m a t r i x elements o f the h y p e r f i n e H a m i l t o n i a n o f the type Aj=±l,AN=AF=0 cause a d e p a r t u r e from coupling.  'pure' case  (b ) SJ  The ground s t a t e wavefunctions become mixed  t h a t the J=N-^,F  2  such  l e v e l t a k e s on some o f the c h a r a c t e r o f the  J=N+^,Fj l e v e l o f the o t h e r f i n e s t r u c t u r e component. degree o f m i x i n g i n the ground s t a t e was  c a l c u l a t e d by  The setting  up 2x2 m a t r i c e s (H) i n the case (b ) b a s i s f o r the p a i r s o f BJ  i n t e r a c t i n g F l e v e l s , u s i n g the s p i n - r o t a t i o n , Fermi c o n t a c t and d i p o l a r m a t r i x elements  (equations 2.23-2.30) and  d i a g o n a l i z i n g t h e s e t o o b t a i n e i g e n v a l u e s (A) and e i g e n v e c t o r s (S) :  H S = A  where S =  cosfl sinfl  -sin* cos»  These e i g e n v e c t o r s were then used t o t a k e a p p r o p r i a t e combinations o f t h e b a s i s s t a t e s  (tf * ) t o form t h e 1  818  (2.33)  linear  49  eigenfunctions  (*  ei8en  ) cos« -sine  = The  matrix of  case  e q u a t i o n 2.32  (b^)  was assumed t h a t  s t a t e so t h a t line  S  strengths  _  = Su p p e r iM  2  the b a s i s  squares  the  i.e.  (2.35)  Sc  b  a unit matrix i n t h i s  are the  g i v e n by  basi .-l lower  no h y p e r f i n e m i x i n g o c c u r s is  upper  (2.34)  basis  i n t h e u p p e r and l o w e r s t a t e s ,  ^ It  1 T  t r a n s i t i o n moments g i v e n b y  was t h e n t r a n s f o r m e d t o  eigenfunctions  ptosis  sine costf  of  the  i n the  case.  elements of  upper  The r e l a t i v e ji  f i n a l  ,  and  it  <•>»/  was  found t h a t  intensity  as  t h e AN/AJ t r a n s i t i o n s  a result  s p i n components. II.  With the  possible  to  2.4  of the hyperfine mixing of  A sample c a l c u l a t i o n i s  a i d of these i n t e n s i t y  assign  hyperfine-induced  acquire considerable  the  'extra*  lines  the  electron  shown i n A p p e n d i x  calculations  it  was  with  the  check t h a t  the  associated  transitions.  Results The  main aim o f t h i s  a n a l y s i s was t o  c a l i b r a t i o n system worked and t o spectroscopic  data that  see  the  t h e m o l e c u l a r beam s y s t e m  produce.  A s m e n t i o n e d i n C h a p t e r 1,  hyperfine  splittings  the  could  ground s t a t e  measured from o u r d a t a a g r e e d v e r y  w i t h t h o s e o b t a i n e d from microwave d a t a . shown i n T a b l e 2 . 1 .  q u a l i t y of  The r e s u l t s  A comparison  are rather impressive,  well is with  50  T a b l e 2.1  Comparison o f microwave and o p t i c a l l y h y p e r f i n e s p l i t t i n g s i n N0 .  measured  2  N  U p p e r F  2  2*5  J  2  N  2k  2  3k  2  L o w e r F 2 ^  T h i s W o r k ( M H z )  J  M i c r o w a v e ( M H z )  3 1 . 9 6  lk 2k  111.12  D a t a  D i f f e r e n c e ( M H z )  3 2 . 6 9  a  + 0 . 7 3  1 7 7 . 6 5  a  - 0 . 0 7  2  lk  2k  2  2k  3k  5 2 . 1 9  5 2 . 5 4  3  + 0 . 3 5  2  lk  lk  2  lk  2k  4 0 . 3 2  4 0 . 3 4  3  + 0.  2  lk  2  lk  lk  2 4  2 4 . 4 4  a  - 0 . 4 4  2 2 1 . 1 6  2 2 0 . 8 8  a  - 0 . 2 8  6 0 . 8 6  b  - 0 . 3 6  0  a b  -  k  k  lk  0  k  k  . 8 8  02  6  6k  6 k  6  6k  5k  6 1 . 2 2  6  6 k  lk  6  6k  6k  1 2 7 . 8 9  1 2 8 .  6  5k  6k  6  6k  lk  1 8 3 . 6 5  1 8 2 . 8 3  b  - 0 . 8 2  6  5k  5k  6  5k  6k  6 4 . 5 2  6 3 . 3 4  b  - 1 . 1 8  -  R e f e r e n c e R e f e r e n c e  T a b l e 2.2  9 8  +  b  1.09  3 2 . 3 3 .  Comparison of a microwave and o p t i c a l l y measured r o t a t i o n a l i n t e r v a l i n N0 along w i t h a c a l c u l a t e d value. 2  U p p e r N F J  2  lk  2  2k  -  k lk  L o w e r N F J  T h i s W o r k ( M H z )  C a l c u l a t e d ( M H z )  M i c r o w a v e ( M H z )  D a t a  D ( m o m  i f i c p t e a  f e r o i c s u  r e n w a v a l l r e d  0  k  k  7 6 2 2 1 . 1 1 ± 2 . 4  7 6 2 1 6 . 5 6  7 6 2 1 6 . 3 3  - 4 . 7 8  0  k  lk  7 5 6 7 3 . 2 1 1 1 . 7  7 5 6 6 7 . 8 2  7 5 6 6 7 . 7 2  - 5 . 4 9  c e e y )  51  the average  d i f f e r e n c e between the o p t i c a l and microwave data  b e i n g j u s t g r e a t e r than 0.5 MHz. s p l i t t i n g between two shown i n T a b l e 2.2.  A comparison of the  r o t a t i o n a l l e v e l s was  a l s o made and i s  The agreement i s not as c l o s e although  the v a l u e s o b t a i n e d are w i t h i n 6 MHz,  which i s  remarkably  good.  These data a l s o served as a check of the ground s t a t e  energy  l e v e l c a l c u l a t i o n s , and one can see t h a t t h e r e i s  agreement t o w i t h i n a few t e n t h s of a MHz  between the  microwave data and our c a l c u l a t e d v a l u e s . The c o n s t a n t s o b t a i n e d from the upper s t a t e f i t s two  subbands are g i v e n i n Tables 2.3  a v a i l a b l e comparisons.  and 2.4,  f o r the  a l o n g w i t h the  The main c o n c l u s i o n t h a t can be drawn  from these r e s u l t s i s t h a t although the upper s t a t e o f N0  2  is  h i g h l y p e r t u r b e d , t h e r e i s enough r e g u l a r i t y f o r meaningful parameters t o be determined.  Looking a t the r e s u l t s o f band  99, where o t h e r data were a v a i l a b l e , one sees a f a i r l y c l o s e agreement between parameters. examined, one of them (N=5)  Of the f i v e r o t a t i o n a l  i s q u i t e b a d l y p e r t u r b e d , as shown  by i t s Fermi c o n t a c t and d i p o l a r c o u p l i n g c o n s t a n t s . 115 the N=3  levels  In band  r o t a t i o n a l l e v e l i s p e r t u r b e d ; i n f a c t the N=3  Fj  s p i n component i s doubled, w i t h the two p a r t s b r a c k e t i n g the F component i n energy.  The h y p e r f i n e s t r u c t u r e has c o l l a p s e d i n  both cases such t h a t the F = 2\, apparent  3\  s p l i t t i n g a t our r e s o l u t i o n  and Ah l e v e l s have no (<2-3  MHz).  The  c o n t a c t parameter i s the most h i g h l y a f f e c t e d a t t h i s p e r t u r b a t i o n ; the d i p o l a r c o u p l i n g c o n s t a n t shows  Fermi  2  T a b l e 2.3a  M o l e c u l a r c o n s t a n t s f o r upper s t a t e o f N0 99.  b N  T  1  1 3 5 7 9  (cm ) X  N  a 16850.319250(12) 16854.559929(10) 16862.124221(10) 16872.985741(10) 16887.051546(10)  T  N» 1 3 5 7 9 a b c d  -  ZZ  <  This Work  M H Z  -9.9(12) -11.7(31) 158.0(47) -28.8(71) -60.0(91)  T h i s Work  Previous  70.7(09) 70.8(17) 137.6(25) 61.2(46) 44.7(46)  66.5  b  b  57.0(52)° 37.9(38)  i  -11.2  Results  This Work  (MHz) Previous Results  1452.07(48) 1411.09(18) 1427.97(11) 1443.65(8) 1451.30(6)  .1409(420)*? 1469(180)° 1379(120) 1444.9° 1451.6  E r r o r l i m i t s a r e l a i n terms o f t h e l a s t d i g i t quoted, R e f e r e n c e 34. R e f e r e n c e 35. R e f e r e n c e 17.  T a b l e 2.3b  Upper s t a t e r o t a t i o n a l c o n s t a n t s f o r N0 , band 9 2  T (cm )  B (cm )  _ 1  16849.4775(65)  -1  * 0.42368(39)  a  P r e v i o u s R e s u l t s 16849.48 ,16849.8 b  D (cm )  X  0  T h i s Work  (MHz)  )  Previous Results  band  2/  C  (0.427,0.421)°,0.4224  a - E r r o r l i m i t s a r e l a i n terms o f t h e l a s t d i g i t quoted, b - Reference 18. c - Reference 15.  6.89(41)xl0~ b  3.5xl0~  5  b  5  53  T a b l e 2.4a  N» 1 3 3 5 7 9  T  M o l e c u l a r c o n s t a n t s f o r upper s t a t e o f N0 115.  (cm )  b^  -1  N  17092.895578(12) 17096.601492(11) 17097.295595(11) 17104.320836(10) 17115.056881(10) 17128.939538(10)  a  a - E r r o r l i m i t s a r e la  T a b l e 2.4b  55.33(96) -51.4(18) -50.4(19) 63.0(27) 89.6(36) 50.4 (92)  T  2 Z  (MHZ)  -13.4(13) -153.9(33) -151.9(35) 2.3(52) 60.3(70) -20.1(92)  i n terms o f l a s t d i g i t  band  7 (MHz) -613.15(49) 6640.64(18) -3763.52(18) 1643.47(11) 1654.74(8) 1307.93(7) quoted.  Upper s t a t e r o t a t i o n a l c o n s t a n t s f o r N0 , band 115. 2  T This  (MHz)  2/  Work  Previous Results  (cm" )  B (cm" )  1  Q  17092.181(62) 17092.3  b  1  0.410(32)  a  (0.434,0.394)  a - E r r o r l i m i t s a r e l a i n terms o f l a s t d i g i t b - Reference 15.  quoted.  b  54  i r r e g u l a r i t i e s f o r a l l t h e observed r o t a t i o n a l electron  spin-rotation  parameter i s most o b v i o u s l y p e r t u r b e d  f o r t h e N=l and N=3 l e v e l s . rotation  l e v e l s , and t h e  The n e g a t i v e e l e c t r o n  parameter f o r N=l i s a r e s u l t o f t h e F  above t h e F  t  2  spin-  level  lying  levels.  In c o n c l u s i o n , t h e use o f N0  2  as a t e s t molecule has shown  t h a t not o n l y does the c a l i b r a t i o n system work t o a h i g h degree o f accuracy, but a l s o t h a t t h e m o l e c u l a r beam apparatus i s capable o f producing v e r y h i g h r e s o l u t i o n  electronic  s p e c t r a which a l l o w one t o see v e r y f i n e d e t a i l s o f t h e energy l e v e l s o f t h e molecule.  55  Chapter 3 A p p l i c a t i o n to  3.l  the  Rotational  S t r u c t u r e of  the  B0  Molecule  2  Introduction  D e s p i t e our success a t making m o l e c u l a r beams of N0  2  it  has proved t o be v e r y d i f f i c u l t t o make m o l e c u l a r beams of u n s t a b l e t r a n s i t i o n m e t a l - c o n t a i n i n g r a d i c a l s such as T i N and NbN,  u s i n g a microwave d i s c h a r g e source, and under c o n d i t i o n s  t h a t normally l e d to strong laser-induced fluorescence spectra.  I t was  p o s s i b l e t o produce  T i N molecules  in a  d i s c h a r g e flame i n f r o n t of the skimmer, but the number d e n s i t y and l i f e t i m e of these molecules was  such t h a t v e r y  few  got through the skimmer and i n t o the d e t e c t i o n r e g i o n ; t h e r e f o r e f l u o r e s c e n c e s i g n a l s were not o b t a i n e d .  The more  s t a b l e molecule B0 , which can be made i n a d i s c h a r g e through 2  BC1  3  and 0 , was 2  chosen  f o r f u r t h e r study of the source  c h a r a c t e r i s t i c s and d i s c h a r g e c o n d i t i o n s necessary t o get molecules o f t h i s type i n t o a molecular beam. proven p o s s i b l e t o o b t a i n LIF s p e c t r a of B0  2  I t has  i n a beam, but  b e f o r e any h i g h r e s o l u t i o n work c o u l d be accomplished found necessary t o do a survey spectrum a n a l y s i s of the B0  2  and  indeed  i t was  rotational  v i s i b l e band systems by u s i n g a lower  r e s o l u t i o n technique. f r e q u e n c i e s of the B0  Not o n l y d i d t h i s study determine 2  the  t r a n s i t i o n s more a c c u r a t e l y , but i t a l s o  p i n p o i n t e d o t h e r areas of i n t e r e s t such as upper s t a t e  56  perturbations.  The a c q u i s i t i o n o f d a t a o v e r a l a r g e  number r e g i o n , test  of  the  The  some 300  2  t r i a t o m i c molecule exceptionally  laser  spectra  an a l m o s t - s t a b l e  w i t h an u n p a i r e d e l e c t r o n , ground" f o r the  work by J o h n s  et  al.  [39].  accuracy ± 0 . 0 2  analysis  cm"  have been t h e  al.  is  al.  work t h e  resolved  assignments  (0,0)  and 2,  transform state,  infrared  1  spectra  i n our t r a n s i t i o n s  again our data extend s t a t e v =l 2  recorded better  (estimated  sensitive  a much more  bands o f  [36];  (total  The i n f r a r e d d i o d e  X ^ g , where c o n s t a n t s  involved  early  times  as  the  complete  t h e A ^ and X ^ 2  l e v e l s were  [39]  we h a v e  band had  however and c a r r i e d  a n g u l a r momentum  laser  [37,38]  involved only the  f o r a l l the v =l 2  and  Two o f t h e  Fourier  ground  vibronic  have been d e t e r m i n e d ,  to higher J .  2  The (0,0)  a l l t h e band head t r a n s i t i o n s  out t o h i g h e r J  quantum n u m b e r ) .  as  diode  done.  b e e n p r e v i o u s l y a n a l y z e d by J o h n s  the  100  with grating spectra  t r a n s i t i o n n e a r 5400 A h a v e b e e n m e a s u r e d .  completely  original  i n the  spectra  about  results,  such  infrared  [37,38]  and a l t h o u g h n o t  o r Maki e t  has been this  achieve  [36]),  1  al.  to  an  theory of  i n 1961,  Our r e s o l u t i o n  linear  is  Fourier transform infrared  t h a n J o h n s was a b l e  Kawaguchi e t  [36]  t a k e n by K a w a g u c h i e t  1 9 8 0 ' s and r e c e n t  In  which i s  The o n l y r o t a t i o n a l a n a l y s e s  photolysis  Maki  system.  good " t e s t i n g  flash  by  s u p p l i e d a f u r t h e r demanding  1  calibration  B0 m o l e c u l e ,  systems.  cm" , h a s  wave  levels  although  four upper  a n a l y z e d by Johns b u t t h e  other  two  57  levels  3.2  have been a n a l y z e d f o r the  Experimental 3.2.1  in Ar carrier  3.1.  free  2  r a d i c a l s had t o  gas,  fluorescence  w i t h B0  experiment,  cell  the  2  (the  fluorescence molecules  chemiluminescence.  18390 cm" a n d 18444 cm" t o  3.2.2  1  Intermodulated  C h a p t e r 2,  BO2 was t h e  first  shown  green  During the running  intensity.  to The  f l o w o f BOj  beams  from  the  2  over the  in  a  l a s e r beams a n d t h e BO2  r a n g e 18215  18500 cm" u s i n g t h e 1  cm"  1  laser  to dye  560.  was d e s c r i b e d . of  a  f o r N 0 , a n d was m o n i t o r e d u s i n g  r i g h t angles to both the  D a t a were c o l l e c t e d  rhodamine  into  p r e s s u r e was k e p t n e a r 700 m T o r r  system used  1  3  a characteristic pale  w i t h two c o u n t e r - p r o p a g a t i n g l a s e r  PMT p l a c e d a t  B C 1 and  as  s p e c t r u m was p r o d u c e d b y c r o s s i n g t h e  same d y e l a s e r  In  actively  'cube'),  p r o d u c e maximum l a s e r - i n d u c e d f l u o r e s c e n c e  flow.  be  t h r o u g h a microwave d i s c h a r g e  The d i s c h a r g e e m i t t e d  glow a s s o c i a t e d the  B0  T h i s was done b y f l o w i n g s m a l l amounts o f  cube-shaped metal  of  work.  2  produced.  Figure  in this  B0 P r o d u c t i o n  2  2  time  Details  U n l i k e N0 , the  0 ,  first  the  Fluorescence  theory of  T h e method o f technique  of  laser-induced  collecting  fluorescence  sub-Doppler spectra  intermodulated fluorescence  d e s c r i b e d b y Sorem and Schawlow  [40].  Figure  3.1  (IMF), shows  pump  PMT  lock-in  HeNe  laser  MicroVAX computer  Ar* i o n  Figure  3.1  to  laser  Block  ring  diagram  showing  laser  intermodulated  BS  calibration  ~7  fluorescence  m  experiment. 00  59  the flow o f B0  2  molecules c r o s s e d by two  l a s e r beams,  one  modulated a t frequency f , and the o t h e r , t r a v e l l i n g i n the t  o p p o s i t e d i r e c t i o n through the sample, modulated a t f . 2  As i n Chapter  molecules  2, the two  l a s e r beams i n t e r a c t  t h a t have o p p o s i t e Doppler  the a x i s of the l a s e r beam. modulated a t e i t h e r  f  t  or f  l i n e p r o f i l e i s obtained.  2  with  s h i f t s with respect to  I f the f l u o r e s c e n c e i n t e n s i t y i s recorded, a  Doppler-broadened  However i f the f l u o r e s c e n c e  i n t e n s i t y modulated a t frequency  (f!+f ) i s recorded,  s i g n a l o b t a i n e d i s a sub-Doppler  IMF  molecules  2  the  s i g n a l because the  i n t e r a c t s i m u l t a n e o u s l y w i t h the f i e l d s of both  l a s e r beams.  