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Measurement of relative transition probabilities in argon using the Faraday effect Stockmayer, Philip Henry 1969

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MEASUREMENT OF RELATIVE TRANSITION PROBABILITIES IN ARGON USING T H E F A R A D A Y E F F E C T by PHILIP HENRY S T O C K M A Y E R B.Sc. , University of British Columbia, 1967 THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E REQUIREMENTS  FOR T H E DEGREE O F  M A S T E R O F SCIENCE in the Department of Physics  We accept this thesis as conforming to the required standard  T H E UNIVERSITY  O F BRITISH COLUMBIA  April,  1969  In  presenting  an  advanced  the  Library  I further for  this degree shall  agree  scholarly  by  his  of  this  written  thesis  in partial  f u l f i l m e n t of  at  University  of  the  make that  i t freely  permission  purposes  representatives. thesis  for  be  may It  financial  available for  of  PHYS/CS  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  by  the  is understood gain  Columbia  for  extensive  granted  permission.  Department  British  shall  the  requirements  Columbia, reference  copying Head  of  and  of my  I agree  that  Study.  this  thesis  Department  that  copying  or  be  allowed  without  not  for  or  publication my  -iiAbstract Relative  transition  probabilities  c o n n e c t i n g t h e 4 s and 4 p s t a t e s d e t e r m i n e d by a r e l a t i v e l y and C u r z o n i n v o l v i n g given  spectral  line  for  of neutral  some s p e c t r a l  lines  a r g o n have  been  new t e c h n i q u e d e v e l o p e d by  Seka  the F a r a d a y emitted  effect.  In  this  method a  by an a r g o n d i s c h a r g e  tube  is  p l a n e p o l a r i z e d and t h e n p a s s e d t h r o u g h an a b s o r p t i o n  tube  embedded i n an a x i a l  of  the  line  are  of  magnetic  p o l a r i z a t i o n , and t h e  m e a s u r e d and f r o m t h e that p a r t i c u l a r used agrees field  line  particularly  the  suited  transition  The r o t a t i o n  of  the  source  absorption coefficient  plasma i s  deduced.  The  This  technique  to strong t r a n s i t i o n s methods.  probabilities  and a g r e e w e l l w i t h v a l u e s  The v a l u e s are q u i t e  which  for  for is  are  difficult  obtained here  accurate  plane  theory  w i t h the o b s e r v e d r o t a t i o n s  f r o m 0 t o 2600 g a u s s .  t o measure by o t h e r the  the  excellently  values  intensity  results in  field.  for  (errors  g i v e n by Weise i n a r e c e n t  < 10%)  compil-  ation. For  the a n a l y s i s  to determine density  of  of  the e f f e c t  lower  the Faraday of  of which first  show a l i n e a r  2000 g a u s s .  the magnetic  excited states  was done u s i n g a r e l a t i v e  rotation  in  field  it  on t h e  necessary number  the a b s o r b i n g plasma.  absorption technique,  decrease  was  the  This  results  i n t h e number d e n s i t i e s  in  the  -iii TABLE  OF  CONTENTS  Chapter  Page Abstract Table of Contents L i s t of F i g u r e s L i s t of .Tables L i s t of R e f e r e n c e d Equations Acknowledgements  ii iii v vi vi vii  I  Introduction  1  II  Theory A. F a r a d a y Rotation B. Relative T r a n s i t i o n P r o b a b i l i t i e s C. Relative A b s o r p t i o n D. S e l f - A b s o r p t i o n Coefficient  7 7 12 12 13  III  Apparatus and P r o c e d u r e A. F a r a d a y Rotation Apparatus (1) Source and A b s o r b e r (2) Magnet (.3) P o l a r i z e r s (4) M o n o c h r o m a t o r and B e a m Divergence (5) P h o t o m u l t i p l i e r and Detection System B. P r o c e d u r e f o r M e a s u r i n g F a r a d a y Rotation C. Relative A b s o r p t i o n Apparatus and P r o c e d u r e D. R a d i a l Gradient of N i n the A b s o r b e r  15 15 15 17 17 18 18 20 20 25  IV..  