This signal  i s a non-linear saturation effect,  and t h e r e i s a l s o a component modulated a t f i ~ f advantage of u s i n g the sum  frequency f + f t  the low frequency amplitude collected  frequency  f o r B0  2  3.2.3 The B0  had l i n e w i d t h s of approximately  spectrum  spaced by 750 MHz 2  100  IMF  data  MHz,  and t h e r e f o r e no h y p e r f i n e  was  of Frequencies scanned i n 3 0 GHz  an example i s shown i n F i g u r e 3.2.  w i t h the B0  The  resolved.  Calculation  2  [19]; the  i s t h a t i t avoids  n o i s e of the l a s e r .  mainly due t o p r e s s u r e broadening, s t r u c t u r e was  2  2  and 150 MHz  i n t e r v a l s , of which  I n t e r f e r o m e t e r markers  were recorded s i m u l t a n e o u s l y  fluorescence signal.  The  150 MHz  markers were  used s o l e l y t o check f o r l a s e r mode hops, which show up i r r e g u l a r i t i e s i n the marker s p a c i n g .  as  Because of the l a r g e  60  Figure  3.2  A s a m p l e 30 GHz B 0 s c a n . T h e 150 MHz m a r k e r s a r e n o t shown due t o t h e i r h i g h d e n s i t y . The a s s i g n e d t r a n s i t i o n s have been l a b e l l e d w i t h t h e v i b r o n i c symmetry , a s u p e r s c r i p t i n d i c a t i n g t h e b o r o n i s o t o p e , and a b r a n c h l a b e l . The b r a n c h l a b e l i s R o r P (AJ=+1 o r A J = - 1 ) w i t h t h e s p i n component as a s u b s c r i p t when r e q u i r e d and t h e a s s o c i a t e d J " o r N" v a l u e i n p a r e n t h e s e s . 2  61  number o f 750 MHz markers, wave numbers from t h e B u r l e i g h wavemeter were o n l y recorded f o r a few markers near t h e b e g i n n i n g and end o f each scan.  The order numbers o f these  markers were then c a l c u l a t e d u s i n g a rearranged v e r s i o n of e q u a t i o n 1.1, a l o n g w i t h t h e wave number o f one marker near the b e g i n n i n g o f t h e scan.  The data channel numbers o f t h e  750 MHz markers were found by t h e computer, which a l s o c a l c u l a t e d t h e i r wave numbers.  These wave numbers were then  f i t t e d t o a seventh degree polynomial i n o r d e r t o determine e q u a t i o n f o r t h e frequency i n terms o f channel number.  an  The  channel numbers o f t h e B0 t r a n s i t i o n s were found u s i n g t h e 2  c u r s o r method (see S e c t i o n 2.2.9) and t h e p o l y n o m i a l was then used t o c o n v e r t these channel p o s i t i o n s i n t o wave numbers. 3.3  Theory 3.3.1 B0  2  Background  i s a l i n e a r symmetric t r i a t o m i c molecule b e l o n g i n g t o  the p o i n t group D^; t h e c h a r a c t e r t a b l e i s g i v e n i n F i g u r e 3.3.  I t has 21 e l e c t r o n s , o f which t h e o u t e r e l e v e n occupy  the f o l l o w i n g o r b i t a l s i n t h e X ^ ground s t a t e 2  ...K) (a ) (7r )V ) 2  2  u  u  In  the f i r s t  to  7r , g i v i n g g  g  3  ...x iy 2  e x c i t e d s t a t e , A!!,,, an e l e c t r o n has moved from ir„ 2  D , ooh  2  +  9+  2 2u g_ 2 nu ng A A  u g U  I  20* oo  2  C  ^ OO  a  h  coC  2  2S <° 2  000  V  2 S  ™ co  i  00  +1  +1  +1  +1  +1  +1  +1  +1  +1  +1  +1  +1  -1  +1  +1  +1  +1 -1  -1  +1  -1 -1  +1  -1 +1  -1 +1  +1  +1  +1  -1  +1  -1  -1  -1  -1  +2  2cos«3  2cos2p  -2  0  0  -2cos^  -2cos2p  +2  +2  2cos^>  2cos2p  +2  0  0  +2cosp  +2cos2y>  -2  +2  2cos2p 2cos4io  +2  0  0  +2cosp  +2cos4*3  +2  +2  2cos2»? 2cos4p  -2  0  0  -2cos2y? -2cos4p  -2  F i g u r e 3.3  The D,^ p o i n t group c h a r a c t e r t a b l e .  63  ... ( o - ) ( a ) ( 7 r ) ( 7 r ) 2  B  The and  'u' and 1  3  2  u  u  3  B  ...A n 2  u  'g' s u b s c r i p t s are a b b r e v i a t i o n s f o r  'ungerade'  g e r a d e , which mean antisymmetric and symmetric w i t h 1  Rough contour diagrams o f the ir  respect to inversion.  a  o r b i t a l s a r e shown i n F i g u r e  stretch, v  g  3.4.  The f o u r v i b r a t i o n a l degrees o f freedom i n B0 symmetric  and ir  are the  2  the antisymmetric s t r e t c h , v , and the  w  3  doubly degenerate bending motion, v ; the s t r e t c h i n g modes are 2  not d e a l t w i t h i n t h i s a n a l y s i s .  When a l i n e a r t r i a t o m i c  molecule bends, the v i b r a t i o n a l degeneracy i s l i f t e d as the D«,  h  symmetry i s broken.  I f t h e r e i s no o r b i t a l degeneracy the two  components o f the bending v i b r a t i o n become the a - a x i s r o t a t i o n and the non-degenerate  bending motion o f the bent molecule.  However i f the e l e c t r o n i c s t a t e i s o r b i t a l l y degenerate the same must occur f o r both o r b i t a l components; the r e s u l t i s t h a t t h e r e a r e two e l e c t r o n i c p o t e n t i a l c u r v e s , one f o r each o r b i t a l component, which touch a t the l i n e a r  configuration,  but which are otherwise s e p a r a t e , as shown i n F i g u r e These two e l e c t r o n i c p o t e n t i a l curves r e p r e s e n t two  3.5. different  e l e c t r o n i c s t a t e s o f the bent molecule, which l i e v e r y c l o s e i n energy and i n t e r a c t s t r o n g l y through the a - a x i s r o t a t i o n . T h i s c o n s t i t u t e s a major breakdown of the Born-Oppenheimer s e p a r a t i o n o f e l e c t r o n and n u c l e a r motion, and i t s r e s u l t s are l o o s e l y c a l l e d the R e n n e r - T e l l e r e f f e c t [41].  Figure  3.4  Rough  contour  diagrams  of  n  g  and n  u  orbitals.  65  Figure  3.5  The p o t e n t i a l energy curves s e p a r a t i n g as a l i n e a r molecule bends.  66  Figure of  B0 .  3.5  represents  The u p p e r c u r v e h a s A  2  notation)  corresponding to  (a ) (b ) (b ) (a ) (a ) (b ) , 2  2  1  2  2  point  2  1  2  2  The l o w e r  2  2  2  2  1  2  2  The d i s c u s s i o n given  here  small, in  so  of  the  assumes t h a t that  F i g u r e 3.5  the  the  electron  configuration  ^  symmetry,  amplitude  splitting  of  the as  u  of  motions, 90°  one  a r i s i n g from t h e i n the  xz p l a n e  phase d i f f e r e n c e ,  a n g u l a r momentum i s internuclear  axis  as  is  bending motion  model  bending motion of  addition of  and one  the  i n the  by 6 and i t s  c a l l e d I.  [41].  a n g u l a r momentum,  The p r o j e c t i o n o f  L onto the  a n d a new quantum number K=A+I i s vibronic If  one  is,  includes  momentum, levels  (that  A+Z+Z,  for a ^  the is  first  electronic electron called  electronic  P.  orthogonal  projection  and v i b r a t i o n a l )  onto  with  discussed  total  with  is  the  the  by Renner  axis label  is  A,  the  energy  levels.  internal  angular  The p a t t e r n o f v i b r o n i c  state  a  electronic  internuclear  the  be  The v i b r a t i o n a l  introduced to  spin,  linear  yz p l a n e ,  In a degenerate  L , as  shown  bending  a  two  is  the  which can  s t a t e t h i s v i b r a t i o n a l a n g u l a r momentum c o u p l e s electronic  curves  for the  shown i n F i g u r e 3 . 6 .  denoted  effects  a perturbation of  a v i b r a t i o n a l a n g u l a r momentum, as  with the  the  two p o t e n t i a l  harmonic o s c i l l a t o r  to  a  molecule  of  corresponding  w i t h Z *.  Associated  thought  i n t h e LV  o r b i t a l a n g u l a r momentum  the  state  symmetry  with  vibration. is  ground  (in  c a n be c o n s i d e r e d  doubly-degenerate  for the  symmetry  and c o r r e l a t e s  2  1  2  curve has  (a ) (b ) (b ) (a ) (a ) (b ), 2  situation  and c o r r e l a t e s  2  1  group.  1  the  shown i n F i g u r e 3 . 7 .  energy The  67  F i g u r e 3.6  The degenerate bending motion o f B0 . A 90° outof-phase a d d i t i o n causes t h i s t o l o o k l i k e v i b r a t i o n a l a n g u l a r momentum. 2  68  F i g u r e 3.7  The p a t t e r n o f v i b r o n i c energy l e v e l s formed when the bending v i b r a t i o n o f a l i n e a r symmetric t r i a t o m i c molecule i n a n e l e c t r o n i c s t a t e i s e x c i t e d [36]. 2  z  69  n o t a t i o n used f o r l a b e l l i n g the v i b r o n i c l e v e l s i s V i V v 2  the v i b r o n i c symmetry ^Kp, i n parentheses. 010 ( E ~) .  The  2  U  and  Therefore  has  l e v e l s such as 000( n 2  The  ) or  i . e . the h i g h e r  are  l e v e l l i e s below  2  level,  3/2g  f i r s t excited electronic states  i n v e r t e d s p i n - o r b i t s t a t e s such t h a t the n the ^  with  the i n v e r s i o n symmetry, g or u,  one  ground and  3  3/2  P s t a t e s l i e lower i n energy.  t r a n s i t i o n s which have been s t u d i e d i n t h i s t h e s i s are  shown i n F i g u r e 3.8; Appendix I I I .  t h e i r l i n e frequencies  The 2~ - S 2  2  are l i s t e d i n  t r a n s i t i o n i s forbidden  +  i n pure  case (b) c o u p l i n g , but the s p i n - o r b i t i n t e r a c t i o n i n the A n 2  s t a t e of B0  2  mixes the v i b r o n i c l e v e l s a s s o c i a t e d w i t h the  p o t e n t i a l s i n F i g u r e 3.5,  w i t h the r e s u l t t h a t the two  2  as n=h  coupling;  - 010 ( S ) t r a n s i t i o n  2  takes on some of the c h a r a c t e r of a h-h coupling,  s t a t e s i n case 2  two  010( S)  s t a t e s are more a c c u r a t e l y d e s c r i b e d i n consequence the 010 ( 2~)  u  (c)  +  t r a n s i t i o n i n case  (c)  so t h a t i t becomes weakly allowed.  3.3.2  Hund's Coupling  Cases  F i v e p o s s i b l e ways i n which the angular momenta of a molecule may represent  be coupled have been d e s c r i b e d by Hund.  These  f i v e c h o i c e s o f b a s i s f u n c t i o n s f o r the c a l c u l a t i o n  of m o l e c u l a r energy l e v e l s , and cases (a) t o  (e) .  The  B0  2  are now  c a l l e d Hund's c o u p l i n g  spectra reported  i n t h i s t h e s i s give  examples o f cases (a) t o ( c ) . Vector Figure  3.9.  diagrams f o r case (a) c o u p l i n g are shown i n I f the v i b r a t i o n a l angular momentum i s equal t o  Figure  3.8  Energy l e v e l studied.  diagram  showing  the  transitions  71  F i g u r e 3.9  V e c t o r diagram of Hund's c o u p l i n g case a) G = 0, b) 6^0.  (a) w i t h  72  zero,  as i n t h e 000( n)-000( n) band, F i g u r e 2  2  3.9a a p p l i e s .  The  e l e c t r o n o r b i t a l and s p i n angular momenta, L and S, a r e s t r o n g l y coupled t o t h e i n t e r n u c l e a r a x i s and then f u r t h e r coupled t o t h e r o t a t i o n a l angular momentum, R, t o form t h e t o t a l a n g u l a r momentum, J . J  The  Thus case (a) can be regarded as (3.1)  = L + S + R  sum o f t h e p r o j e c t i o n s o f L and S i s A+S=n, t h e p r o j e c t i o n  o f J onto t h e i n t e r n u c l e a r a x i s .  In F i g u r e  3.9b, t h e  v i b r a t i o n a l angular momentum i s not zero ( i . e . when t h e bending v i b r a t i o n i s e x c i t e d )  and t h e o r b i t a l , v i b r a t i o n a l and  e l e c t r o n s p i n angular momenta a r e a l l coupled t o t h e internuclear axis.  The p r o j e c t i o n quantum numbers K and P  (described previously)  a r e shown.  As i n F i g u r e  3.9a, these  a n g u l a r momenta couple w i t h t h e r o t a t i o n t o g i v e t h e t o t a l a n g u l a r momentum, J , i n t h i s case w r i t t e n as J = L + G + S + R  Both e q u a t i o n s 3.1 and 3.2 r e p r e s e n t the c o u p l i n g  case (a) c o u p l i n g ,  o f t h e angular momenta t o t h e i n t e r n u c l e a r  t a k e s precedence over any c o u p l i n g s themselves.  (3.2)  The b a s i s f u n c t i o n s  where axis  between t h e momenta  f o r case (a) can be w r i t t e n  |nA;SS;GZ;JP> where n i s t h e e l e c t r o n i c s t a t e . In Hund's c o u p l i n g  case (b), t h e e l e c t r o n s p i n angular  momentum i s o n l y weakly coupled t o t h e i n t e r n u c l e a r a x i s such t h a t i t s p r o j e c t i o n can no longer be d e f i n e d .  A vector  73  diagram i s shown i n F i g u r e  3.10 and t h e c o u p l i n g may be  written L  The  + G + R = N  b a s i s f u n c t i o n s can be w r i t t e n  t o S e l e c t r o n i c s t a t e s where t h e r e  N + S =  (3.3)  J  |t?;NKSJ>.  Case (b) a p p l i e s  i s no o r b i t a l  angular  momentum t o couple t h e s p i n t o t h e a x i s , o r i n t h e case o f B0 , 2  o f L and G i s such t h a t a S v i b r o n i c s t a t e  when t h e c o u p l i n g arises. case  However, case  (b) a l s o can a r i s e a t l a r g e J v a l u e s i n  (a) s t a t e s , as t h e e l e c t r o n s p i n uncouples from t h e  i n t e r n u c l e a r axis with increasing r o t a t i o n . Hund's case coupling  (c) c o u p l i n g  a r i s e s when t h e s p i n - o r b i t  i s so l a r g e t h a t t h e o r b i t a l , v i b r a t i o n a l and  e l e c t r o n s p i n a n g u l a r momenta couple t o g e t h e r t o form what w i l l be termed J  a  as shown i n F i g u r e  3.11.  This  intermediate  a n g u l a r momentum then couples w i t h t h e r o t a t i o n a l a n g u l a r momentum t o g i v e t h e t o t a l angular momentum, J .  This i s  w r i t t e n as L  One can no l o n g e r  + G + S = J  strictly  a  J  a  + R = J  d e f i n e t h e p r o j e c t i o n s A, I  however t h e t o t a l angular momentum s t i l l i n case  (a) c o u p l i n g .  (3.4)  o r E;  has a p r o j e c t i o n P as  74  Figure  3.10  V e c t o r diagram o f Hund's c o u p l i n g case ( b ) .  75  F i g u r e 3.11  V e c t o r diagram of Hund's c o u p l i n g case ( c ) .  76  3.3.3  B0  2  R o t a t i o n a l L e v e l s and S e l e c t i o n  i s s i m i l a r t o N0  2/  Rules  having two e q u i v a l e n t  oxygen atoms  w i t h zero n u c l e a r s p i n which l e a d t o t h e absence o f a l t e r n a t e rotational levels.  For the 2 v i b r o n i c l e v e l s t h i s  implies  t h a t o n l y odd o r even v a l u e s o f N e x i s t ; these a r e shown i n Figure  3.12 f o r t h e v =l l e v e l s o f t h e ground s t a t e . 2  The  o p p o s i t e occurs i n t h e upper s t a t e because t h e symmetry w i t h respect the  t o i n v e r s i o n i s g r a t h e r than u.  The F i / F n o t a t i o n i s 2  same as t h a t used f o r N0 w h i l e t h e e / f n o t a t i o n 2  labels  l e v e l s o f even and odd p a r i t y f o l l o w i n g t h e c o n v e n t i o n introduced  by Brown e t a l . [42].  This notation  because i t l e a d s t o s t r a i g h t f o r w a r d  i s useful  s e l e c t i o n r u l e s , and t h e  H a m i l t o n i a n m a t r i x f a c t o r s i n t o two submatrices which a r e i d e n t i f i a b l e w i t h t h e e and f l e v e l s .  The s e l e c t i o n r u l e s f o r  e l e c t r i c d i p o l e t r a n s i t i o n s a r e [42] AJ=0, e-H-f AJ=±1, e«e, whereas p e r t u r b a t i o n s  (3.5)  f«f  follow AJ=0, e<+e, f ~ f  For n and A v i b r o n i c l e v e l s , i t i s found t h a t lambda  (3.6) alternate  (A)-doubling components a r e m i s s i n g , such t h a t only one  l e v e l e x i s t s f o r each J v a l u e .  A-doubling i s a f u r t h e r  breakdown o f t h e Born-Oppenheimer approximation; i t a r i s e s when r o t a t i o n does not a l l o w t h e o r b i t a l angular momentum t o  77  N .it/a  F  <e)  t  F 7/a B/2  S/2 1/2  2v  3  ( f )  Fl Fa  (e) (f)  Fi F  (e)  (f)  2  -t>/2  F  x  •7/2  F  2  6/2  F  (  f  )  (e)  t  8/2  F  2  1/2  F  t  (f) (e)  (  f  "  F i g u r e 3.12  R o t a t i o n a l l e v e l s which e x i s t i n the ground S vibronic levels.  