A n a l y s i s and. Results A. F a r a d a y Rotation Data Reduction B. A Check F o r a R a d i a l Gradient i n N C. A n a l y s i s of Relative A b s o r p t i o n D. C u r v e Fitting E. S e l f - A b s o r p t i o n and Relative Populations i n the Source  27 27 28 29 34  D i s c u s s i o n and Conclusion. A. E x p e r i m e n t a l E r r o r s (1) F a r a d a y Rotation (a) Random E r r o r s (b) Systematic E r r o r s (c) A c c u r a c y of the ^ (2) Relative A b s o r p t i o n B. E v a l u a t i o n of Technique  41 41 41 41 42 46 48 53  V  C.  Conclusion  r  e  i  s  37  55  -iv TABLE  OF  CONTENTS  (continued)  Chapter  Page Bibliography Appendices I. Calculations Involving the Use of a D i g i t a l Computer A. C a l c u l a t i o n of k l B y Curve F i t t i n g to F a r a d a y Rotation Data B. C a l c u l a t i o n of k l F r o m Relative Absorption Data II. E f f e c t on Anomalous D i s p e r s i o n of Colli.sional Broadening of the Absorption Line  57 58 58  Q  58  Q  61 63  -V-  LIST OF  FIGURES  Figure  Page  1  Partial  Term  D i a g r a m of N e u t r a l A r g o n  2  Absorption, D i s p e r s i o n , and E m i s s i o n P r o f i l e s f o r a Normal  2  Zeeman L i n e  3  3  E x p e r i m e n t a l Geometry  4  B l o c k D i a g r a m of E x p e r i m e n t a l A r r a n g e m e n t f o r Faraday  8  Rotation M e a s u r e m e n t s  16  Tube  16  5  A b s o r p t i o n Discharge  6  Dynode Voltage D i v i d e r of P h o t o m u l t i p l i e r Tube  19  7  Typical Recorder  21  8  B l o c k D i a g r a m of E x p e r i m e n t a l A r r a n g e m e n t for Relative A b s o r p t i o n M e a s u r e m e n t s R e c o r d e r T r a c e s of Transmittance V e r s u s H for Line at 7515 A* Deviations i n Qe.*p for L i n e at 8115 A Due to a R a d i a l Gradient "of N-. ;  29  P l o t s of Relative A b s o r p t i o n Coefficients to Determine Magnetic F i e l d E f f e c t on N  32  9 10  11  12  Traces'for Faraday  Rotation  23  a  23  T y p i c a l E x a m p l e s of Qexp and the T h e o r e t i c a l Best F i t s  36  13  P l o t of -^-/U  V e r s u s k l and T r a n s m i t t a n c e  Al  Flow C h a r t of C u r v e Fitting C a l c u l a t i o n  59  A2  F l o w C h a r t of Relative A b s o r p t i o n  62  A3  The E f f e c t of C o l l i s i o n a l Broadening Dispersion  o  on  50  Anomalous 65  -vi LIST O F  TABLES  Table  Page  1  F i n a l Values f o r y  33  2  C o m p a r i s o n of E x p e r i m e n t a l A b s o r p t i o n Coefficients  34  3  E x p e r i m e n t a l Values of A  38  4  F i n a l V a l u e s for JJL  39  5  Analysis  49  r e  ^  of the S o u r c e s of E r r o r  LIST O F R E F E R E N C E D  Equations  EQUATIONS  Page  2.8, 2.9  9  2. 13, 2. 16  10  2. 18, 2.20, 2.21  II  2.23, 2.24  12  2.26, 2.27, 2.29  13  2.30  14  3.1, 3.2, 3.5  24  5. 8  49  - V l l  -  A cknowledgement  I would Curzon, work.  like to thank my  supervisors,  for their guidance and patience S p e c i a l thanks i s also due to Mr.  helpful discussions.  Dr.  W.  Seka and Dr.  F.  throughout the course of thi J. C.  Burnett  for many  -1-  CHAPTER  I  INTRODUCTION  In the branches of physics where spectroscopic analysis i s used, knowledge of atomic t r a n s i t i o n p r o b a b i l i t i e s i s most valuable.  For instance, i n astrophysics, determinations  of  electron density and temperature, number density of atoms and degree of ionization, and the abundance of elements i n a s t e l l a r atmosphere a l l depend e x p l i c i t l y on r e l i a b l e values for the t r a n s i t i o n p r o b a b i l i t i e s .  A similar dependence occurs  in plasma physics, where spectroscopic analysis i s a major diagnostic t o o l , and i n laser research, where emphasis i s placed on e x c i t a t i o n processes. Theoretical c a l c u l a t i o n s , using approximate wave-functions, y i e l d doubtful values.  