state  )  78  be  s t r i c t l y q u a n t i z e d along t h e i n t e r n u c l e a r  a x i s , so t h a t t h e  +A and -A components o f a degenerate s t a t e have s l i g h t l y d i f f e r e n t energies.  Each J l e v e l t h e r e f o r e s p l i t s i n t o two A-  components, one o f which i s m i s s i n g because o f t h e zero s p i n s of t h e e q u i v a l e n t oxygen atoms.  The r o t a t i o n a l l e v e l s which  e x i s t f o r t h e ground s t a t e a r e shown i n F i g u r e 3.13. The absence o f a l t e r n a t e the  A-components l e a d s t o a " s t a g g e r i n g " i n  appearance o f some o f t h e branches i n t h e B0 spectrum. 2  3.3.4  Case  (a)  H a m i l t o n i a n and M a t r i x E l e m e n t s  For t h e ground s t a t e A v i b r o n i c vibronic  l e v e l s case (a) b a s i s  l e v e l s and a l l t h e n  f u n c t i o n s were used, w i t h t h e  e f f e c t i v e Hamiltonian operator being A  A  H ff  =  e  A  Hgo + H  rot  A  +  HCD  A  +  HCJKJ)  In t h i s e q u a t i o n Hgo i s t h e s p i n - o r b i t r o t a t i o n a l Hamiltonian, correction  LD  A  + H  (3.7)  LDCD  interaction, H  i s the c e n t r i f u g a l  rot  i s the  distortion  t o t h e r o t a t i o n a l Hamiltonian, Hs^ i s t h e  centrifugal distortion correction A  interaction, H  t o the spin-orbit  ,  LD  A  i s t h e A-doubling o p e r a t o r and H  centrifugal distortion The  A  + H  spin-orbit  i s its  correction.  i n t e r a c t i o n , H^ = AL-S, Ha  where t h e c o n t r i b u t i o n s  LDCD  = AL S Z  can be taken as  2  (3.8)  from t h e x and y components o f L a r e  dropped as they a r e o f f d i a g o n a l i n e l e c t r o n i c s t a t e .  The  79  J  J 11/2  9/2 _  (e)  7/2  (f)  5/2 3/2 1/2  —(e)  (f)  9/2  (e)  7/2  —————(f)  5/2  (•)  3/2  (f)  13/2  1 (  e  1  /  (  1  /  ">  2  )  <> f  2  9/2  (  e  7  /  (e) (f)  2  *'*  )  C> (f)  3/2  7/2  _ _ _ _ _ (  5/2  (  2 *S/2 Figure  3.13  f  e  )  (e)  9/2 1  f  )  )  2 A  3/2  R o t a t i o n a l l e v e l s which e x i s t i n t h e ground n and A v i b r o n i c l e v e l s . The d a s h e d l i n e s represent the missing l e v e l s .  state  80  s p i n - o r b i t c o n s t a n t i s A. The  r o t a t i o n a l term i s A  H  An  A  A  A  A  o  = B R = B(J-L-G-S)  W T  where B i s t h e r o t a t i o n a l c o n s t a n t .  (3.9)  A f t e r some a l g e b r a , t h e  terms t h a t a r e d i a g o n a l i n A and t h e v i b r a t i o n a l quantum numbers may be w r i t t e n A  /N O  H  A O  A9  A O  A A  = B(J -J +S -3 ) 2  2  2  2  Z  Z  rot  A /\  - B(J S.+J.S ) +  +  (3.10)  The c e n t r i f u g a l d i s t o r t i o n H a m i l t o n i a n i s A  A  A  A  A  A  Ho = -D(J-L-G-S)  (3.11)  where t h e m a t r i x r e p r e s e n t a t i o n o f t h e a n g u l a r momentum term i s taken as t h e square o f t h e c o e f f i c i e n t m a t r i x f o r B from e q u a t i o n 3.10. The c e n t r i f u g a l d i s t o r t i o n c o r r e c t i o n t o A, d e s c r i b e d w i t h a parameter A , a r i s e s because t h e s p i n - o r b i t c o u p l i n g i s D  a f u n c t i o n o f t h e bond l e n g t h ; t h i s c o r r e c t i o n i s  HSOCD  The  = ^A [ ( J - L - G - S ) , L S ] 2  D  Z  Z  +  (3.12)  square b r a c k e t s and p l u s s i g n i n d i c a t e t h a t t h e  anticommutator,  [a,b]  +  = ab+ba, must be taken i n o r d e r f o r t h e  H a m i l t o n i a n m a t r i x t o remain The  Hermitian.  A-doubling i n a n s t a t e can be t r e a t e d by assuming 2  t h e r e i s an e f f e c t i v e o p e r a t o r  81  HLD =  a c t i n g between the  -h (p+2q) (J+S_+J_S+) +  and  |A=+I>  d e t e r m i n a b l e c o n s t a n t s i n case  ^q(J?+J- ) 2  (3.13)  |A=-l> components.  The  (a) are q and p+2q.  The c e n t r i f u g a l d i s t o r t i o n c o r r e c t i o n t o the A - d o u b l i n g i s determined  i n the same manner as the c o r r e c t i o n t o the  s p i n - o r b i t c o u p l i n g , t h a t i s by t a k i n g the anticommutator  of  the A - d o u b l i n g o p e r a t o r w i t h the r o t a t i o n a l o p e r a t o r , such that  HLDCD =  -  hqnt (J+2+J.2) , (J-L-G-S) 2 ]+ A  A A  A  A  A  (3.14) A  o  A  h (p +2qD) [ (J S.+J.S ) , (J-L-G-S) ] 2  D  +  +  +  The same combination o f c o n s t a n t s i s d e t e r m i n a b l e :  qj) and  p +2q . D  D  The H a m i l t o n i a n m a t r i c e s f o r the n and A v i b r o n i c 2  2  are shown i n T a b l e s 3.1 and 3.2 r e s p e c t i v e l y . f u n c t i o n s , which are l i n e a r combinations number b a s i s f u n c t i o n s |A 2 J P>,  The  states  basis  o f the s i g n e d quantum  have been w r i t t e n i n  shorthand n o t a t i o n , they a r e :  e/f> = | n e/f> = e/f> = e/f> = 2  3/2  (1/72)( |1 (1/72){ |1 (1/72) { |1 (1/72) { |1  j  -k r  *2  -k k  j j J  k>  +  3/2>  +  3/2>  +  5/2>  +  I"! I"! I"! I"!  The e and f p a r i t y l a b e l s can be dropped s i n c e no A-doubling i s r e s o l v e d .  k l ^2  k -k  J J J J  (3.15)  -k>) -3/2>  }  -3/2>  }  -5/2>  }  from the A f u n c t i o n s 2  82  Table 3.1  The H a m i l t o n i a n m a t r i x f o r the 000 (^J v i b r o n i c s t a t e s o f B0 . The ± r e f e r t o t h e e and f l e v e l s respectively. The term v a l u e , T , r e p r e s e n t s the o r i g i n o f the v i b r o n i c s t a t e 2  0  iXa  e/f>  | X e/f>  T +*A+B[(J+J ) -21  - D[((J+^) -2r+(J+^) -l] (  + T  -B[ ( J + ^ ) - l ] * + 2 D [ ( J + ! s ) - l ] 2  a  0  2  !  ±  hA [(J+h)-2] D  hqs>{J+h)  [ (J+h)  -1]  2  2  ] (J+^) -l]  ±  hq»(J+\)  ±  \ ( p D + 2 q „ ) (J+h)  2  T -U+B(J+^) -D[  3 / 2  [  (J+h)-!]  (J+!i)V(J+^) -l]  2  2  0  - %Aa(J+5|) T 5 ( P + 2 q ) (J+'i) =F *qb(J+*) [ ( J + ^ r - 1 ] T i(PD+2qD) ( J + * ) 2  symmetric  !  l  T a b l e 3.2  3 / 2  a«q(J+^) [ ( J + * s ) - i ] \ „  O T  The H a m i l t o n i a n m a t r i x f o r t h e X ^, 010 ( A) v i b r o n i c s t a t e o f B0 . The ± r e f e r t o t h e e and f l e v e l s respectively. The term v a l u e , T , r e p r e s e n t s the o r i g i n o f the v i b r o n i c s t a t e . 2  2  2  0  |  2  A  5  /  2  |  e/f>  2  A  3  /  2  e  /f>  T +^A+B[ ( J + y - 6 ] - D[(J+^) -ll(J+^) +32] 2  0  < A 2  5  /  2  e/f|  J  2  -B[ (J+^) -4]*+2D[ ( J + h ) - 4 ] 2  +  ijAflt (J+^) - l l ( J +  1  ) +32] 2  i  2  T -hA+B[ ( J + 3 ) - 2 ] - D(J+!0 [ (J+^) -3] 2  symmetric  0  2  -  -  2  hhoiiJ+hy-2]  *AH(J+fc) [(J+5jr-3] a  3 / 2  83  T h e H a m i l t o n i a n and M a t r i x E l e m e n t s c a s e (b)  3.3.5  The v = l E v i b r o n i c 2  2  f o r Hund's  l e v e l s o f t h e ground s t a t e were  a n a l y z e d u s i n g Hund's case  (b) b a s i s f u n c t i o n s .  The e f f e c t i v e  Hamiltonian operator i s A  A  H ff e  =  H  rot  A  A  A  + Hrj) + HgR + Hsuco  A  where H  (3.16)  A rot  i s t h e r o t a t i o n a l Hamiltonian, Ho, c o n t a i n s  quartic  A  and s e x t i c c e n t r i f u g a l d i s t o r t i o n c o r r e c t i o n s , r o t a t i o n i n t e r a c t i o n and Hgo i s i t s c e n t r i f u g a l  Hgj i s t h e s p i n distortion  correction. The r o t a t i o n a l Hamiltonian takes t h e form H  = BN  (3.17)  2  rot  with the c e n t r i f u g a l d i s t o r t i o n c o r r e c t i o n s Ho, = -DN + HN 4  The s p i n - r o t a t i o n between t h e e l e c t r o n a n g u l a r momentum.  6  being (3.18)  constant, 7 , r e p r e s e n t s t h e i n t e r a c t i o n  s p i n angular momentum and t h e r o t a t i o n a l  The o p e r a t o r form o f t h i s i n t e r a c t i o n i s HSR = 7 N - S  (3.19)  The c e n t r i f u g a l d i s t o r t i o n c o n s t a n t f o r t h e s p i n - r o t a t i o n interaction,  T , has t h e simple  form  d  A  A  A  AO  HSRCD = T (N'S)ir D  84  s i n c e N and 8 commute. The H a m i l t o n i a n matrix i s shown i n T a b l e 3.3.  The b a s i s  f u n c t i o n s w r i t t e n as | F > and | F > are t  |Fj> =  |F > 2  2  | 2, N 2  0 S J =  (3.21)  = | E, N 0 S J = N-%> 2  I t i s i n t e r e s t i n g t o note t h a t i n case generated  H+h>  (b), t h e 2x2 m a t r i x  i s completely d i a g o n a l whereas i n case  (a), t h e  m a t r i x has o f f - d i a g o n a l elements which r e q u i r e t h a t t h e matrix be d i a g o n a l i z e d b e f o r e energy 3.3.6  The H a m i l t o n i a n Elements  l e v e l s can be o b t a i n e d .  for  A !^, v =l, 2  2  and i t s  Matrix  The most i n t e r e s t i n g r e s u l t i n t h i s t h e s i s i s t h e complete a n a l y s i s o f t h e m a n i f o l d o f s p i n and v i b r o n i c b e l o n g i n g t o t h e bending A!!,, s t a t e o f B0 2  2  fundamental o f t h e A n  i s unique  2  u  state.  levels The  among t h e t r i a t o m i c molecules t h a t  have been s t u d i e d so f a r because i t s s p i n - o r b i t c o u p l i n g i s v a s t l y l a r g e r than the e l e c t r o s t a t i c  (or R e n n e r - T e l l e r )  s p l i t t i n g between t h e o r b i t a l components.  The consequences o f  t h i s a r e h i g h l y unusual, and a l l o w much more i n f o r m a t i o n t o be e x t r a c t e d from t h e spectrum  than i s normally p o s s i b l e .  A very  d e t a i l e d d e r i v a t i o n o f t h e Hamiltonian and i t s m a t r i x elements w i l l t h e r e f o r e be g i v e n . O r b i t a l angular momentum e f f e c t s are b e s t i n t r o d u c e d u s i n g t h e language f o r a bent molecule.  Omitting e l e c t r o n  85  Table 3 . 3  The Hamiltonian matrix f o r the X n , 0 1 0 ( Z ) v i b r o n i c s t a t e s o f B 0 . The term v a l u e , T , r e p r e s e n t s the vibronic origin. 2  2  G  2  0  I Fi>  | F> 2  T +BN(N+1) -DN (N+1) 2  2  0  + HN^N+l) +  3  0  35TDN (N+1) 2  T +BN(N+1)-DN (N+1) + HN (N+1) 2  0  0  3  - ^(N+l) „ -  JS  7 D  N(N+1)  2  2  86  s p i n f o r t h e moment, t h e t o t a l angular momentum, J , i s t h e sum o f t h e r o t a t i o n a l angular momentum, R, and t h e e l e c t r o n o r b i t a l angular momentum, L, J = R + L  (3.22)  R - J - L  (3.23)  which g i v e s  The r o t a t i o n a l H a m i l t o n i a n i s then H  rot  = AR =  2 2  + BRV + C % 2  (AJ +B^ +c5 ) 2  z  2  2  x  (3.24)  2  y  (2I&jL +2B3jt +2c3yi\) z  x  (AL,, + BL^ +  +  2  2  CL\ ) 2  Because t h e a - a x i s r o t a t i o n o f a bent molecule t u r n s i n t o one component o f t h e degenerate molecule,  bending v i b r a t i o n i n t h e l i n e a r  i t forms a s p e c i a l case, which must be s e p a r a t e d  from t h e b- and c - a x i s r o t a t i o n s which a r e s t i l l p r e s e n t i n the l i n e a r l i m i t .  The f u l l Hamiltonian should then be w r i t t e n /\  A  A  A  H— (Hbending+Hg-axis rotation) "'"H^- and -axis rotations c  (3.25)  Now, as e x p l a i n e d above, an e l e c t r o n i c n s t a t e o f a l i n e a r molecule  can be d e s c r i b e d as two Born-Oppenheimer  p o t e n t i a l curves, r e p r e s e n t i n g two d i f f e r e n t  electronic  s t a t e s , which touch a t t h e l i n e a r c o n f i g u r a t i o n (see F i g u r e 3.5).  Each o f these two bent-molecule  Born-Oppenheimer s t a t e s  i s an ' o r d i n a r y ' non-degenerate e l e c t r o n i c s t a t e , w i t h i t s own  87  bending  Hamiltonian  H = hP  2  + ^ AQ  2  + . . .  (3.26)  In t h i s e q u a t i o n Q i s the bending c o o r d i n a t e  (essentially,  180° minus the bond angle a t the c e n t r a l atom, m u l t i p l i e d by a mass f a c t o r ) , P i s i t s conjugate momentum and A i s a f o r c e constant.  The c o m p l i c a t i o n caused by the o r b i t a l  angular  momentum i s t h a t these two Born-Oppenheimer s t a t e s are coupled by the c r o s s - t e r m A  A  ^electronic-rotation  from e q u a t i o n 3.24 T h i s may  =  ~2hJ L z  A  (3.27)  z  which has m a t r i x elements between them.  seem unimportant  a t f i r s t s i g h t , but the term becomes  enormous near the l i n e a r l i m i t , s i n c e the r o t a t i o n a l constant A i s i n f i n i t e when the molecule i s e x a c t l y  linear.  The bending and a - a x i s r o t a t i o n a l motions of the  two  e l e c t r o n i c s t a t e s t h a t touch i n the l i n e a r l i m i t t o become a n s t a t e must be r e p r e s e n t e d by a 2x2  matrix,  |*> H =  |*">  iP^AV+Afltf+A )  The b a s i s f u n c t i o n s | factors:  -2AKA  2  -2AKA of equation 3.28  (3.28)  ^P +^A"Q +A(K +A ) 2  2  2  2  are p r o d u c t s of t h r e e  88  i)  An  e l e c t r o n i c f a c t o r f o r the  o r b i t a l motion of  the  electrons, *** = where * i s the  HTTP  i(  ± e™)  and  differences  group, and  (3.29) the  factors  symmetry o p e r a t i o n s of  same as t h a t  of l^,  The  the  irreducible  which i s why  iii)  An  |v>,  which  and  a-axis r o t a t i o n a l |K>  where x i s the  the  couples them.  A harmonic o s c i l l a t o r v i b r a t i o n a l f u n c t i o n ,  be w r i t t e n  and  |-A>)  averaged p o s i t i o n of  i n f a c t the product of t h e i r  o p e r a t o r of equation 3.27  can  ±  are needed so t h a t these  r e p r e s e n t a t i o n s i s the  ii)  (|A>  the plane of the bent molecule.  t r a n s f o r m c o r r e c t l y under the point  (2)-\  =  angle between the  unpaired electron sums and  (e *  =  factor  (27r)~V * K  (3.30)  angle between the plane of the bent molecule  an a r b i t r a r y r e f e r e n c e p l a n e . At t h i s p o i n t  the R e n n e r - T e l l e r e f f e c t can  be  seen t o  a r i s e n a t u r a l l y as an a - a x i s r o t a t i o n a l c o u p l i n g between  two  bent-molecule e l e c t r o n i c s t a t e s t h a t happen t o become degenerate when the molecule i s l i n e a r . electron  s p i n i s c o n s i d e r e d the  d i a g o n a l i n the b a s i s A  spin  "  orbit  3.8).  <L >2 t o the z  However, when  spin-orbit coupling i s o f f -  of equation 3.28,  and  adds a term  electronic-rotational coupling  I t then becomes p r e f e r a b l e  t o t r a n s f o r m the  (see  equation  electronic  89  factors  9  £ E L  t o a new b a s i s  | A  (27r)"V  = ±1> =  by t a k i n g t h e i r sums and d i f f e r e n c e s .  i #  (3.31)  I n t h e new b a s i s ,  e q u a t i o n 3.28 becomes h^+k  H =  (A +A~)Q +A(K+A) +  2  ^ (A -A')Q +  h (A*-A~)Q  2  hP +h (A + +A-)Q 2 +A(K-A) 2  2  2  The terms i n e q u a t i o n 3.32 a r e t h e s e . bending  (3.32)  2  h^+h (A*+A~)Q i s t h e 2  H a m i l t o n i a n f o r a non-degenerate e l e c t r o n i c  state  where t h e p o t e n t i a l i s t h e mean o f t h e two Born-Oppenheimer p o t e n t i a l s forming t h e n s t a t e ; ^(A -A")Q +  2  i s the d i f f e r e n c e  between t h e two Born-Oppenheimer p o t e n t i a l s , and t h e terms A(K±A)  2  a r e t h e angle-dependent terms t h a t c o n v e r t t h e one-  d i m e n s i o n a l bending H a m i l t o n i a n i n t o t h e two-dimensional harmonic o s c i l l a t o r H a m i l t o n i a n w i t h t h e v i b r a t i o n a l  angular  momentum quantum number I g i v e n by I  = K-A  (3.33)  The new b a s i s f u n c t i o n s can t h e r e f o r e be w r i t t e n as products of  |A=±I>  e l e c t r o n i c f a c t o r s and e i g e n f u n c t i o n s f o r t h e two-  d i m e n s i o n a l harmonic o s c i l l a t o r , course diagonal i n  K=Z+A,  |vZ>; t h e m a t r i x 3.32 i s o f  because i t i s a transformed v e r s i o n  of e q u a t i o n 3.28. Returning t o the v =l l e v e l of the 2  are two d i f f e r e n t  s t a t e o f B0 , t h e r e 2  |K| v a l u e s , 0 and 2, which a r i s e when A=±l  90  i s added t o Z = ± l . required.  