Experimental  determinations  show large  deviations dependent on the measuring technique employed, and u n t i l these show considerably more o v e r a l l consistency, their r e l i a b i l i t y w i l l also be i n doubt. The present experiment measures r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s of some spectral l i n e s occuring between the 4s and 4p states of neutral argon by a r e l a t i v e l y new involving the Faraday e f f e c t .  technique  In t h i s method, plane polarized  l i g h t from an argon source i s transmitted through an  absorbing  argon plasma which i s embedded i n a longitudinal magnetic f i e l d . The l i g h t coming from a given l i n e i s then isolated with a  -2Pi iL  S  Noieitlon  S„  Zf>,  ft  1. •Zpc  'AIn  T  St  to  ft:  6ft  OS  6t>  On  *  OK  SM OS  •2p«  N Oo o  IV  N  No  5 0*  8  N  Oo  5*  <0  ls 3-r)"  t  Is.  3p«,  * denotes lines f o r which r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s were found i n t h i s experiment  Figure 1  P a r t i a l Term Diagram of Neutral Argon  -3-  monochromator.  The a b s o r p t i o n p r o f i l e  by t h e Zeeman e f f e c t  and i s  different  for for  left  and  Because  the anomalous d i s p e r s i o n a s s o c i a t e d w i t h  circularly in  this  suffer  the  polarized incident  region. a net  circularly X is  the  the wavelength  magnetic  which  field,  of  is  H . a  called  1 is  for  the  lines  light  the  line,  the  field.  are  and  left  absorber,  the  axial  T h i s net  due t o t h e Zeeman  rotation,  same a v e r a g e  and  X  f r e q u e n c y as  the  split  incident  effect. to the c o r r e s p o n d i n g a b s o r p t i o n  transition  and f  and s i n c e k  is  t h e number d e n s i t y  of  is  then the r e l a t i v e  The t h e o r y  for  the  oscillator  transition  o r i g i n a t i n g f r o m t h e same l o w e r  The r o t a t i o n s they  will  strength of  the  discussed i n Chapter  since  right  k± by t h e K r a m e r s - K r o n i g r e l a t i o n  s t a t e bf  magnetic  line  d e p e n d e n t upon t h e  d e t e r m i n e d by s t u d y i n g t h e r o t a t i o n of  for  length of  p r o p o r t i o n a l t o N f l where N i s  strength of  different  . . . l . i  incident  S i n c e nj. c a n be r e l a t e d  lower  right  rotation  the Faraday  the  given  and  the  line,  directly  left  is  the  the  of  coefficients  an  a l s o be  from the  i n d i c e s of r e f r a c t i o n  absorption line is  2).  ^  ~  polarized light,  in radians,  light  for  split  rotation,  n_ a r e  +  radiation will  Thus i n c i d e n t  A where n ,  Figure  index of r e f r a c t i o n  is  right  polarized incident  absorption resonance,  (see  line  circularly of  radiation  this  probabilities  s t a t e may be  f o r each  line  the experiment  as a is  function  further  II.  s t u d i e d are q u i t e  measured f o r  values  large  of H  a  (up t o ^ ~ 2 7 0 ° )  and  r a n g i n g from z e r o  u p t o s t r e n g t h s where t h e Zeeman components a r e  d i s p l a c e d by  -5-  several Doppler half-widths, the amount of data that can be correlated i s quite large. This technique was o r i g i n a l l y developed by Seka and Curzon who used i t to determine r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s i n neon.  Using neon glow discharges as source and absorber the  polarization of o r i g i n a l l y plane polarized l i g h t transmitted through the absorber was analysed as a function of longitudinal magnetic f i e l d on the absorber.  A r a d i a l gradient of N  and a linear decrease of N as a function of magnetic were assumed.  field  By using least squares curve f i t t i n g i n several  parameters, among which was the peak absorption c o e f f i c i e n t , k l , good agreement was found between theory and experiment Q  when a l l the Zeeman components of the absorption l i n e were displaced beyond the half-width of the source l i n e .  Although  in natural neon the analysis i s complicated by the presence of two isotopes i t was found that the rotation was i n s e n s i t i v e to a l l parameters other than N f l . The present experiment uses the same general data gathering and analysis techniques.  