Two  separate m a t r i c e s are t h e r e f o r e  The d i a g o n a l elements are v e r y simple, b e i n g  the e i g e n v a l u e s of the two-dimensional  harmonic  just  oscillator  Hamiltonian, E .i = (v +l)« v  where w i s the v i b r a t i o n a l  (3.34)  a  frequency i n cm"  1  units.  The o f f -  d i a g o n a l elements are f a i r l y c o m p l i c a t e d because, i n o r d e r t o connect the v i b r a t i o n a l b a s i s f u n c t i o n s | v ' , Z ' = K - A > |v,Z=K+A>,  and  the c o o r d i n a t e o p e r a t o r has t o be a v i b r a t i o n a l  l a d d e r o p e r a t o r , which i s a l s o assumed t o a c t between the e l e c t r o n i c f u n c t i o n s |A=+1> and A = - l > . calculation will purposes  be found i n r e f e r e n c e [43], though f o r the  of t h i s work the o n l y important m a t r i x element i s  <A=l;v,Z=K-l| \ (  In  The d e t a i l s of the  A  +  - A ~ ) Q  e q u a t i o n 3.35  2  |  A=-l;v,Z=K+l> = \zw [ ( v + l ^ - K ] * 2  (3.35)  the experimental parameter e i s d e f i n e d as e =  ( A  +  - A ~ ) / ( A  +  + A ~ )  .  (3.36)  The elements o f f - d i a g o n a l i n v can be t r e a t e d by p e r t u r b a t i o n theory  [43], although f o r K=0  energy  expression  they s i m p l i f y t o g i v e the exact  E(v,K=0) = w(v+l)(l±e)* E q u a t i o n 3.37 the two  shows t h a t the K=0  energy  (3.37)  l e v e l s correspond t o  Born-Oppenheimer p o t e n t i a l s making up the n e l e c t r o n i c  91  s t a t e , so t h a t f o r t h i s s p e c i a l case t h e r e i s no v i b r o n i c coupling  (as can a l s o be seen from equation  eigenvalue  of J  2  3.27,  s i n c e the  i s zero) .  The p a r t of equation 3 . 3 2 t h a t i s d i a g o n a l i n v f o r K=0 3 . 3 4 and  i s g i v e n by equations  3.35:  |A=l;v=l,Z=-l> =  H(K=0)  |A=-l;v=l,Z=l> (3.38)  2u> 2w  The m a t r i c e s  f o r K=±2,v=l  collapse to l x i ' s ,  no b a s i s f u n c t i o n s w i t h v = l , | Z | = 3 .  since there  are  The v i b r o n i c l e v e l s  |K|=2,v=l are t h e r e f o r e not a f f e c t e d by v i b r o n i c c o u p l i n g except  i n h i g h e r order, as a r e s u l t of the elements o f f -  d i a g o n a l i n v, and  l i e near the c e n t r e of the m a n i f o l d  states i n this vibrational The  next  level.  s t e p i s t o i n t r o d u c e the s p i n - o r b i t c o u p l i n g .  Another f a c t o r ,  |S2>,  where Y>=±\,  i s added t o the b a s i s s e t ,  and the e f f e c t s of the s p i n - o r b i t o p e r a t o r , e q u a t i o n seen t o be as f o l l o w s . but  identical,  3 . 3 8 becomes two  Equation  s  =  3.8,  are  separate,  matrices  A = ± l ; v = l , Z = Tl;Z= ±*s>  H(K=0, 2 )  of  I A=T1;V=1,Z=±1;S=±^>  2u/+^A  (3.39)  w h i l e the e n e r g i e s of the K=2 ( A) components are E ( K = 2 , A)  = 2u.±^A  (3.40)  92  A l r e a d y i t can be seen t h a t i f A i s much l a r g e r than e w , as i n the A n 2  n  s t a t e o f B0 , t h e r e w i l l be two groups o f s p i n - v i b r o n i c 2  l e v e l s , each c o n s i s t i n g o f a Z s t a t e and a component o f t h e 2  2  A, s e p a r a t e d by t h e s p i n - o r b i t c o u p l i n g (see F i g u r e 3.7) F i n a l l y , t h e b- and c - a x i s r o t a t i o n s , from e q u a t i o n 3.25,  are i n c l u d e d .  In the l i n e a r l i m i t the r o t a t i o n a l  Hamiltonian  is written A  H  A rot  A  A  A . 9  = B(J-L-G-S)  2  as i n e q u a t i o n 3.9, and i t s m a t r i x elements d i a g o n a l i n A, v and  I come from t h e s i m p l i f i e d form, e q u a t i o n 3.10. As b e f o r e  the r-dependence o f t h e r o t a t i o n a l and s p i n - o r b i t  parameters,  B and A, produces c e n t r i f u g a l d i s t o r t i o n e f f e c t s , which have been d e s c r i b e d i n equations 3.11 and 3.12. The major d i f f e r e n c e i n t h e r o t a t i o n a l l e v e l s t r u c t u r e o f t h e bending fundamental compared t o t h e z e r o - p o i n t l e v e l l i e s i n t h e Atype and Z-type d o u b l i n g s .  The A-type d o u b l i n g Hamiltonian,  e q u a t i o n 3.13, i s s t i l l p r e s e n t b u t i t s e f f e c t s a r e u s u a l l y masked because i t i s d i a g o n a l i n Z and t h e r e f o r e a c t s between the A component and t h e two S components.  Similar  c o n s i d e r a t i o n s u s u a l l y apply t o t h e Z-type d o u b l i n g , b u t t h e A !^ s t a t e o f B0 i s not a " u s u a l " case. 2  2  Z-type d o u b l i n g has not been d i s c u s s e d b e f o r e i n t h i s t h e s i s ; i t i s a v i b r a t i o n a l C o r i o l i s e f f e c t which i s caused by the x- and y-components o f t h e v i b r a t i o n a l a n g u l a r momentum, G.  The expansion  o f t h e r o t a t i o n a l Hamiltonian, e q u a t i o n 3.9,  93  produces many terms which a r e omitted from e q u a t i o n 3.10 because they a r e o f f - d i a g o n a l i n A, v and Z . are of  - B ( J G _ + J _ G \ ) and B ( G \ § . + G _ S ) . +  +  Among these  The m a t r i x elements o f G\ a r e  t h e t y p e A V = A V = ± 1 , A Z = ± 1 , so t h a t i n second o r d e r they 2  3  connect t h e two degenerate b a s i s l e v e l s f o r t h e bending fundamental,  |v=l,Z=l>  and |v=l,Z=-l>,  through t h e  antisymmetric s t r e t c h i n g v i b r a t i o n , v . 3  fundamental  In t h e bending  o f a S molecule such as C0 t h e r e s u l t i s a l  2  s p l i t t i n g o f t h e r o t a t i o n a l l e v e l s , a c c o r d i n g t o t h e formula Aw = q ^ f j + l ) where q  v  i s t h e Z-type d o u b l i n g c o n s t a n t .  (3.41) ( S t r i c t l y , of  c o u r s e , what i s observed i n COo, i s a s t a g g e r i n g , s i n c e h a l f t h e l e v e l s a r e m i s s i n g because o f t h e zero n u c l e a r s p i n s o f t h e oxygen atoms.)  The Z-type d o u b l i n g can be c o n s i d e r e d as  a r i s i n g from a transformed H a m i l t o n i a n , H elements  , whose m a t r i x  [44] a r e  <v ,Z |H |v ,Z±2> =?iq y(v TZ) (v ±Z+2) V J ( J + l ) - P ( P ± 1 ) (2)  2  2  v  x  2  2  (3.42)  yj(j+i)-(p±i)(P±2)  T h i s e q u a t i o n i s d e r i v e d by t h e a p p l i c a t i o n o f second-order degenerate p e r t u r b a t i o n t h e o r y t o t h e m a t r i x elements o f -B(J*G_+J_G ) ; the derivation +  a l s o g i v e s an e x p r e s s i o n f o r q i n v  terms o f t h e v i b r a t i o n a l f r e q u e n c i e s and C o r i o l i s c o n s t a n t s ,  94  ->t>2  q  =  v  T  2B w  [1 + 4 2 k=l ,3  2  J  2  2  — w  k  k  2 w  2  —]  -w  (3.43)  2  Another m a t r i x element, s i m i l a r t o e q u a t i o n 3.42, a r i s e s i n m u l t i p l e t e l e c t r o n i c s t a t e s [44] from t h e c r o s s - t e r m between A  A  A  A  A  A  A  A  -B(J G_+J_G ) and B(G S.+G.S ) : +  +  <v ,Z;Z±l|H 2  (2)  +  +  |v2,Z±2;i:>  (3.44)  =-*sq y(V =FZ) (v ±Z+2) v  2  2  x yj(j+i)-(p±i)(p±2) ys(s+i)-s(s±i) Both these elements a r e p r e s e n t i n ^  s t a t e s though, as  e x p l a i n e d above, they a r e not normally needed. The  f u l l H a m i l t o n i a n m a t r i c e s r e q u i r e d f o r t h e v =l 2  o f t h e A ^ s t a t e o f B0 a r e g i v e n i n T a b l e 3.4. 2  levels  There a r e  e i g h t b a s i s f u n c t i o n s f o r each J i n t h e g e n e r a l case, b u t t h e 8x8 m a t r i x t h a t r e s u l t s can be f a c t o r i z e d i n t o two 4x4 m a t r i c e s by t r a n s f o r m i n g t o t h e p a r i t y b a s i s |JP±> = 2"*[|A v I S;JP> ± | —A v -I  -2;J,-P>]  The two 4x4 m a t r i c e s correspond t o t h e e and f p a r i t y  (3.45) levels,  and o n l y one o f them i s needed f o r any p a r t i c u l a r J v a l u e , because o f t h e zero s p i n s o f t h e oxygen n u c l e i . i n d e t a i l the eight basis functions are  W r i t t e n out  95  T a b l e 3.4  The Hamiltonian matrix f o r t h e A ^ 010A and S vibronic states. The ± r e f e r t o t h e e and f p a r i t y levels respectively. 2  ±>  1^3/2  (B +%AD[ (J+is) -6] - ( P L + ^ A H ) r (J+ii) -11(J+^) +32]  | Slo«r  ±>  2  2  2  |  ±>  2  S  u  p  p  e  r  ± >  2  4  4  [(J+^) -4]* { B - 2 D J (J+ -!) - 4 ] } 2  1  t  »iq»[(J+^) -i * a  ]  ±^q[(J+^) ri]* [(J+*) - 4 ] * 2  [(J+»0 -•*]  + i A(l-3/4e" ) s  i  (B^-'iADtCJ+is) -^] - ( D - J i A a ) (J+H) [(J+*)-3] +T +AT - iA(l-3/4e ) 2  a  1  CE  [  [ (J + ! s ) - l ] * (±!sq(J+4)-qv) 2  (J+S5) -l]*[±^(P+2q) + sqv(J+ s) ] 2  !  i  s  B»(J+*) -Dj.[(J+*)* + (J+ s) ] +TQJ + ! j A ( l - % € ) ± S7 .(J+^)+a7 . a  !  - [ * ( B . + B ) ] (J+h) + (D .+Dj;_) ( J + * ) ±eu B  f r  3  E  3  !  symin* 2 t r i c  r  a :  B -(J+! ) -D .[(J+i ) +(J+%) ] +T -!sA(l-v«e ) 5  I  2  £  S  :l  eB;  a = !s (J+'s) {J-h)* = -*(J+%) ( J + 3 / 2 )  for e levels for f levels  b = -\(J+h) (J+3/2) f o r e levels = h(J+h) (J-h) for f levels 2  2  4  96  |  2  |  2  A  A  5 / 2  3  /  ,±>  =  2~*{|1  1  1  ±>  =  2"*{|1  1  1  2  /  h;J  -\;3  5/2> ±  | - 1 1 - 1  3/2> ±  | - 1 1 - 1  -h  ; J  k;J  -5/2>}  -3/2>}  (3.46)  I Slower,±> = 2"*{ 1 1 l - l | 2 per/±> = 2 " * { | i 2  up  The n o t a t i o n  2  conventional  2  S  l  h;J  - l -h;J -\> ± | - l l  and Ei 2  u p p e r  owe  X f and S* because 2  instead  o f t h e more  a r e mixed s o  t h e group t h e o r y  2  a s n=h s t a t e s  labels i n st  (c) c o u p l i n g . An i n t e r e s t i n g  necessary diagonal  because  i t was  T h i s must r e p r e s e n t t h e electronic  o f thedata,  s t a t e which,  can be separated  from  apparent s p i n - r o t a t i o n i n t e r a c t i o n i nt h e S 2  states  electrostatic  that  i s caused by t h e presence  splitting  Normally  i nthe off-diagonal  i t i s not possible  AD a n d i s e p a r a t e l y almost t o t a l l y  provides  work i s t h a t  f o r the ^ states.  o f t h ecompleteness  vibronic  of this  7N«S i n t e r a c t i o n o f t h e  the very large  them.  result  t o add a s p i n - r o t a t i o n i n t e r a c t i o n term t o t h e elements  intrinsic  are  h;J h>)  t h e two s t a t e s  i n a p p r o p r i a t e ; t h e S s t a t e s behave  case  l  r h a s been used  strongly by s p i n - o r b i t coupling that are  l - % ; J -^>}  h> ± | - i l  element  squares because  [45]; the A ^ state  o f B0  2  one o f t h e r a r e examples  between  t o determine t h e parameters  i na ^ state by l e a s t correlated  of the  o f where t h i s  they  2  correlation i s  broken. The v i b r o n i c in  t h e energy  necessary  energies  level  TQ^ a n d A T  S A  diagram o f F i g u r e  to refer a l l vibronic  in  Table  3.4 a r e d e f i n e d  3 . 1 4 ; i t h a s been  energies  t o the X !^ 0 1 0 2  F i g u r e 3.14  D e f i n i t i o n of the v i b r o n i c used i n Table 3.4  e n e r g i e s TQS and  98  state  as  zero,  c o u l d be u s e d naturally, constant level,  t h e measured t r a n s i t i o n  least  for a l l  to  fitting. use  frequencies  It  a single  f o u r s p i n - v i b r o n i c components  because of  p r e c i s i o n of  the  data,  small higher-order  would, B rotational of  this  the  has  A ^ 010 2  not  been  effects.  Results  The body o f  L e a s t Squares F i t t i n g  goal  of  data to  a spectroscopic  Hamiltonian matrix.  improved upon.  This type  Hamiltonian into  their  analysis  is  to  a s m a l l e r number o f p a r a m e t e r s  begins with estimates of  the  squares  have been p r e f e r a b l e  3.4.1  the  i n the  but with the  possible  3.4  so t h a t  'skeleton'  This the  is  an i t e r a t i v e  that  of problem i s the  solved  a  large  appear  process  constants which are  two p a r t s :  matrices  reduce  which  gradually  by  separating  molecular parameters  which c o n t a i n t h e i r  in  and  a n g u l a r momentum  coefficients,  H = X • B ss ~ ss In  this  equation  associated  X is  'skeleton'  the vector matrix.  (3.47)  of parameters  The energy  levels  a r e o b t a i n e d by d i a g o n a l i z a t i o n o f H a c c o r d i n g ss U H T  SS  where E c o n t a i n s eigenvectors.  the  of  the  system  (3.48)  SS  e i g e n v a l u e s and U i s  the  to  U = E  SS SS  The c a l c u l a t i o n s  and B i s  a matrix  a r e done b y a  of  computer  99  programme which g i v e s U, E and t h e r e s i d u a l s A E (observed minus c a l c u l a t e d e n e r g i e s f o r each data p o i n t ) .  These r e s i A X , by  d u a l s a r e used t o o b t a i n c o r r e c t i o n s t o t h e parameters, making use o f t h e Hellmann-Feynman theorem.  SEi/SX =  T h i s s t a t e s [46]  /¥ *(6"H7cSX)* dr i  (3.49)  i  or, f o r a s i n g l e parameter, X , D  cSEi/cSX  n  =  [U (6H/5X,)U] T  1 1  = D  (3.50)  i B  The d e r i v a t i v e s matrix, D , g i v e s t h e dependence o f t h e energy l e v e l s on changes i n the parameters,  according t o  AE = D • AX  The  (3.51)  l e a s t squares c o r r e c t i o n s t o t h e e s t i m a t e s o f t h e  parameters,  A X , as a f u n c t i o n o f D and t h e r e s i d u a l s A E , a r e r-j  r^j  AX =  (D D) T  _ 1  • D  T  • AE  (3.52)  where t h e e x t r a matrix a l g e b r a i s needed because D i s a r e c t a n g u l a r m a t r i x which cannot be i n v e r t e d d i r e c t l y .  These  c o r r e c t i o n s , A X , can be added t o t h e a p p r o p r i a t e parameters t o be used  i n t h e next i t e r a t i o n .  The process i s c o n t i n u e d  until  the r e s i d u a l s a r e reduced t o t h e l e v e l o f t h e experimental precision.  How w e l l a s e t o f e n e r g i e s o r t r a n s i t i o n s f i t s a  model i s measured by the standard d e v i a t i o n , a, f o r unweighted data d e f i n e d by  100  (AE) (AE) T  a =  (3.53)  n-m  where n i s t h e number o f independent measurements and m i s the number o f parameters b e i n g determined. The f i t t i n g o f the data was done i n two s t e p s . the ground s t a t e combination d i f f e r e n c e s o b t a i n ground s t a t e c o n s t a n t s .  Firstly  (A F") were f i t t e d t o 2  A combination d i f f e r e n c e  A F"(J) i s the d i f f e r e n c e i n energy between an R ( J - l ) 2  t r a n s i t i o n and a P(J+1) t r a n s i t i o n which g i v e s t h e energy s e p a r a t i o n o f the l e v e l s J - l and J + l i n t h e lower s t a t e ; i s shown i n F i g u r e 3.15.  this  With the ground s t a t e c o n s t a n t s  determined, t h e ground s t a t e e n e r g i e s can be c a l c u l a t e d and the l i n e f r e q u e n c i e s used t o determine the upper  state  constants. 3.4.2  Numerical R e s u l t s and D i s c u s s i o n  The r o t a t i o n a l c o n s t a n t s f o r the X ^ 000 l e v e l , as 2  determined from f i t s t o t h e combination d i f f e r e n c e s , a r e g i v e n i n T a b l e 3.5 f o r the boron-11 i s o t o p e and T a b l e 3.6 f o r the boron-10 i s o t o p e .  Constants determined by p r e v i o u s workers  are a l s o g i v e n f o r comparison.  I t i s d i f f i c u l t t o make d i r e c t  comparisons as the m a t r i x elements used i n p r e v i o u s works [3739] d i f f e r from ours.  Our d i a g o n a l B parameter corresponds t o  t h e i r B+^q, which means t h a t the v a l u e s agree t o w i t h i n experimental e r r o r .  