The experimental arrange-  ment i s described i n d e t a i l i n Chapter I I I .  Modifications  which were made to the technique of Seka and Curzon include the measurement of the r a d i a l gradient of N, which, for the . argon discharge used, was found to be very small and to have a n e g l i g i b l e e f f e c t on the r e s u l t s , the measurement of the decrease i n N as a function of magnetic f i e l d by a r e l a t i v e absorption technique, and the change of the method of numerical integration used i n c a l c u l a t i n g the t h e o r e t i c a l curve from Gauss-Hermite to Simpson's r u l e .  The l a t e r modification  -6e n a b l e d the field  theoretical  on t h e a b s o r b e r .  absorption in the  curve  Evidence  of r e l a t i v e l y  t h e s o u r c e was a l s o  strong  transition agreement  of  self-  f o u n d and a c c o u n t e d f o r  the data which  probabilities  is  o b t a i n e d between  transition  probabilities  within  the quoted e r r o r  the  recent  compilation.  The u s e o f strongly  this  in  level  accuracy  of  good and t h e v a l u e s (see  the v a l u e s  the  is  T a b l e 3)  the measurement,  ment b e y o n d t h e h a l f - w i d t h  of  large,  of  individual  the  increasing  and s e c o n d l y , the s o u r c e  last  chapter  is  the  relative  f o r Zeeman d i s p l a c e -  line  the r o t a t i o n  and  line  further  shapes.  assessment of  work.  is  their  presented i n Chapter V.  for  a  absorption  summarizes the accomplishments o f  e x p e r i m e n t and p r o p o s e s a r e a s  a  for  decreased to  a s w e l l a s an o v e r a l l  t e c h n i q u e compared t o o t h e r s  relative  well  advantages:  the e x p e r i m e n t a l e r r o r s  importance,  the  the  were  o n l y w e a k l y d e p e n d e n t upon t h e s o u r c e and a b s o r b e r A d i s c u s s i o n of  The  o b t a i n e d by W i e s e ^ i n  absorption is  quite  IV.  model a n d  by t h e Zeeman s p l i t t i n g o f  and t h e r o t a t i o n  relative  of Chapter  method h a s two m a j o r  absorbing lines  reasonable  t o the  theoretical  obtained of  leads  the s u b j e c t  e x p e r i m e n t a l d a t a was q u i t e  this  zero  analysis. The a n a l y s i s  line  t o be c a l c u l a t e d down t o  the  the  Also  -7-  CHAPTER  II  THEORY A.  Faraday  Rotation  The t h e o r y developed i n t o rotation. wave o f  indices  the  Faraday  a theoretical  effect  s t r e n g t h E(V>)  field, of  H , a  which  n  +  respectively  due t o  field)  ly  circularly  strength  k  +  and k_.  £"+ = where x and  E(v>)  unit  vectors  gives  the  analysing the  circularly  ( w h i c h have d i f f e r e n t  values  +  and E_,  into  two  each w i t h a  oppositefield  x  = " H r ^ f e -(-(f) . . . 2-1  coordinates  (see  t h e x-component o f  of  Figure  a 3).  E_ and Im(E + ) =  y-component.  After  of  = E  and l e f t  has  a l o n g t h e x- and y - a x e s  Note Ey  right  The p l a s m a  axial  that:  system of  Re(E+)  plane  p r o p a g a t e d a l o n g an  (-2L * i # ) , E-  are  observed  polarized  c a n be d i v i d e d  right-handed Cartesian that  is  and  and c o r r e s p o n d i n g a b s o r p t i o n  p o l a r i z e d beams, E  o f E(L>)/2 s o  the  linearly  and n _ f o r  polarized radiation  coefficients  presented here  embedded i n a p l a s m a .  refraction  the magnetic  is  expression for  C o n s i d e r a monchromatic  field  magnetic  of  travelling nicol  incident  becomes:  set  at  beam,  through length an a n g l e Sl_ t o the  right  1 of the  circularly  p l a s m a and an plane  of  polarization  p o l a r i z e d component  -8-  column  Figure 3  =  Experimental Geometry  em  j,- ¥x+  -'  i(Jl  ...2.2  e)  where £ i s the angle through which the f i e l d d i r e c t i o n has rotated i n length 1 and i s given by:  C  . . . 