The form o f the o f f - d i a g o n a l  elements  F i g u r e 3.15  A A F"(J') combination 2  difference.  102  T a b l e 3.5  M o l e c u l a r c o n s t a n t s f o r t h e X ^ 000 l e v e l o f B 0 , i n cm" . 2  U  2  1  Constant  Maki e t a l . T391  T h i s Work  A  [-148.58]  [-148.58]  B  0.3293697(18)  D  1.3207(53)x(10)~  3  -3.12(22)xl0~  D q p+2q A  -5.15(65)X10~  0.3293926(73)  b  1.3117(130)xl0~  7  5  5  5.76(17)xl0~  7  -1.322(166)xlO  -5  -4.29(135)xl0~  5  5.670(235)xl0~  3  a - F i x e d t o v a l u e g i v e n i n Reference [ 3 6 ] . b - E r r o r l i m i t s a r e 3a i n u n i t s o f t h e l a s t f i g u r e  T a b l e 3.6  3  3  quoted.  M o l e c u l a r c o n s t a n t s f o r t h e X ^ 000 l e v e l o f BO , i n cm" . 2  10  2  1  Constant  Maki e t a l . T391  T h i s Work  [-148.58]  A  [-148.58]  B  0.3293521(17)  D  1.3148(56)X10~  D q p+2q A  3  -3.65(15)xl0~ -5.12(64)X10" 5.84(14)X10~  5  3  5  3  0.3293815(15)  b  7  1.297(29)xlO"  7  -1.401(263)xl0~ -5.46(254)X10~ 5.70(38)xlO"  a - F i x e d a t v a l u e g i v e n i n Reference [ 3 6 ] . b - E r r o r l i m i t s a r e 3a i n u n i t s o f t h e l a s t f i g u r e  5  5  3  quoted.  103  i n v o l v i n g B i s s l i g h t l y d i f f e r e n t ; t h i s confuses t h e correspondance  o f t h e AJ/AD terms (AD=2AJ) i n such a way t h a t ^A  D  s h o u l d be somewhat more n e g a t i v e than t h e i r Aj, which i s as we find i t .  Given our l a r g e r range o f J v a l u e s and extremely low  s t a n d a r d d e v i a t i o n s o f 0.00030 cm" and 0.00026 cm" f o r B 0 and 1  1  U  2  10  BO r e s p e c t i v e l y , our r o t a t i o n a l c o n s t a n t s f o r t h e 000 i !!) 2  2  l e v e l s a r e a d e f i n i t e improvement over p a s t f i t s .  Our data do  not however a l l o w t h e s p i n - o r b i t parameter t o be r e f i n e d and thus Johns' v a l u e o f -148.5 cm" remains t h e b e s t 1  8  available.  The upper s t a t e f i t s f o r A ^, 000 (hi) were not as s t r a i g h t 2  forward and t h e standard d e v i a t i o n s c o u l d n o t be reduced as far  as those o f t h e ground s t a t e .  The f i n a l B 0  f i t has a =  n  2  0.00154 cm" ; t h a t f o r BO2 i s s l i g h t l y b e t t e r , a t 1  0.00141 cm" . 1  10  The f i n a l e f f e c t i v e c o n s t a n t s a r e g i v e n i n  T a b l e s 3.7 and 3.8 f o r B 0  and B02 r e s p e c t i v e l y , a l o n g w i t h  U  a v a i l a b l e comparisons.  10  2  The r e s i d u a l s o b t a i n e d a r e shown  p l o t t e d a g a i n s t J i n F i g u r e 3.16.  S e v e r a l data p o i n t s a r e  w i d e l y s c a t t e r e d and were not i n c l u d e d i n t h e f i t ; these a r e not i n c o r r e c t l y a s s i g n e d l i n e s b u t r e f l e c t random perturbations i n the  s t a t e caused by h i g h e r v i b r a t i o n a l  l e v e l s o f t h e ground s t a t e .  The d e n s i t y o f ground s t a t e  l e v e l s was c a l c u l a t e d , u s i n g H a a r h o f f ' s  formula  [47], t o be  0.86 |K|-states p e r wavenumber a t 18,300 cm" , which i s e a s i l y 1  h i g h enough t o cause t h e number o f random p e r t u r b a t i o n s shown i n F i g u r e 3.16.  Although  the c e n t r i f u g a l  distortion  c o r r e c t i o n s t o t h e A-doubling parameters a r e n o t w e l l  104  M o l e c u l a r c o n s t a n t s f o r t h e A X 000 l e v e l o f B 0 , i n cm" .  T a b l e 3.7  n  2  1  Constant  T h i s Work  o  18291.59659(56)  T  a  18291.5  g  -101.3  Q  A  -101.28126(94)  B  0.31073912(95)  0.3106  D  1.2375(28)xl0  1.2xl0~  -3.589(65)xl0"  D q p+2q A  P  Johns [36]  + 2 C D  _V  5  -3.06(58)xl0 1.1826(76)xl0~  3  1.1(19)xl0~  9  -7.1(35)xl0~  8  2  D  a - E r r o r l i m i t s a r e 3a i n u n i t s o f t h e l a s t f i g u r e  T a b l e 3.8  7  quoted.  M o l e c u l a r c o n s t a n t s f o r the A X 000 l e v e l o f BO , i n cm" . 10  2  1  Constant T  o  T h i s Work 18313.23753(53)  A  -101.35670(89)  B  0.3107332(10)  D  1.2345(37)xio"  D q p+2q A  -3.951(73)xl0~  a  7  5  -3.04(19)xl0 1.1798(36)xl0"  2  a - E r r o r l i m i t s a r e 3a i n u n i t s o f the l a s t f i g u r e  quoted.  105  a) ® a  0.0010  «  >  •o 0.0005 *> a 3 U  -H  «  0.0000  > u V  »  •0.000*  O  ©  -0.0010  ©  «  a:  19.5  9.5  29.5  39.5  49.5  59.5  S -o.oio 59.5  Figure  3.16  R e s i d u a l s f r o m t h e u p p e r s t a t e f i t o f t h e ^1 r o v i b r o n i c l e v e l s p l o t t e d a g a i n s t J ' f o r a) B0 and b ) B O 2 . C i r c l e d data points i n d i c a t e a s s o c i a t e d t r a n s i t i o n s which nave been e x c l u d e d from the f i n a l f i t . T h e c r o s s e s r e p r e s e n t t h e TI / r e s i d u a l s w h i l e t h e d i a m o n d s r e p r e s e n t t h e *n residuals. 10  3  l/2  2  2  106  determined i n these s e e n by t h e The in  they  constants  Table  spectra  fact  3.9.  fits,  f o r the  Only the  The d a t a s e t  S , +  2  U  with the  S " and A  because  the  5 / 2  u  states 2  2  -  + g  2  the  2  5  2  3 / 2 . U  2  A6/  2 | 0  of  the  constant fit  2  done f r o m  upper l e v e l of  c a n be  are the  position 2  A  3 / 2  band has  local  U  the  2  E  and A  +  2  g  separations  separations However t h i s  of  the  „ level  was  2  A  of  level  3 / 2 n  later  of  the  5/2i  g upper  float  these  state  f i t  fixed.  lower  state in  to  the  X ^ , 010 2  the  other  T h e r e a s o n why t h i s that  not  d e t e r m i n e d b y an  with respect  i s  of  does  t o be  energies of the  vibronic levels.  the  levels  data set  involved f i x i n g the  which allowed the  interacts  the  S " b a n d and b e c a u s e A ^ 010  the  obtained  and i n t r o d u c i n g a p a r a m e t e r ,  t h e upper  in  doublings which appear i n both  subbands;  rotational levels to  effect  2  g  the  states,  corresponding J l e v e l s  2  lower s t a t e s .  6 / 2 u  ground s t a t e v =l be  of  given  rotational  four s p i n - v i b r o n i c  strong 2 ~ -  procedure; t h i s  upper s t a t e a  A  relative  spin-orbit  2  -  2 i g  doubled l i n e s  position  iterative  the  These s e p a r a t i o n s  the  are  boron-10  g  2  allow the The  from t h e  of  (as  was a n a l y z e d a s  f o r a complete  p e r t u r b a t i o n s between t h e  and  states  2  2  S * and A /  + U  A and 2  w e a k l y a l l o w e d A - X 010-010 Z ~ -  U  S  2  required  deviation).  isotope  consisted  cause v a r i o u s l i n e  groups of  2  separations  same u p p e r s t a t e a s rotational  X ^ , 010  allow  states.  2  U  standard  boron-11  combination differences together  are d e f i n i t e l y  lower the  recorded d i d not  analysis.  2  they  has  to  although t h e A / g 2  3  w i t h t h e nearby X " upper l e v e l 2  g  t h e p e r t u r b a t i o n i s t o p r o d u c e anomalous  2i  the  K-type  only  107  M o l e c u l a r c o n s t a n t s f o r t h e X*n 010 l e v e l s i n cm" .  T a b l e 3.9  of B0 , n  s  2  1  Constant  V  T h i s Work  B  0.3306791(34)  D  1.420(30)xl0~  H  -9.3(80)X10~  7  0.15225(14)  "  1.244(48)X10~ -1.17(19)xio"  7  0.15367(15)  a b c d  -  !  7  d  0.33037696(69) 1.326(80)  7  0.15348(17) 6  d  -5.92(77)X10"  7  d  111.3599(25)  A  t-144.6721]  B  0.3306500(21)  D  1.3599(80)xio"  AH  8  12  1.922(67)X10~  D  3  7  -5.34(84)xl0"  d  0.3303520(39)  H  A  -  0.15222(115)  D  Q  1.53(13)xl0~  6  228.1937(36)  T  7  0.330700(25)  a  7  0 B  T  3  13  2.160(64)X10~ 2  r  Kawaauchi e t a l .  [-146.5]°  b  -9.77(42)X10~  5  1.32(16)xio"  0.330662(28) 7  1.39(24)xlO -3.5(ll)xl0~  -7  5  8  E r r o r l i m i t s a r e 3a i n u n i t s o f t h e l a s t f i g u r e quoted, O r i g i n a l l y allowed t o v a r y ; then f i x e d a t t h i s v a l u e , F i x e d t o v a l u e g i v e n i n Reference [ 3 6 ] . There i s a f a c t o r o f -4 d i f f e r e n c e i n m a t r i x element a l o n g w i t h d i f f e r e n t N-dependence.  108  doublings, the  and no e x t r a  K-type doublings  separation is  of  described It  states  the  2  turns  out  of  zero,  ^3/2 s t a t e the  its  is  not  as  is  exist  as  the  listed  fixed  as  With the  level the  B0 )  of  X ^,  the level  taken  2  as  state  levels,  f o r the  The  2  2  because  information  studied  V e r y few the  analysis.  It  is  state  2  suitable  comparisons  upper S s t a t e s ;  A vibronic transitions  were t o o  seen t h a t  very different  at  his  overlapped  the  intrinsic  from t h e The  to  apparent residuals  a r e shown i n F i g u r e 3 . 1 7 ; a g a i n random p e r t u r b a t i o n s  standard deviation  t o be comparable t o  (a = 0 . 0 0 0 4 4  ^  000 level  2  a s we h a v e no  g r o u n d s t a t e have had t o be e x c l u d e d  precision  J=3/2  ground  other  s p i n - r o t a t i o n p a r a m e t e r d e t e r m i n e d by J o h n s .  by t h e  o r i g i n of  A ^ , 0 1 0 A and S u p p e r  for the  s p i n - r o t a t i o n parameter i s  obtained  2  i n Appendix I I I .  the  Johns' value  A !^, 0 1 0  the  0 0 0 ( ^ ^ J state,  2  1  i n Table 3 . 1 0 .  permit r o t a t i o n a l  (This  it.  o n l y Johns has  resolution  X ^,  the  s p i n - o r b i t constant  The c o n s t a n t s  all  n e a r 6 3 0 cm" .  these are  refine  are given  to  locate  r o v i b r o n i c energies of  which has been t a k e n a t with which to  to  lowest r e a l energy  accurately  the  be d e t e r m i n e d .  [ 3 8 ] have d e t e r m i n e d t h e  ^^.g b a n d , (the  the  below.)  be p o s s i b l e  al.  relative  energy  levels  3 / 2 i U  nevertheless  information for  states to  2  u  c o u l d be c a l c u l a t e d ; level  enough  2 ~ and A  to  000  2  the  c a n be a s s i g n e d ;  accurately with respect  010 2" -  gl  carry  i n more d e t a i l  s i n c e Kawaguchi e t  xhi  lines  cm" ). 1  T h e A !^, 2  the  i n order for  the  experimental  010 levels  are  very  109  T a b l e 3.10  Molecular constants f o r the A n i n cm" . 2  010 l e v e l s o f B 0 U  u  1  Constant T AT  OS  TSA 03/2 A 2  6 W  T h i s Work  P r e v i o u s Work  18434.97225(18)  a  18435.l  h Q  2.51275(22) 0.14208(36) -100.85127(25) -13.89603(83)  -13.l  b  B  ss+  0.3116669(10)  0.3124 °  B  0.31160984(77)  0.3108^^  A  0.31157124(58)  ss  1.2423(82)xlO  B  D  D  A  %  p+2q q 7  S+  7  S-  7  DS+  7  DE-  A  D  —i  1.2764(41)xl0  +  D  Q  1.2487(28)X10  _*7  -4 4.7459(52)xlO -2 d [1.1826x10 ] -5 d [-3.06X10  4.714X10  -4 f  ]  -2.107(31)xl0~  3  0.474  e  -1.443(29)X10~  3  0.444  e  1.091(23)xl0 9.50(32)xl0 1.77(12)xl0 -6.81(59)xl0  —  R  Q  E r r o r l i m i t s a r e 3a i n u n i t s o f t h e l a s t f i g u r e quoted, Reference [ 3 6 ] . Reference [36] a b s o l u t e v a l u e ±0.001. F i x e d a t v a l u e from 000 upper s t a t e as not enough information t o obtain, e - Apparent s p i n - r o t a t i o n parameters, from Reference [ 3 6 ] . f - C a l c u l a t e d from e q u a t i o n 3.43. a b c d  -  2 /  110  0.0030  0.0020  •a 01  0.0010  •o 01  0.0000  V) XI o  -0.0010 3  •o  -0.0020  49.5  b) ®  0.020  id >  •o  0.010  ©  01  a  ®  3 O  a  ©  0.000  ® ©  ©  ©  13 01  ii  ©  ©  ® ©  -0.010  © 3  -0.020  ® 9.5  F i g u r e 3.17  19.5  49.5  39.5  29.5  R e s i d u a l s from t h e upper s t a t e f i t o f t h e A and S r o v i b r o n i c l e v e l s p l o t t e d a g a i n s t J f o r a) A (represented by crosses) and S (represented by diamonds) and b) A (represented by c r o s s e s ) and 2 pper (represented by diamonds) . C i r c l e d data p o i n t s i n d i c a t e a s s o c i a t e d t r a n s i t i o n s have been excluded from f i n a l f i t .  2  1  5/2  2  lower  2  3/2  2  U  Ill  interesting  i n that  this  is  the  first  avoided c r o s s i n g with doubled l i n e s resonance. 2  2  and A  5 / 2  rotational F  2  l e v e l s were  s p i n components  No  doubled l i n e s  n e a r J=h, state  that  induced at possible  to  actually  before  the  l e v e l s with reasonable that  the  since  with the  upper The  2  state A  between t h e  the  rotational  3 / 2  F  (see  relative  Figure  crossing level  3.18)  positions  precision.  Once i t  levels  X ] ^ , 000  of  a l l the  (^^g)  2  J=3/2  of  of  occurs the  makes the  2  2  A  3 / 2  is  it  £ ~ and  had been the  and  x  situation.  anomalous K - t y p e d o u b l i n g t h a t  them w i t h h i g h a c c u r a c y .  r o v i b r o n i c energy  studied,  lie  s t a t e s were v i r t u a l l y d e g e n e r a t e ,  was a b l e t o p l a c e the  first  higher J values  calculate  the  a h i g h l y unusual  identified the  of  each o t h e r .  found t o  2  however t h e  the  to  t h e 2f s t a t e ,  c o u l d be  is  appears;  of  relative  an  seen i n a K - t y p e  T h i s allowed accurate placement vibronic levels  2  recorded case of  2  A  3 / 2  determined  f i t t i n g program  Appendix V l i s t s  upper v i b r o n i c  level  taken as  states  zero  energy. It  is  possible  fitting  to  d e t e r m i n e t h e more f u n d a m e n t a l  Specifically, levels,  to manipulate the  using the  (shown  i n Figure  p r o j e c t i n g back t o the obtain values vibronic  separations  of  energy  3.19),  *J=-\  w , e and g . 2  levels of  K  a ^  of the  in  the  constants. 'rotationless'  which a r e o b t a i n e d by  energy  %  parameters used  level,  it  is  possible  One u s e s e x p r e s s i o n s s t a t e which are  [43]  for  the  to  112  18897.00  18895.00 i  3/2  188 93.00  18691.00E (cm"')  A  5/2  18792 .002  V  •  18790.00  18788.00  18786.002v  9.5  19.5  29.5  39.5  •  49.5  J  F i g u r e 3.18 A graph o f the r o t a t i o n a l energy l e v e l s o f t h e A X v =l v i b r o n i c states. The c r o s s e s r e p r e s e n t the A l e v e l s w h i l e the c i r c l e s r e p r e s e n t the S levels. The Z f and A e n e r g i e s have been s c a l e d by a f a c t o r o f -0.3115J(J+l) w h i l e the 2* and A have been s c a l e d by a f a c t o r o f -0.31J(J+1). T h i s s c a l i n g i s done t o ensure t h a t the d e t a i l s of the energy l e v e l s can be seen c l e a r l y . 2  2  2  2  2  3 / 2  2  2  5 / 2  113  18892.2984 18892.2866  5  -upper,g  6  V,  18790.2531 18787.7084  5  1  =  5/2, g  7  loiter ,g  A !!, 2  18415.87!  18313.9679  9  V  2  v  2  0  =  '3/2,u  E(cm ) 633.1480 8 588.2067  9  442.0725  0  404.9550  8  = 1  x 147.92  -1.31524  3.19  n  3  V  Figure  2  l  2  =  0  3/2.g  The " r o t a t i o n l e s s " e n e r g i e s o f a l l t h e v i b r o n i c l e v e l s s t u d i e d with r e s p e c t t o a zero at the XTi , 0 0 0 ( ^ 3 / 3 ) J=3/a level. g  114  E (2)  = 2w -he w -5/64e w -e A / (16w ) ±\ [ A ( 1 - h e ) +4e u> ( l + e / 3 2 ) ] *  E (A)  = 2w -3/4e w -2i/64e a; -3e A /(16w )±^A(l-3/4e )+2g  E(n)  =  2  4  2  2  2  [48]  2  2  2  2  4  2  2  to  allow  2  2  and a  2  The g  theoretically  3.4.3  2  for  all  the  for  the  ground s t a t e  for  the  effective  It  It  experimentally  arises  from  s t a t e w i t h A and 2 e l e c t r o n i c i n Table  3.11  along with  states. values  term being the v i b r a t i o n a l  rotational analysis  vibronic levels  example seen.  fits  This  has  2  fits  constants  standard  deviations  that  those  a r e l e s s t h a n 0 . 0 0 1 5 cm" . 1  an a v o i d e d c r o s s i n g indicates  010  been  1  i n a K-type  situations  when t h e  resonance  such as  this  s p i n - o r b i t parameter  R e n n e r - T e l l e r parameter.  molecular transitions  being used  B0  and  a r e l e s s t h a n 0 . 0 0 0 3 cm" w h i l e  i n other molecules  numbers f o r t h e  of  000  rotational  i n v o l v e d ; the  upper s t a t e of  f o r the  states  has produced a c c u r a t e  much l a r g e r t h a n t h e  presently  for.  2  out.  must e x i s t  allowed  for  i n t r o d u c e d b y Brown  which had been  X ^ a n d A !!,, e l e c t r o n i c  carried  has been  p a r a m e t e r was  K  solve  Conclusion  the  The f i r s t  K  B).  A v e r y complete of  final  2  K  f o u r t h o r d e r a n d c a n be u s e d t o  for discrepancies  (this  2  2  obtained are given  correction to  levels  2  2  mixing of a n e l e c t r o n i c The v a l u e s  2  2  2  2  constants.  2  2  2  2  2  above  ew  2  w -^e w -7/64e w -e A /(16w )±^A(l-^e )+g  noted but not  for  2  2  4  2  These are c o r r e c t t o the  2  2  were  A c c u r a t e wave  o b t a i n e d and a r e  i n m o l e c u l a r beam work a t UBC a i m e d  at  is  115  T a b l e 3.11  Fundamental c o n s t a n t s from f i t t e d parameters i n B0 , i n cm" . 1  2  A n  state  2  Constant  u  w  2  -0.0291-. 14 477.29  31  [-13.896 ] ' 04 0.00087  State  2  Previous Results  T h i s Work e  X n T h i s Work -0.1939 484  448.18  a  -13.I  a  1  19  -86.907  Previous Results  Q  449.9(6)  b  -86.9(3)  b  03  *K  2  a  2  8 3  a - Reference [ 3 6 ] . b - Reference [ 4 9 ] .  ?  46 0.00119^^  0.0011(2)  b  116  measuring t h e h y p e r f i n e s t r u c t u r e . The c a l i b r a t i o n system has a g a i n proven i t s e l f t o be a c o n v e n i e n t and a c c u r a t e method o f c a l i b r a t i n g s p e c t r a l d a t a . For example, t h e assignment o f the 010-000 2" - ^Z' t r a n s i t i o n s 2  was done u s i n g A F  1  2  2~ - S  2  2  +  transitions.  combination d i f f e r e n c e s o b t a i n e d from the The energy s e p a r a t i o n o f the S 2  +  and 2~ 2  lower s t a t e s i s over 200 cm" , but y e t t h e combination 1  d i f f e r e n c e s were never worse than 0.0008 cm" w i t h an average 1  v a l u e b e i n g 0.0003 cm" . 1  Thus the c a l i b r a t i o n system has shown  i t s e l f t o be not o n l y u s e f u l f o r a c c u r a t e measurements o f h y p e r f i n e s p l i t t i n g s but a l s o convenient f o r extended coverage of e l e c t r o n i c  transitions.  117  References 1.  J.O. Schroder, B. Z e l l e r and W.E. E r n s t , J . Mol. Spec. 127, 255 (1988).  2.  S. Gerstenkorn and P. Luc, A t l a s du S p e c t r e d A b s o r p t i o n de l a Molecule d'lode. CNRS Ed., P a r i s , France, 1978.  3.  W.H. Hocking, M.C.L. Gerry and A . J . Merer, 57, 54 (1979).  4.  S. Gerstenkorn, P. Luc and A. P e r r i n , J . Mol. Spec. 64, 56 (1977).  5.  J . L . H a l l and S.A. Lee, A p p l . Phys. L e t t . 29, 367 (1976).  6.  F.V. Kowalski, R.E. T e e t s , W. Demtroder and A.L. Schawlow, J . Opt. Soc. Am. 68, 1611 (1978).  7.  Javan, J . Opt. Soc. Am. 67, 1413 (1977).  8.  H.J. Foth, H.J. Vedder and W. Demtroder, J . Mol. Spec. 88. 109 (1981).  9.  Coherent Inc. Autoscan brochure and p r i v a t e communications w i t h Coherent I n c .  10.  Grant Instrument Buyers Guide, B a r r i n g t o n W6-KA bath system.  11.  M. Hercher, A p p l . Opt. 7, 951 (1968).  12.  D. Brewster, Trans. Roy. Soc. (Edinburgh) 12., 519 (1834).  13.  H.J. Foth and H.J. Vedder, J . Mol. Spec. 102. 148 (1983).  14.  F. B y l i c k i and H.G. Weber, Chem. Phys. L e t t . 79, 517 (1981).  15.  R.E. Smalley, L. Wharton and D.H. Levy, J . Chem. Phys. 63, 4977 (1975).  16.  D.K. Hsu, D.L. Monts and R.N. Zare, S p e c t r a l A t l a s o f N i t r o g e n D i o x i d e 5530 t o 6480A. Academic P r e s s , 1978.  17.  T. Tanaka, R.W. 188 (1975).  18.  C.G. Stevens and R.N. Zare, J . Mol. Spec. 56, 167 (1975).  19.  W. Demtroder, L a s e r Spectroscopy. S p r i n g e r - V e r l a g , New York, 1981.  1  Can. J . Phys.  F i e l d and D.O. H a r r i s , J . Mol. Spec. 56,  118  20.  R.B. B e r n s t e i n , Chemical Dynamics V i a M o l e c u l a r Beam and L a s e r Techniques. Oxford U n i v e r s i t y P r e s s , 1982.  21.  G. Herzberg, M o l e c u l a r S p e c t r a and S t r u c t u r e I I I n f r a r e d and Raman S p e c t r a o f Polyatomic M o l e c u l e s . D. Van Nostrand Company Inc., 1966.  22.  G. Herzberg, M o l e c u l a r S p e c t r a and S t r u c t u r e I I I E l e c t r o n i c S p e c t r a and E l e c t r o n i c S t r u c t u r e o f Polyatomic M o l e c u l e s . D. Van Nostrand Company Inc., 1966.  23.  C H . Townes and A.L. Schawlow, Microwave Spectroscopy. McGraw-Hill, New York, 1965.  24.  I.e. Bowater, J.M. Brown and A. C a r r i n g t o n , Proc. R. Lond. A 333, 265 (1973).  25.  J.K.G. Watson i n V i b r a t i o n a l S p e c t r a and S t r u c t u r e (J.R. D u r i g , Ed.), V o l . 6, E l s e v i e r S c i . Pub. Co., New York, 1977.  26.  K.-E.J. H a l l i n and A . J . Merer, Can. J . Phys. 54/ (1976).  27.  J.H. Van V l e c k , Rev. Mod.  28.  L a n d o l t - B o r n s t e i n , Numerical Data and F u n c t i o n a l R e l a t i o n s h i p s i n S c i e n c e and Technology, V o l . 6. M o l e c u l a r Constants, e d i t e d by K.H. Hellwege and A.M. Hellwege, S p r i n g e r - V e r l a g , New York, 1974.  29.  E. H i r o t a , High R e s o l u t i o n Spectroscopy o f T r a n s i e n t M o l e c u l e s . S p r i n g e r - V e r l a g , New York, 1985.  30.  A. P e r r i n , J.-M. 63, 791 (1988).  31.  J . I . S t e i n f e l d , M o l e c u l e s and R a d i a t i o n : An I n t r o d u c t i o n t o Modern M o l e c u l a r Spectroscopy. The MIT P r e s s , Massachusetts, 1978.  32 .  W.C. Bowman and F.C. DeLucia, J . Chem. Phys. 77, (1982).  33.  P.A. Baron, P.D. Godfrey and D.O. 60. 3723 (1974).  34.  G. Persch, H.J. Vedder and W. 123. 356 (1975).  35.  T. Tanaka and D.O.  36.  J.W.C. Johns, Can. J . Phys. 39, 1738  Phys. 23, 213  Soc.  1157  (1951).  F l a u d and C. Camy-Peyret, Molec.  Phys.  92  H a r r i s , J . Chem. Phys.  Demtroder, J . Mol.  H a r r i s , J . Mol. Spec. 59, 413 (1961).  Spec. (1976).  119  37.  K. Kawaguchi, E. H i r o t a and C. Yamada, Mol. Phys. 44, 509 (1981).  38.  K. Kawaguchi and E. H i r o t a , J . Mol. Spec. 116, 450 (1986).  39.  A.G. Maki, J.B. Burkholder, A. Sinha and C.J. Howard, J . Mol. Spec. 130. 238 (1988).  40.  M.S. Sorem and A.L. Schawlow, Opt. Commun. 5, 148 (1972).  41.  R. Renner, Z. P h y s i k 92, 172 (1934).  42.  J.M. Brown, J.T. Hougen, K.-P. Huber, J.W.C. Johns, I. Kopp, H. L e f e b v r e - B r i o n , A . J . Merer, D.A. Ramsay, J . Rostas and R.N. Zare, J . Mol. Spec. 55, 500 (1975).  43.  J.M. Brown and F. J^irgensen, Adv. Chem. Phys. 52, 117 (1983).  44.  A . J . Merer and J.M. A l l e g r e t t i , Can. J . Phys. 49, 2859 (1971).  45.  J.M. Brown and J.K.G. Watson, J . Mol. Spec. 65, 65 (1977).  46.  H. L e f e b v r e - B r i o n and R.W. F i e l d , P e r t u r b a t i o n s i n t h e S p e c t r a o f Diatomic M o l e c u l e s . Academic Press Inc., F l o r i d a , 1986.  47.  P.C. Haarhoff, Mol. Phys. 7, 101 (1963).  48.  J.M. Brown, J . Mol. Spec. 68, 412 (1977).  49.  K.G. Weyer, R.A. Beaudet, R. S t r a u b i n g e r and H. Walther, Chem. Phys. 47, 171 (1980).  120  APPENDIX I . T r a n s i t i o n f r e q u e n c i e s f o r N0 . APPENDIX IA. Band 99. 2  N'  J'  F'  N"  !  1.5 1.5 1.5 1.5 1.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5  0.5 1.5 2.5 0.5 1.5 1.5 0.5 1.5 0.5 2.5 1.5 1.5 0.5 2.5 2.5 2.5 1.5 1.5 1.5 0.5 0.5 1.5 1.5 0.5 1.5 0.5 1.5 1.5 0.5 3.5 2.5 2.5 4.5 3.5 3.5 2.5 2.5 1.5 2.5 1.5 1.5 3.5 2.5 3.5 2.5 3.5 2.5 1.5 3.5 4.5 2.5  0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  J"  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 2.5 2.5 2.5 2.5 2.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 4.5 4.5 4.5  F"  Frequency (cm" )  1.5 1.5 1.5 0.5 0.5 1.5 1.5 0.5 0.5 2.5 1.5 2.5 1.5 3.5 2.5 1.5 2.5 1.5 0.5 1.5 0.5 0.5 1.5 0.5 2.5 1.5 2.5 1.5 1.5 2.5 1.5 2.5 3.5 3.5 2.5 2.5 1.5 2.5 2.5 1.5 0.5 2.5 1.5 3.5 3.5 2.5 1.5 1.5 3.5 4.5 3.5  16850.290581 16850.291784 16850.293821 16850.297983 16850.299192 16850.364864 16850.366369 16850.372205 16850.373726 16847.768572 16847.767550 16847.766448* 16847.766448* 16847.762744 16847.761017 16847.759663 16847.758949 16847.757600 16847.756774 16847.756387 16847.755536 16847.829730 16847.830590 16847.831216 16847.831919* 16847.831919* 16847.839579 16847.840648 16847.842206 16851.961138* 16851.961138* 16851.960212 16851.957115 16851.955518 16851.953805 16851.952588 16851.951221 16852.120254 16852.119324 16852.118903 16852.118001* 16852.118001* 16852.118001* 16852.119719 16852.121047 16852.125634 16852.128062 16852.129000 16846.055270* 16846.055270* 16846.054044  8  1  APPENDIX I , continued, Band 99. N«  J'  3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 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 7  3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 7.5  F' 3.5 4.5 4.5 3.5 3.5 3.5 2.5 2.5 3.5 3.5 2.5 1.5 2.5 3.5 3.5 3.5 2.5 5.5 6.5 4.5 5.5 4.5 4.5 4.5 5.5 3.5 4.5 3.5 5.5 4.5 5.5 5.5 4.5 3.5 6.5 5.5 4.5 6.5 5.5 4.5 4.5 5.5 5.5 6.5 4.5 3.5 5.5 4.5 5.5 5.5 4.5 7.5  N"  J"  4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6  4.5 4.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 4.5 4.5 4.5 4.5 6.5 6.5 6.5 6.5 6.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 6.5 6.5 6.5  Y»  4.5 5.5 4.5 4.5 3.5 2.5 3.5 2.5 2.5 3.5 2.5 2.5 3.5 4.5 4.5 3.5 3.5 5.5 5.5 4.5 4.5 3.5 3.5 4.5 4.5 2.5 3.5 3.5 4.5 4.5 5.5 4.5 4.5 3.5 7.5 6.5 5.5 6.5 5.5 4.5 5.5 5.5 6.5 6.5 4.5 4.5 5.5 5.5 6.5 6.5 5.5 7.5  Frequency (cm") 16846.053623 16846.050290 16846.046604 16846.044980 16846.043208 16846.041926* 16846.041926* 16846.040721 16846.206032 16846.207429* 16846.207429* 16846.208381 16846.208675 16846.209176 16846.217734 16846.219395 16846.220751 16853.564708 16853.566227 16853.568370 16853.569685 16853.570087 16853.557867 16853.559705 16853.560995 16853.820776 16853.821662 16853.822068 16853.822969 16853.823536 16853.826677 16853.831583 16853.833819 16853.834199 16844.285431 16844.288166 16844.288891 16844.289620 16844.290165 16844.272646 16844.274191 16844.275541 16844.277711 16844.279236 16844.536390 16844.536767 16844.537331 16844.537885 16844.539440 16844.549986 16844.552604 16855.096152 3  1  APPENDIX I , continued, Band 9 9 .  N' 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9  J  1  F'  7.5 8.5 7.5 6.5 7.5 7.5 7.5 6.5 7.5 6.5 7.5 6.5 7.5 7.5 6.5 5.5 6.5 6.5 6.5 7.5 6.5 5.5 6.5 6.5 6.5 7.5 6.5 7.5 6.5 6.5 7.5 8.5 7.5 7.5 7.5 6.5 7.5 8.5 7.5 7.5 7.5 6.5 7.5 6.5 7.5 7.5 7.5 7.5 7.5 8.5 6.5 7.5 6.5 6.5 6.5 7.5 6.5 5.5 6.5 6.5 6.5 7.5 6.5 7.5 6.5 6.5 9.5 9.5 9.5 10.5 9.5 8.5 9.5 9.5 9.5 8.5 9.5 8.5 9.5 9.5 8.5 7.5 8.5 8.5 8.5 9.5 8.5 7.5 8.5 8.5 8.5 9.5 8.5 8.5 9.5 10.5 9.5 9.5 9.5 8.5 9.5 8.5 9.5 8.5  F" N" J" 6 6.5 7.5 6 6.5 6.5 6 6.5 6.5 6 6.5 5.5 6 5.5 5.5 6 5.5 6.5 6 5.5 6.5 6 5.5 4.5 6 5.5 5.5 6 5.5 6.5 6 5.5 5.5 6 5.5 6.5 6 6.5 7.5 6 6.5 6.5 6 6.5 5.5 8 8.5 9.5 8 8.5 8.5 8 8.5 7.5 8 8.5 8.5 8 8.5 7.5 8 7.5 6.5 8 7.5 7.5 8 7.5 7.5 8 7.5 8.5 8 7.5 8.5 8 7.5 6.5 8 7.5 6.5 8 7.5 7.5 8 7.5 6.5 8 7.5 7.5 8 7.5 8.5 8 8.5 8.5 8 8.5 7.5 8 8.5 9.5 8 8.5 9.5 8 8.5 8.5 8 8.5 8.5 8 8.5 7.5 8 7.5 7.5 8 7.5 8.5 8 7.5 6.5 8 7.5 7.5 8 7.5 8.5 8 7.5 7.5 8 7.5 8.5 8 8.5 8.5 8 8.5 7.5 10 10.5 11.5 10 10.5 10.5 10 10.5 9.5 10 9.5 8.5 10 9.5 9.5  Frequency" (cm" ) 1  16855.097567 16855.099277 16855.100507 16855.101363 16855.086628 16855.088785 16855.090002 16855.449037 16855.449417 16855.450211 16855.450544 16855.451518 16855.456355 16855.460639 16855.463976 16842.445048 16842.447661 16842.448624 16842.448963 16842.449852 16842.429197 16842.431366 16842.432180 16842.434871 16842.436243 16842.791263 16842.792457 16842.792857 16842.793577 16842.794147 16842.795157 16842.807617 16842.811167 16856.460598 16856.461847 16856.463375 16856.464481 16856.465551 16856.448318 16856.451788 16856.909071 16856.909858 16856.910566 16856.910848 16856.911915 16856.923159 16856.926659 16840.441541 16840.443924 16840.445117 16840.422909 16840.424677  123 APPENDIX I , continued, Band 99. N'  9 9 9 9 9 9 9 9  J'  9.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5  APPENDIX IB. N  1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3  J  1  1.5 1.5 1.5 1.5 1.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 2.5  F'  9.5 8.5 9.5 7.5 8.5 9.5 9.5 8.5  -  N"  10 10 10 10 10 10 10 10  J"  F"  9.5 9.5 9.5 9.5 9.5 9.5 10.5 10.5  0 .5 8 .5 9 .5 8 .5 9 .5 0 .5 0 .5 9 .5  N"  J"  pii  0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 2.5 2.5 2.5 2.5 2.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 2.5 1.5 1.5 1.5 1.5  1 .5 1 .5 1 .5 0 .5 0 .5 1 .5 1 .5 0 .5 0 .5 2 .5 1 .5 2 .5 1 .5 3 .5 2 .5 1 .5 2 .5 1 .5 0 .5 0 .5 0 .5 1 .5 0 .5 2 .5 1 .5 2 .5 1 .5 1 .5 2 .5 1 .5 2 .5 3 .5 3 .5 3 .5 2 .5 2 .5 1 .5 2 .5  Frequency* (cm ) -1  16840 .428374 16840 .884345 16840 .884858 16840 .885525 16840 .886197 16840 .887292 16840 .902904 16840 .906503  Band 115. F  1  0.5 1.5 2.5 0.5 1.5 1.5 0.5 1.5 0.5 2.5 1.5 1.5 0.5 2.5 2.5 2.5 1.5 1.5 1.5 0.5 1.5 1.5 0.5 1.5 0.5 1.5 1.5 0.5 3.5 2.5 2.5 4.5 3.5 2.5 3.5 2.5 2.5 2.5  -  Frequency (cm" ) 8  1  17092 .901736 17092 .902771 17092 .904302 17092 .909095 17092 .910156 17092 .872271 17092 .873630 17092 .879338 17092 .880801 17090 .379283 17090 .378819 17090 .377781* 17090 .377781* 17090 .373284 17090 .371524 17090 .370068* 17090 .370068* 17090 .368638 17090 .367838 17090 .366754 17090 .337255 17090 .338065 17090 .338668 17090 .339415* 17090 .339415* 17090 .347144 17090 .348156 17090 .349605 17094 .956324 17094 .957344 17094 .956324 17094 .950441* 17094 .950441* 17094 .950441* 17094 .948691* 17094 .948691* 17094 .947344 17094 .509674  124  APPENDIX I , continued, Band 115. N 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  1  J»  2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5  F' 1.5 1.5 3.5 2.5 3.5 2.5 3.5 2.5 1.5 2.5 2.5 3.5 2.5 3.5 4.5 2.5 3.5 2.5 3.5 4.5 2.5 3.5 4.5 4.5 4.5 3.5 3.5 3.5 2.5 2.5 4.5 3.5 3.5 2.5 1.5 2.5 3.5 3.5 3.5 2.5 3.5 2.5 2.5 3.5 4.5 3.5 4.5 4.5 4.5 3.5 2.5 3.5 4.5  N" J" 2 1.5 2 1.5 2 1.5 2 1.5 2 2.5 2 2.5 2 2.5 2 2.5 2 2.5 2 2.5 2 2.5 2 2.5 2 2.5 2 2.5 2 2.5 2 1.5 2 1.5 2 1.5 4 4.5 4 4.5 4 4.5 4 4.5 4 4.5 4 4.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 4.5 4 4.5 4 4.5 4 3.5 4 3.5 4 . 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 4.5 4 4.5 4 4.5 4 4.5 4 4.5 4 4.5  Frequency (cm") 17094.509181 17094.508435* 17094.508435* 17094.508435* 17094.510252 17094.511462 17094.516222 17094.518513 17094.519293 17093.742844 17093.741733* 17093.741733* 17093.735835* 17093.735835* 17093.735835* 17093.734084* 17093.734084* 17093.732542 17087.050346* 17087.048691* 17087.050436* 17087.048691* 17087.043767 17087.050346* 17087.040007* 17087.040007* 17087.038166* 17087.036873* 17087.038166* 17087 .036873* 17087.038166* 17088.596636 17088.597830* 17088.597830* 17088.598647 17088.599075 17088.599647 17088.608326 17088.610010 17088.611143 17087.822135* 17087.822135* 17087.823466* 17087.823466* 17087.825333* 17087.825333* 17087.825333* 17087.829030 17087.833992* 17087.833992* 17087.835462* 17087.835462* 17087.835462* 8  1.5 0.5 2.5 1.5 3.5 3.5 2.5 1.5 1.5 1.5 2.5 2.5 3.5 3.5 3.5 2.5 2.5 1.5 3.5 4.5 3.5 4.5 5.5 3.5 4.5 4.5 3.5 2.5 3.5 2.5 3.5 2.5 3.5 2.5 2.5 3.5 4.5 4.5 3.5 3.5 2.5 2.5 3.5 3.5 3.5 4.5 4.5 5.5 4.5 4.5 3.5 3.5 3.5  1  125  APPENDIX I, continued, Band 115. N'  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 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7  J" 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7.5  F 5.5 6.5 4.5 5.5 4.5 4.5 4.5 5.5 3.5 4.5 3.5 5.5 4.5 5.5 5.5 4.5 6.5 5.5 4.5 6.5 5.5 4.5 4.5 5.5 5.5 6.5 4.5 3.5 5.5 4.5 5.5 5.5 4.5 7.5 8.5 6.5 7.5 6.5 6.5 6.5 7.5 5.5 6.5 7.5 5.5 6.5 7.5 7.5 6.5 6.5 5.5 8.5 1  J"  •p»  4 4.5 4 4.5 4 4.5 4 4.5 4 4.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 3.5 4 4.5 4 4.5 4 4.5 6 6.5 6 6.5 6 6.5 6 6.5 6 6.5 6 5.5 6 5.5 6 5.5 6 • 5.5 6 5.5 6 5.5 6 5.5 6 5.5 6 5.5 6 5.5 6 6.5 6 6.5 6 6.5 6 6.5 6 6.5 6 6.5 6 6.5 6 5.5 6 5.5 6 5.5 6 5.5 6 5.5 6 5.5 6 5.5 6 5.5 6 6.5 6 6.5 6 6.5 6 6.5 6 6.5 8 8.5  5.5 5.5 4.5 4.5 3.5 3.5 4.5 4.5 2.5 3.5 3.5 4.5 4.5 5.5 4.5 3.5 7.5 6.5 5.5 5.5 5.5 4.5 5.5 5.5 6.5 6.5 4.5 4.5 5.5 5.5 6.5 6.5 5.5 7.5 7.5 6.5 6.5 5.5 5.5 6.5 6.5 4.5 5.5 6.5 5.5 6.5 7.5 6.5 5.5 6.5 5.5 9.5  N"  Frequency (cm" ) 8  1  17095.743526 17095.744680 17095.747358 17095.748523 17095.749025 17095.736877 17095.738722 17095.739810 17096.039388 17096.039759 17096.040521* 17096.040521* 17096.041562 17096.044234 17096.049291 17096.052105 17086.463691 17086.466756 17086.467744 17086.470068 17086.468803 17086.451781 17086.453215 17086.454249 17086.456379 17086.457618 17086.754726 17086.755596 17086.755230 17086.756224 17086.757397 17086.767665 17086.770768 17097.142429 17097.143604 17097.145602 17097.146690 17097.147620 17097.133259 17097.135343 17097.136390 17097.547993 17097.548648 17097.549914 17097.549451 17097.550816 17097.556083 17097.560365 17097.563306 17097.561333 17097.564104 17084.491739  APPENDIX I, continued, Band 115.  N« 7 7 7 7 7 7 7 7 7 7 7 7 7 7 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9  J' F' 7.5 7.5 7.5 6.5 7.5 7.5 7.5 6.5 7.5 6.5 7.5 7.5 7.5 7.5 7.5 8.5 6.5 6.5 6.5 5.5 6.5 6.5 6.5 7.5 6.5 7.5 6.5 6.5 9.5 9.5 9.5 10.5 9.5 8.5 9.5 9.5 9.5 8.5 9.5 8.5 9.5 9.5 8.5 7.5 8.5 8.5 8.5 9.5 8.5 9.5 8.5 8.5 8.5 9.5 8.5 8.5 9.5 10.5 9.5 9.5 9.5 8.5 9.5 8.5 9.5 10.5 8.5 7 . 5 8.5 8.5 8.5 9.5 8.5 9.5 8.5 8.5  N" 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10  J" 8.5 8.5 8.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 7.5 7.5 7.5 7.5 7.5 8.5 7.5 8.5 8.5 10.5 10.5 10.5 9.5 9.5 9.5 9.5 9.5 10.5 10.5  F"  8.5 7.5 7.5 6.5 7.5 7.5 8.5 8.5 6.5 6.5 7.5 8.5 8.5 7.5 9.5 9.5 8.5 8.5 7.5 7.5 8.5 6.5 7.5 8.5 9.5 8.5 8.5 7.5 11.5 10.5 9.5 8.5 9.5 8.5 9.5 10.5 10.5 9.5  Frequency (cm" ) 8  1  17084.494334 17084.495446 17084.496523 17084.476514 17084.478232 17084.479309 17084.481624 17084.482906 17084.891638 17084.892321 17084.893252 17084.894722 17084.907528 17084.910643 17098.369950 17098.370967 17098.372980 17098.373816 17098.375158 17098.358057 17098.361232 17098.772589 17098.773367 17098.774619 17098.783349 17098.775649 17098.787359 17098.790635 17082.350949 17082.353529 17082.354948 17082.338188 17082.336936 17082.749614 17082.750509 17082.751870 17082.767376 17082.770668  a) The l a s t d i g i t i s not s t r i c t l y a s i g n i f i c a n t figure however they were i n c l u d e d i n t h e f i t t i n g p r o c e s s . * I n d i c a t e s a blended l i n e .  127  APPENDIX I I .  Sample i n t e n s i t y c a l c u a t i o n .  A sample c a l c u l a t i o n f o r t h e P(6) t r a n s i t i o n s i s shown f o l l o w e d by a t a b l e c o n t a i n i n g t h e measured and p r e d i c t e d intensities.  line  P(6) t r a n s i t i o n s have N"=6 w i t h  J" =  13/2  J" =  11/2  F" =  (FJ  F" =  (F ) 2  11/2  13/2 15/2 11/2  B/2  There w i l l be mixing between t h e two  13/2  F"=n/2  l e v e l s and t h e two  F o r t h e 13/2 l e v e l , t h e 2x2 matrix, i n u n i t s o f  F"=i3/2 l e v e l s . MHz, i s  |F >  |Fi>  2  H  103. 399~  85.044  =  -143.407  103.399 D i a g o n a l i z a t i o n leads t o A  =  124.893  0  0  -183.256  S =  0.933  -0.360  0.360  0.933  Two example m a t r i x elements ( i n t h e n o t a t i o n  <5  9/2 11/21 n I 6  <5  9/2 11/2  11/2 i 3 / 2 >  I /i I 6  =  13/2 1 3 / 2 >  -2  =  .  523  0.0  The r o t a t i o n a l e i g e n f u n c t i o n s basis  |NJF> ) a r e  allowed t r a n s i t i o n forbidden t r a n s i t i o n a r e l i n e a r combinations o f t h e  functions ^basls  =  |g  1 1 / 2 1 3 / 2 >  «  basis  D , , u a  _  J g  1 3 / 2  i3/2>  128  and are o b t a i n e d by the t r a n s f o r m a t i o n *1  eigen eigen  which g i v e s v* *™  =  eigen  _  1  _  0.933  0.360  -0.360  0.933  0.933  *  1  0.360  *  1  B A S I S  B A S I S  | fj, | *° z isen  basis I I *  =  |<0.933*  *  2  + 0.933  *  2  =  |0.933(-2.523)  lnt  2  a |<*  M  + 0|  |<-0.3 6 0 4 *  =  |-0.360(-2.523)  B A S I S 1  M 8 1 S 1  2  >  I B  B A S I S  are then  ,  basis i . . i -  „-  + <0 . 3 6 0 * 2  - , ~ . . basis^ i 2 T  | M | 0. 360*  2  > I  = 5.54  2  |M|-0.360* +  0|  2  B A S I S 1  >  + <0 . 9 3 3 *  B A  "  L B  | M | 0 . 9 3 3*  B A 8 L 8 2  > |  2  = 0.82  allowed and P ( 6 )  FJFJ  "  >|  ei8en  2  =  The P ( 6 )  _ _ _ _ basis^  | |*  eisen  2  B  1 2  |A*| 0 . 9 3 3 *  D A S X S 1  basis  + 0.360  The i n t e n s i t i e s o f the t r a n s i t i o n s e i g e n  basis  F ^  f o r b i d d e n i n t e n s i t i e s are  t a b u l a t e d below and have been s c a l e d such t h a t the s t r o n g e s t intensity J' 11/2 11/2 11/2 11/2 11/2 11/2 11/2 11/2 11/2 11/2 11/2  is 1. i  J"  F"  9/2 9/2 11/2 11/2 13/2 13/2 11/2 9/2 13/2 11/2 13/2  11/2 11/2 11/2 11/2 11/2 13/2 13/2 13/2 13/2 13/2 13/2  9/2 11/2 11/2 13/2 13/2 15/2 13/2 11/2 13/2 11/2 11/2  F  a - Measured from band 9 9 d a t a .  a Measured 0.061 0.168 0.168 0.412 0.153 1.000 0.779 0.718 0.076 0.107  Predicted 0.009 0.056 0.020 0.100 0.003 1. 0 0 0 0.755 0.674 0.009 0.011 0.00003  The agreement  i s by no means p e r f e c t ( p o s s i b l y because the  c o r r e s p o n d i n g e f f e c t i n the upper s t a t e has been  ignored,  b e i n g s m a l l e r ) but the r e l a t i v e i n t e n s i t i e s are p r e d i c t e d c l o s e l y enough t o a i d i n the assignment.  130 APPENDIX I I I . Observed t r a n s i t i o n s system of B O 2 .  ( i n cm") f o r the A^-X^TI, 1  000 - 000 *n ( B0 ) U  2l2  J  R BRANCH  1.5 2.5 3.5 4.5 5.5 6.5 7.5 3.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 24 . 5 23.5 25.5 26.5 27 . 5 2S.5 29.5 30.5 31.5 32.5 33.5 34 . 5 35.5 36.5 37.5 38.5 39.5 40.5 41.5 42.5 43.5 44.5 45.5 46.5 47.5 48.5 49.5  18316.75649 18317.28153 18317.76921 18318.21962 18318.63078 18319.00627 18319.34293 18319.64391 18319.90183 18320.13087 18320.31156 18320.46324 18320.56978 18320.64081 18320.67788 16320.67301 18320.63523 18320.55474 18320.44119 18320.28651 18320.09611 18319.86402 18319.29745 18319.60096 18318.95358 18318.57332 18318.15811 18317.70248 18317.21253 18316.68280 18316.11482 18315.50658 18314.86536 18314.18336 18313.46868 18312.70675 18311.91702 18311.08194 18310.21559 18309.30482 18308.36606 18307.38126 18306.35590 18305.30818 18304.21412 18303.08137 18301.91093 18300.70437 18299.45276  P BRANCH 16313.56427 18312.81264 18312.02365 18311.19681 18310.33250 18309.43010 18308.49074 18307.51306 18306.49985 18305.44382 18304.35902 18303.22528 18300.85533 18299.61335 18298.33592 18297.01803 18295.66637 18294.27304 18292.84567 18291.37851 18288.33005 18289.87431 18286.75349 18285.13845 18283.48092 18281.78972 18280.06135 18278.29520 18276.49232 18274.65228 18272.77190 18270.85435 18268.90021 18266.90964 18264.88217 18262.81223 18260.71046 18258.56788 18256.38956 18254.17226 18251.92195 18249.63118 18247.29433 18244.94136 18242.53620 18240.09972 18237.61817  2  Q BRANCH 18315 18315 18314 18314 18314 18314 18314 18313 18313  .20763 .11318 .98144 .81192 .60465 .35949 .07717 .75651 .39819  APPENDIX I I I , c o n t i n u e d ,  000  -  000  J  R BRANCH  P BRANCH  5C.5 51.5 52.5 53.5 54.5 55.5 56.5 57.5 58.5 59.5 60.5 61.5 62.5 63.5 65.5  18298.17997 18296.86583 18295.50555 18294.11617 18292.67910 18291.22174 18289.70461 18288.16789 18286.57941 18284.97131 18283.30735 18281.62022 18279.89254 18278.12548 18274.47865  18235.10809 18232.54612 18229.97078 18227.34659 18224.68477 18221.98578 18219.24801 18216.48139  000 - 000 \ J  R BRANCH  1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5 32.5  18269.44305 18269.94862 18270.89507 18271.35974 18271.69378 18272.09788 18272.34550 18272.84978 18273.13C74 18273.20658 18273.41697 18273.57307 18273.47848 18273.57307 18273.38486 18273.42461 18273.15980 18273.13074 18272.77849 18272.68861 18272.24832 18272.09788 18271.57605 18271.35974 18270.75425 18270.48793 18269.78036 18269.44305 18268.66464  131  ("BOO) .  ^3/2  Q BRANCH  ( B0 ) U  /  2  P BRANCH 18266.22004 18265.47671 18264.67259 18263.86779 18262.97800 18262.11168 18261.13622 18260.20843 18259.14766 18258.15813 18257.01165 18255.96098 18254.72871 18253.61639 18252.29862 18251.12541 18249.72298 18248.48722 18246.99038 18245.69968 18244.12744 18242.76871 18241.10885 18239.68993 18237.94121 18236.46230 18234.63284 18233.08797 18231.17575 18229.58156 18227.56680  2  Q BRANCH 18267.85621 18267.80660  APPENDIX I I I ,  continued,  R BRANCH  000 -  U  2  P BRANCH  3 3 . 5  1 8 2 6 8 ,  26:,.79  1 8 2 2 5 . 9 0 2 1 9  3 4 . 5  1 8 2 6 7 .  3 9 8 8 0  1 8 2 2 3 . 8 1 7 5 5  3 5 . 5  1 8 2 6 6 ,  9 3 7 7 3  1 8 2 2 2 . 0 8 8 1 5  3 6  1 8 2 6 5 .  9 8 4 3 2  1 8 2 1 9 . 9 1 8 8 0  3 7  1 8 2 6 5 .  4 6 0 7 5  1 8 2 1 8 . 1 3 1 6 7 1 8 2 1 5 . 8 7 1 6 4  3 8  1 8 2 6 4 .  4 2 6 9 8  3 9  1 8 2 6 3 .  8 3 8 6 3  4 0  1 8 2 6 2 ,  7 1 9 2 6  41.5  1 8 2 6 2 .  0 6 7 4 2  4 2 . 5  1 8 2 6 0 .  8 6 3 4 5  4 3  1 8 2 6 0 .  1 5 1 0 4  4 4  1 8 2 5 8 ,  8 5 9 8 0  4 5  1 8 2 5 8 ,  0 8 3 0 9  4 6  1 8 2 5 6 ,  7 1 1 1 4  4 7  1 8 2 5 5 ,  8 7 1 6 3  1 8 2 5 4 ,  4 1 2 1 9  1 8 2 5 3 .  5 1 1 0 4  1 8 2 5 1 ,  9 6 7 2 0  4 8 , 4 9 , 5 0 , 5 1 , 5 2 , 5 3 ,  1 8 2 5 1 ,  0 0 2 8 7  1 8 2 4 9 ,  3 7 3 0 1  1 8 2 4 8 ,  3 4 7 6 2  5 4 ,  1 8 2 4 6 ,  6 3 4 7 4  5 5  1 8 2 4 5 ,  5 4 3 5 5 7 4 6 4 7  5 6  1 8 2 4 3 ,  5 7 ,  1 8 2 4 2 ,  5 9 2 4 5  5 8 ,  1 8 2 4 0 ,  7 1 4 6 8  Q BRANCH  1 8 2 0 2 . 8 6 0 7 0  000 R BRANCH  132  000 X t f ( B 0 ) .  - 000 X / o ( BO ) 10  2  P BRANCH  Q BRANCH  1 . 5  1 8 3 3 8 ,  3 6 0 0 5  2 .  5  1 8 3 3 8 .  8 8 5 5 9  1 8 3 3 5 .  1 6 8 4 9  1 8 3 3 6 . 7 1 7 3 9  3 .  1 8 3 3 6 . 8 1 1 4 9  5  1 8 3 3 9 ,  3 7 3 1 8  1 8 3 3 4 ,  4 1 7 0 2  1 8 3 3 6 . 5 8 5 3 8  4 . 5  1 8 3 3 9 ,  8 2 3 2 3  1 8 3 3 3 ,  6 2 8 0 5  1 8 3 3 6 . 4 1 5 9 6  5 .  5  1 8 3 4 0 ,  2 3 5 3 7  1 8 3 3 2 ,  8 0 0 8 2  1 8 3 3 6 . 2 0 8 6 3  6 .  5  1 8 3 4 0 ,  6 0 9 9 0  1 8 3 3 1 ,  9 3 6 4 3  1 8 3 3 5 . 9 6 3 7 5  7 .  5  1 8 3 4 0 ,  9 4 6 8 2  1 8 3 3 1 ,  0 3 4 5 0  1 8 3 3 5 . 6 8 0 8 6  8.  5  1 8 3 4 1 ,  2 4 5 8 7  1 8 3 3 0  0 9 4 7 1  1 8 3 3 5 . 3 6 0 9 8  9 ,  5  1 8 3 4 1 ,  5 0 7 4 3  1 8 3 2 9  1 1 7 3 8  1 8 3 3 5 . 0 0 2 6 0  1 0 .  5  1 8 3 4 1 ,  7 3 1 1 2  1 8 3 2 8  1 0 2 4 2  1 8 3 3 4 . 6 0 7 6 3  1 1 ,  5  1 8 3 4 1 ,  9 1 7 3 7  1 8 3 2 7 ,  0 4 9 7 9  1 8 3 3 4 . 1 7 3 8 9  1 2 .  5  1 8 3 4 2 ,  0 6 5 3 1  1 8 3 2 5 ,  9 5 9 6 3  1 8 3 3 3 . 7 0 3 6 9  1 3 ,  5  1 8 3 4 2 ,  1 7 6 4 0  1 8 3 2 4  8 3 1 7 2  1 8 3 3 3 . 1 9 3 6 1  , 5  1 8 3 4 2 ,  2 4 9 0 8  1 8 3 2 3  6 6 6 5 0  1 5 ,  5  1 8 3 4 2 ,  2 8 4 6 3  1 8 3 2 2  4 6 3 3 7  1 6 ,  5  1 8 3 4 2  2 8 1 9 9  1 8 3 2 1  2 2 2 6 9  1 7 ,  5  1 8 3 4 2  2 4 1 7 3  1 8 3 1 9  9 4 4 3 1  1 8  5  1 8 3 4 2  1 6 4 2 8  1 4  1 9  5  1 8 3 4 2  0 4 9 5 7  1 8 3 1 7 . 2 7 4 8 6  2 0  5  1 8 3 4 1 ,  8 9 5 8 4  1 8 3 1 5 . 8 8 4 3 1  2 1 ,  5  1 8 3 4 1 ,  7 0 5 6 6  1 8 3 1 4 . 4 5 5 8 8  APPENDIX  III,  continued,  J  R BRANCH  22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5 32.5 33.5 34.5 35.5 36.5 37.5 38.5 39.5 40.5 41.5 42.5 43.5 44.5 45.5 46.5 47.5 48.5 49.5 50.5 51.5 52.5 53.5 54.5 55.5 56.5 58.5 59.5  18341.47664 18341.21166 18340.90688 18340.56629 18340.18646 18339.77134 18339.31629 18338.82547 18338.30068 18337.72842 18337.12190 18336.48141 18335.80038 18335.08487 18334.32679 18333.53696 18332.70358 18331.83910 18330.93046 18329.99212 18329.00606 18327.99445 18326.93420 18325.84582 18324.71055 18323.54811 18322.33814 18321.10015  1. 5 2. 5 3. 5 4 .5 5. 5 6. 5 7. 5 8. 5 9. 5 10. 5  000  X/a  133  ( B02) . 10  Q BRANCH  18312.98970 18311.48592 18309.94499 18308.36606 18306.75036 18305.09639 18303.40554 18301.67671 18299.91173 18298.10743 18296.27285 18294.38767 18292.47182 18290.51870 18288.52896 18286.50120 18284.43495 18282.33319 18280.19269 18278.01633 18275.80088 18273.55144 18271.25923 18268.93634 18266.57074 18264.17137 18261.73183 18259.25842 18256.74499 18254.19691 18251.60991 18248.98943 18246.32473 18243.63125 18240.89203 18235.31128 18232.46877  18314.32003 18312.86076  R BRANCH 18291 18291 18292 18292 18293 18293 18293 18294 18294 18294  -  P BRANCH  000  J  000  .12209 .62661 .15390 .57299 .03830 .37246 .77668 .02458 .36731 .51974  -  000  X/a  ( BO2) 10  P BRANCH 18287 18287 18286 18285 18284 18283 18282 18281 18280  .89808 .15484 .35116 .54671 .65681 .79010 .81562 .88789 .82764  Q BRANCH  APPENDIX I I I ,  continued,  000  -  000  J  R BRANCH  P BRANCH  11. 5 12. 5 13. 5 14. 5 15. 5 16. 5 17. 5 18. 5 19. 5 20. 5 21. 5 22. 5 23. 5 24. 5 25. 5 26. c 27. 5 28. 5 29. 5 30. 5 31. 5 32. 5 33. 5 34. 5 35. 5 36. 5 37. 5 38. 5 39. 5 40. 5 41. 5 42. 5 43. 5 44. 5 45. 5 46. 5 47. 5 48. 5 49. 5 50. 5 53. 5 55. 5  18294 .80838 18294 .88672 18295 .10554 18295 .09870 18295 .25066 18295 .16044 18295 .25632 18295 .06901 18295 .10838 18294 .84166 18294 .81376 18294 .46206 18294 .37146 18293 .93316 18293 .78275 18293 .26898 18293 .05563 18292 .45716 18292 .16310 18291 .46238 18291 .13120 18290 .35835 18289 .95270 18289 .08997 18288 .62756 18287 .67616 18287 .15484 18286 .12217 18285 .53228 18284 .41089 18283 .76110 18282 .55645 18281 .84832 18280 .55842 18279 .78551 18278 .41532 18277 .58079 18276 .11368 18275 .20517 18273 .66566 18270 .05443 18267 .25270  18279 .83844 18278 .68261 18277 .64002 18276 .41011 18275 .29724 18273 .98235  010 J  2.5 3.5 4.5 5.5 6.5  134  C^BO?.) . Q BRANCH  18271 .40535 18270 .17082 18268 .67444 18267 .38487 18265 .81011 18264 .45267 18262 .79324 18261 .37320 18259 .62730 18258 .14849 18256 .32731 18254 .78587 18252 .87999 18251 .25890 18249 .25101 18247 .59286 18245 .51312 18243 .78125 18241 .61197 18239 .82386 18237 .56641 18235 .71931 18233 .38107 18231 .4 6657 18229 .03959 18227 .06622 18224 .55674 18222 .52625  -  010  2  Ae  /2  ("B0 ) 2  R BRANCH  P BRANCH  Q BRANCH  18350.18298 18350.66988 18351.11814 18351.52774 18351.89887  18345.69942 18344.90510 18344.07291 18343.20157  18348.00873 18347.87396 18347.70128 18347.48961 18347.23931  APPENDIX I I I ,  continued,  R BRANCH 7.5 8.5 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26.5 27.5 28 29 30 31 32 33 34 35 36 37 38 39 40  18352 18352 18352 18352 18353 18353 18353 18353 18353 18353 18353 18353 18353 18353 18352 18352 18352 18352 18351 18351 18350 18350 18349 18349 18348 18348 18347 18347 18345 18345 18344 18344  .23140 .52594 .78175 .99941 .17795 .31922 .42029 .48557 .50892 .49981 .44429 .36219 .22581 .07407 .85360 .63962 .32789 .06563 .64879 .36478 .81712 .55149 .83049 .61776 .69150 .55036 .39872 .33831 .95438 .97906 .35665 .46797  18341. 18340. 18339. 18338. 18337. 18336. 18334, 18333. 18332. 18331. 18329, 18328, 18326, 18325, 18324, 18322, 18320, 18319, 18317, 18316, 18314, 18312, 18310, 18308, 18307, 18305, 18303, 18301, 18299, 18297, 18295, 18292, 18291,  R BRANCH 18305.70469 18306.23834 18306.72771 18307.18618 18307.59910 18307.98574 18308.32247 18308, 18309, 18309, 18309, 18309, 18309,  88528 12530 47145 57020 66400 68217  - 010 A  n  A s / 2  2  3 / 2  18346.95107 18346.62432 18346.25878 18345.85505 18345.41349  ("B0 )  P BRANCH 18301.72774 18300.93568 18300.09898 18299.23318 18297, 18296, 18295, 18293, 18292, 18290 18289,  Q BRANCH  34399 35750 33255 26934 16813 02774 85014 63234 37870 08370 75585 38189 98142 52691 05737 51918 98730 35812 77829 04447 44343 57831 99575 95833 42946 18723 72987 26226 88621 18582 89568 95904 75484  2  135  ( B0 ) .  2  P BRANCH  010 J 1. 2. 3. 4. 5. 6.5 7.5 8.5 9.5 10.5 12.5 13.5 14.5 15.5  010 - 010  37977 39153 37142 21753 07121 91336 68735  2  Q BRANCH  APPENDIX  J 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5 32.5 33.5 34.5 35.5 36.5 37.5 38.5 40.5 42.5 44.5 46.5 48.5  III,  continued,  010  R BRANCH  010  2  A  136  ( B0 ) . U  3 / 2  2  P BRANCH  18309.71121 18309.64669 18309.60605 18309.45836 18309.33783 18309.10866 18308.92674 18308.62450 18308.36606 18307.97988 18307.65528 18307.18618 18306.79156 18306.24335 18305.78101 18305.12392 18304.61431 18303.89159 18303.29920 18302.48618 18301.83410  Q BRANCH  18288.45611 18285.85356 18284.46506 18283.09986 18281.62802 18280.18323 18278.62995 18277.12429 18275.49919 18273.91529 18272.20694 18270.55909 18268.76695 18267.04948 18265.17905 18263.39438 18261.41523 18259.58375 18257.53922 18255.62575 18253.49190 18251.51918 18247.26264 18242.85814 18238.30372 18233.59862 18228.74824  18300.21843 18298.45354 18296.53790 18292.25536  010  -  -  010  2  S *- S * 2  ( B0 )  J  R BRANCH  P BRANCH  0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5  18383.57888 18384.48523 18384.32099 18385.83061 18384.90336 18387.01652 18385.32638 18388.04336 18385.58974 18388.91017 18385.69212 18389.61836 18385.63546 18390.16691 18385.41761 18390.55660 18385.04041 18390.78690 18384.50105  18382.16834 18380.42447 18381.02593 18378.52114 18379.72572 18376.45873 18378.26662 18374.23676 18376.64763 18371.85497 18374.86998 18369.31390 18372.93299 18366.61293 18370.83706 18363.75181 18368.58201 18360.73156  U  2  Q BRANCH  APPENDIX I I I , J 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5 32.5 33.5 34.5 35.5 36.5 37.5 39.5 41.5 43.5  continued,  R BRANCH 18390.85737 18383.79818 18390.76915 18382.92852 18390.52275 18381.88505 18390.11629 18380.65449 18389.55092 18379.22487 18388.82708 18377.60183 18387.94434 18386.90268 18385.70371 18384.34542 18382.82819 18381.15333 18379.31993  010 J  R BRANCH  0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5  18487.83985 18488.27139 18491.30208 18488.54975 18492.86475 18488.68710 18494.28151 18488.68006 18495.55217 18488.52692 18496.67667 18488.22949 18497.65497 18487.78822 18498.48717 18487.19879 18499.17161 18486.46895 18499.70935 18485.59300 18500.10156 18484.57316 18500.34674 18483.40970  010 - 010 ^  +  - S 2  +  P BRANCH  ( B0 ) . n  2  Q BRANCH  18366.16876 18357.54979 18363.59638 18354.20528 18360.86571 18350.69470 18357.97620 18347.01095 18354.92867 18343.14100 18351.72229 18339.07290 18348.35862 18334.81224 18344.83676 18330.37188 18341.15710 18325.76664 18337.31956 18333.32461 18329.17162 18324.86158  - 010 ^ " - Z 2  +  ( B0 ) U  P BRANCH  2  Q BRANCH 18487.15444  18486.92591 18484.68627 18486.13494 18482.47133 18485.19719 18480.10533 18484.11484 18477.59790 18482.88655 18474.94535 18481.51231 18472.14825 18479.99192 18469.20720 18478.32522 18466.12238 18476.51346 18462.89011 18474.55399 18459.51788 18472.44945 18456.00058. 18470.19823 18452.33992  APPENDIX J 25.5 26.5 27.5 28.5 29.5 30.5 31.5 32.5 33.5 34.5 35.5 36.5 37.5 38.5 38.5 39.5 40.5 42.5 44.5 46.5 48.5 50.5  III,  continued,  010 - 010 Z ~ - 2 2  R BRANCH  J  R BRANCH  24.5 26.5 28.5 30.5 32.5 34.5 36.5 38.5 40.5 42.5 44.5 46.5 48.5 50.5  18351.61420 18350.97012 18350.13219 18349.10556 18347.90297 18346.53884 18345.01437 18343.33230 18341.49452 18339.50102 18337.35215 18335.04893 18332.59017  R BRANCH  18382.33641 18381.04953 18379.64433 18378.11447 18376.44603 18374.62883 18372.65947  U  2  Q BRANCH  18465.25786 18462.56834 18459.73199 18456.74960 18453.62104 18450.34675  18446.92359  -  010  2  2  +  - A5 2  ( B0 ) U  / 2  2  P BRANCH  Q BRANCH  18314.04862 18310.57656 18306.91695 18303.08137 18299.08631 18294.93103 18290.61909 18286.15279 18281.53154 18276.75593 18271.82695 18266.74337  2  J  ( B0 ) .  18467.80110  010 - 010 'Ae/a- ^* 24.5 26.5 28.5 30.5 32.5 34.5 36.5  +  P BRANCH  18500.44493 18482.10265 18500.39628 18480.65110 18500.20027 18479.05490 18499.85694 18477.31922 18499.36699 18475.43696 18498.73135 18473.41613 18497.94458 18471.22481 18471.26849 18497.01873 18468.94292 18466.48765 18463.89286 18461.13139 18458.20703 18455.08600  010  2  ("BOa)  P BRANCH 18347.46258 18343.53583 18339.49217 18335.32464 18331.01914 18326.56594  Q BRANCH  APPENDIX I I I , J  continued,  R BRANCH  38. 5 40. 5 42. 5 44 . 5 46. 5 48. 5 50. 5  18370 18368 18365 18363 18360 18357 18354  Jr .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 .5 .5 .5 .5 .5 .5  R BRANCH 18260 18261 18261 18262 18261 18264 18262 18265 18262 18265 18262 18266 18262 18267 18262 18267 18261 18268 18261 18268 18260 18268 18260 18268 18259 18267 18258 18267 18257 18266 18255 18266 18254 18265 18252 18264 18251 18263 18263 18261  .38587 .60499 .08204 .90518 .63590 .06578 .04647 .08344 .31393 .95864 .43822 .69019 .41950 .27982 .25796 .72307 .95245 .02712 .50412 .18659 .91282 .20272 .17820 .07574 .30041 .80660 .27948 .39195 .11498 .83403 .80735 .13582 .35660 .29151 .76461 .30496 .02484 .15206 .19534 .90522  010 X / a - ^ * 2  P BRANCH 18312.29144 18307.22615 18296.62304  -  010  ^  ~ -'z  ~  ("BO )  P BRANCH 18258 18255 18256 18253 18255 18251 18254 18249 18252 18247 18250 18244  ("BOo) .  Q BRANCH  18321.96287  .53455 .25419 .81963 .22432 .47461 .56949 .51046  010  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 38 40  010 -  18248 18242 18246 18239 18244 18236 18242  .02030 .91507 .82684 .96882 .48357 .87961 .00187 .64855 .37682 .27377 .60945 .75636 .69924 .09625 .64692 .29429 .44914 .34845 .11122  18230 18237 18226 18234 18223 18231 18219 18228 18215 18225  .02970 .00595 .65644 .23965 .14104 .33070 .48317 .27797 .68224 .08205  18221 . 7 4 7 3 6 18218 .26651  2  139  APPENDIX I V .  C a l c u l a t e d ground s t a t e e n e r g y l e v e l s o f wave numbers (cm" ) . 1  J 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27 . 5 28.5 29.5 30.5 31.5 32.5 33.5 34 . 5 35.5 36.5 37.5 38.5 39.5 40.5 41.5 42.5 43.5 44.5 45.5 46.5 47.5 48.5 49.5 50.5 51.5  0.0 1.64312 3.94351 6.90108 10.51594 14.78789 19.71720 25.30340 31.54710 38.44746 46.00544 54 . 2 1 9 6 4 63.09196 72.61983 82.80634 93.64762 105.14821 117.30260 130.11719 143.58434 157.71279 172.49234 187.93452 204.02605 220.78182 238.18489 256.25408 274.96821 294.35065 314.37532 335.07083 356.40548 378.41386 401.05791 424.37894 448.33175 472.96522 498.22614 524.17179 550.74013 577.99771 605.87274 634.44198 663.62293 693.50355 723.98962 755.18131 786.97168 819.47413 852.56792 886.38081  148.25574 149.24895 150.88510 153.21601 156.16108 159.82385 164.07779 169.07233 174.63510 180.96126 187.83279 195.49039 203.67062 212.65943 222.14827 232.46803 243.26537 254.91579 267.02152 280.00226 293.41625 307 . 7 2 6 9 4 322.44904 338.08926 354.11932 371.08862 388.42645 406.72436 425.36977 444 . 9 9 5 7 5 464.94855 485.90204 507.16200 529.44240 552.00929 575.61597 599.48953 624.42181 649.60179 675.85896 702.34506 729.92638 757.71832 786.62300 815.72046 845.94769 876.35033 907.89925 939.60674 972.47646 1005.48843 1039.67802  n  B 0  2  140 in  APPENDIX I V , c o n t i n u e d ,  J  52.5 53.5 54.5 55.5 56.5 57.5 58.5 59.5 60.5 61.5 62.5 63.5 65.5  J 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17 . 5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5 32.5 33.5 34.5 35.5  ground s t a t e  ll  energy  levels.  u  3 / 2  920.77712 955.90010 991.59799 1028.03069 1065.02920 1102.77125 1141.06938 1180.12039 1219.71709 1260.07664 1300.97086 1342.63853 1427.80450  l/2  1073.99409 1109.50260 1145.12238 1181.94881 1218.87188 1257.01519 1295.24113  V 405.54032 405.76869 408.69479 409.22775 414 . 4 9 4 5 2 415.33223 422.93932 424.08208 434.02897 435.47716 447.76318 449.51730 464 . 1 4 1 6 2 466.20225 483.16388 485.53174 504.82952 507.50540 529.13804 532.12285 556.08886 559.38362 585.68138 589.28720 617.91492 621.83301 652.78874 657.02043 690.30205 694.84977 730.45401 735.31728 773.24370 778.42516 818.67016 824.17155  633.22490 634.97648 635.36069 639.44761 640.13931 646.56133 647 . 5 6 0 7 3 656.31745 657.62485 668 . 7 1 5 7 5 670 . 3 3 1 5 4 683.75589 685.68062 701 . 4 3 7 7 5 703.67186 721 . 7 6 0 7 5 724 . 3 0 4 9 8 744 . 7 2 4 5 3 747.57964 770.32863 773.49548 798.57252 802 . 0 5 2 0 4 829.45562 833.24886 862 . 9 7 7 3 0 867.08539 899.13688 903.56104 937.93363 942.67517 979.36674 984 . 4 2 7 0 7 1023.43537 1028.81599 1075.84110  141  APPENDIX I V , c o n t i n u e d ,  36. 37. 38. 39. 40, 41 , 42 43 44 46  J 1 .5 2 .5 3 .5 4 .5 5 .5 6 .5 7 .5 8 .5 9 .5 10 . 5 11 . 5 12 . 5 13 . 5 14 . 5 15 . 5 16 . 5 17 . 5 18 . 5 19 . 5 20 . 5 21 . 5 22 . 5 23 . 5 24 . 5 25 . 5 26 . 5 27 . 5 28 . 5 29 . 5 30 . 5 31 . 5 32 . 5 33 . 5 34 . 5 35 . 5 36 . 5 37 . 5 38 . 5 40 . 5  ground s t a t e energy  866. 872. 917. 923. 970. 977. 1026. 1033. 1085. 1146.  2 a  73276 55551 42921 57607 75955 23216 72217 52270 31579 53907  5/2  4 4 5 . 04417 447 . 33312 450. 32176 4 5 3 . 95006 458. 23803 4 6 3 . 18562 4 6 8 . 79283 4 7 5 . 05962 4 8 1 . 78598 4 8 9 . 57186 497 . 8 1 7 2 3 5 0 6 . 72206 5 1 6 . 28633 5 2 6 . 50992 5 3 7 . 39287 5 4 8 . 93509 5 6 1 . 13654 5 7 3 . 99715 5 8 7 . 51688 6 0 1 . 69567 6 1 6 . 53343 6 3 2 . 03012 6 4 8 . 18566 6 6 4 . 99997 6 8 2 . 47299 7 0 0 . 60463 7 1 9 . 39481 7 3 8 . 84344 7 5 8 . 95044 7 7 9 . 71572 8 0 1 . 13917 8 2 3 . 22072 8 4 5 . 96024 8 6 9 . 35766 8 9 3 . 41284 9 1 8 . 12569 9 4 3 . 49609 9 9 6 . 20909  levels.  1070 . 1 3 8 6 1 1119.47549 1125.50154 1171.44497 1177.79636  *3/2  589 591 593 596 599 604 609 614 621 628 635 644 653 662 672 683 695 707 720 734 748 763 778 795 812 829 847 866 886 906 927 948 971 993 1017 1041  .38959 .04682 .36693 .34991 .99574 .30440 .27587 .91013 .20714 .16686 .78927 .07432 .02196 .63215 .90483 .83996 .43747 .69730 .61939 .20366 .45005 .35847 .92885 .16110 .05513 .64087 .82820 .70704 .24729 .44883 .31156 .83538 .02016 .86578 .37213 .53907  1091.85424 1144.81020  142  APPENDIX  IV,  continued,  J  42.5 44.5 46.5 48.5  ground s t a t e  %>/2  1051.55089 1109.52048 1170.11679 1233.33974  energy  levels. 2 A  A  3/2  1200.40584 1258.63995 1319.51130  143  APPENDIX V .  The upper s t a t e energy l e v e l s numbers (cm" ) .  o f " B O i n wave 2  1  J  1  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  45 46 47 48 49 50 51  .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 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5  ^3/2 18315 18316 18318 18321 18325 18329 18333 18339 18344 18351 18358 18366 18374 18383 18393 18403 18414 18425 18437 18450 18463 18477 18492 18507 18523 18539 18556 18574 18592 18611 18631 18651 18671 18693 18715 18737 18761 18784 18809 18834 18860 18885 18913 18940 18968 18997 19027 19057 19087 19118 19150  .20739 .75632 .92469 .71273 .12055 .14701 .79415 .06015 .94728 .44910 .57847 .31712 .68288 .66170 .26080 .48417 .32062 .78350 .85736 .55842 .87085 .80886 .35623 .53540 .32343 .73520 .75807 .41210 .67061 .56318 .05794 .18571 .91216 .27919 .24134 .84751 .03844 .88225 .30804 .38732 .04497 .36385 .25405 .79787 .93105 .71759 .07120 .09227 .67603 .92691 .74789  18417 18418 18420 18423 18427 18431 18435 18441 18446 18453 18460 18468 18476 18486 18495 18506 18516 18528 18540 18553 18566 18580 18595 18610 18626 18643 18660 18678 18696 18715 18734 18755 18775 18797 18819 18842 18865 18889 18914 18939 18965 18991 19018 19046 19074 19104 19133 19163 19194 19225 19257  .10515 .69236 .83370 .69165 .05597 .18380 .77145 .16995 .98053 .64852 .68243 .62076 .87709 .08442 .56399 .04121 .74418 .48948 .40650 .42675 .57628 .85783 .22785 .77822 .36765 .18658 .00256 .08391 .12416 .48366 .72885 .34484 .82674 .70416 .40821 .55359 .47364 .88257 .02877 .69759 .06433 .99381 .58146 .77504 .58026 .03078 .06148 .77088 .01893 .98750 .45563  APPENDIX V , c o n t i n u e d ,  upper s t a t e  J 1  52, 53, 54 , 55, 5£. 57, 58, 59, 60, 61, 62, 63, 64 , 65,  3/2  19183.24666 19216.28271 19250.01636 19284.27715 19319.25254 19354.73382 19390.93915 19427.64879 19465.09170 19503.02444 19541.69687 19580.86340 19620.76402 19661.14674  ^6/2  2. 3. 4. 5. 7. 8. 9. 10. 11 . 12. 13. 14 . 15. 16. 17. 18. 19. 20. 21. 22. 23. 24, 25. 26, 27, 28, 29, 30. 31, 32, 33, 34 35 36  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  37.5  18793.05272 18795.22703 18798.02300 18801.43972 18810.13680 18815.41710 18821.31857 18827.84120 18834.98537 18842.74981 18851.13645 18860.14230 18869.77172 18880.01881 18890.89253 18902.37921 18914.49851 18927.22277 18940.59087 18954.54928 18969.17300 18984.35804 19000.25150 19016.64893 19033.83860 19051.42175 19069.94637 19088.67399 19108.56846 19128.40759 19149.68864 19170.61968 19193.29868 19215.31177 19239.39164  energy  levels.  %/2  19290.68089 19323.36710 19357.85022 19391.75712 19427.49236 19462.61835 19499.60765 19535.95581  ^3/2  18895.09429 18897.28520 18900.09450 18903.53666 18912.28984 18917.59834 18923.53810 18930.09224 18937.29182 18945.09299 18953.54546 18962.59199 18972.29593 18982.58682 18993.55084 19005.08413 19017.30326 19030.07712 19043.54141 19057.55857 19072.28512 19087.55306 19103.52648 19120.03489 19137.26596 19155.01413 19173.49827 19192.49040 19212.22962 19232.43522 19253.44943 19274.91137 19297.16472 19319.85817 19343.37317  145  APPENDIX V , c o n t i n u e d ,  upper s t a t e  energy  levels.  ^6/2  38.5 39.5 41.5 43.5 45.5 47.5 49.5  J 0 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  .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 .5 .5 .5 .5 .5  19262.48266 19287.96386 19392.54009 19448.83721 19507.01387 19567.95829  A  3/2  19392.07257 19443.26368 19496.94353 19553.10974 19611.76657  4  18787 . 93703 18789 . 11923 1 8 7 9 0 . 25308 1 8 7 9 3 . 01572 1 8 7 9 5 . 05815 1 8 7 9 9 . 39797 1 8 8 0 8 . 26562 1 8 8 1 2 . 12511 1 8 8 1 9 . 61843 18824 . 38730 18833 . 45541 1 8 8 3 9 . 13545 1 8 8 4 9 . 77694 1 8 8 5 6 . 36898 1 8 8 6 8 . 58141 18876. 08819 1 8 8 8 9 . 86976 1 8 8 9 8 . 46196 1 8 9 1 3 . 63887 1 8 9 2 2 . 98011 1 8 9 3 9 . 88685 1 8 9 5 0 . 15284 1 8 9 6 8 . 60976 1 8 9 7 9 . 80958 1 8 9 9 9 . 79965 1 9 0 1 1 . 94920 1 9 0 3 3 . 44264 1 9 0 4 6 . 57120 1 9 0 6 9 . 52682 1 9 0 8 3 . 67588 1 9 1 0 8 . 05595 1 9 1 2 3 . 26177 1 9 1 4 9 . 04299 1 9 1 6 5 . 32825 1 9 1 9 2 . 49910 1 9 2 0 9 . 87517 1 9 2 3 8 . 42762 1 9 2 5 6 . 90081  18892 18893 18895 18896 18900 18903 18911 18918 18922 18931 18936 18946 18952 18963 18970 18984 18992 19006 19015 19031 19041 19059 19070 19089 19101 19122 19134 19157 19170 19195 19209 19235 19250 19277 19294 19322 19340 19373 19388  .69466 .38074 .36257 .96597 .52976 .04441 .62663 .36388 .70855 .02959 .29005 .19411 .37103 .85711 .95190 .01883 .02832 .67698 .60680 .83267 .68196 .48532 .25474 .63390 .32470 .27796 .89170 .41675 .95309 .04897 .50909 .17436 .56305 .79224 .10744 .90279 .14727 .50003 .65381  146  APPENDIX V ,  continued,  J 39 40 41 42 43 44 45 47  .5 .5 .5 .5 .5 .5 .5 .5  upper s t a t e  2  Z  19286. 19306. 19337. 19358. 19391. 19412. 19446. 19505.  +  82833 40402 70318 38488 05375 84263 87260 16551  energy  levels.  2  S"  19388.69729 19420.58650 19439.70202  147  

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