2.3  c  where V* i s the frequency of the incident  l i g h t , }? i s the 0  frequency of the emission l i n e centre, and c i s the speed of light.  S i m i l a r l y the l e f t c i r c u l a r l y polarized  where  corresponds to £ and i s given by: "l  ~  C  =  beam becomes:  ... 2.5  Thus the average intensity of the l i g h t of frequency j-> coming  -9-  through the analysing n i c o l i s :  =  —+-IJ.  -—^rb  +  +-  1  £  1  +• £  +2*-  (/  cert,  ft  sj  ...  (ZSL+f)J  2.6  where *p i s the phase angle given by:  To compare theory and experiment the above expression for must be integrated over a l l frequencies of the incident beam.  We assume that the incident l i g h t comes from a homogen-  eous source with plane p a r a l l e l geometry and that only the Doppler broadening mechanism i s s i g n i f i c a n t  so that the l i n e  p r o f i l e , considering self^absorption, i s given by^:  £ (») - B ( l - t - ^ ' ^ )  ... 2.8  X  where B i s an intensity constant, K a self-absorption coeff i c i e n t and ui i s a scaled frequency given by: ~  W  where All  ( ^AJ£)  _  2  9  i s the Doppler half-width of the source l i n e given by: =  XV.  ,  -  ...  2.10  k i s the Boltzmann's constant, m^ the mass of the argon atom and T  s  i s the absolute temperature of the emitting gas.  Thus the integrated intensity of the l i g h t over a p a r t i c u l a r spectral l i n e becomes:  -10-  ... 2.11 To eliminate B from t h i s equation we repeat the measurement of the t o t a l intensity with the analysing n i c o l removed. results i n an observed  This  intensity of:  •too  /,  -KJt~ \f-JkJL / -JkJL u,x  oo  -A_l )) .  ' t o o  ... 2.12  By eliminating B the measured quantity becomes:  ... 2.13 To complete the d e f i n i t i o n of Q we require expressions f o r k and n*. Different  temperatures are assumed f o r the source  +  and absorber gases and the r a t i o of the source temperature to the absorber temperature i s given by R.  Also the absorber  l i n e i s considered to be only Doppler broadened with h a l f width:  ^ o  For the normal Zeeman e f f e c t we get: Jk  ±  Jsxfi [-R(ouT  A,  =  6)*]  ...2.15  with the maximum absorption c o e f f i c i e n t when Doppler broadening alone i s present, k , (a purely ideal quantity) being Q  given by : 4  K  ~  1^TT~  T%  ~*nc~  /  V  ... 2.16  A which assumes the population density of the upper state of the t r a n s i t i o n i s negligible compared to N, the density of atoms in the lower state of the u n s p l i t absorption l i n e :  where e i s  -lithe electronic charge, ro the electronic mass and f the o s c i l l a t o r strength of the t r a n s i t i o n .  The parameter £ i s the r a t i o  of the normal Zeeman s p l i t t i n g to the Doppler width of the source, i . e . =  S  ^  ... 2.17  where a = 1.3996x10® Hz gauss" a x i a l magnetic  1  and H  a  i s the strength of the  field.  The corresponding expressions for n+• are derived from the Kramers-Kronig relations by Fork and B r a d l e y and are given by: 5  1+. - I- &h. r where F(w)  =£,''  This expression assumes Doppler  J£*oLy-.  broadened l i n e s only i n the absorber. =  r  Thus:  ( f i r ( w - s ) ) - r ( / w ( ^ s f j .  ...  2.19  The anomalous Zeeman e f f e c t can be taken into account quite e a s i l y as follows.  I f , f o r a certain l i n e , there are n  pairs of Zeeman components corresponding to Am = ± 1  g  transitions  (m = magnetic quantum number), and the s-th component i s displaced from the central frequency by an amount a A.V S/2/In s  0  2  and has a r e l a t i v e intensity B , then: s' A  9  ±  =  A|  = 4^  5  2  A  Jtpf-PdoToijS)'-]  ...2.20  ( « > - * > * ) ) - r ( & * ) ) ] . . . .  The r e l a t i v e s p l i t t i n g factors, a , s  used here were  2.2i  calculated  from the g values given by Moore and the r e l a t i v e i n t e n s i t i e s , B , were computed i n the usual manner (see Condon and Shortly?). 6  -12B. R e l a t i v e T r a n s i t i o n P r o b a b i l i t i e s I t can be seen from s e c t i o n A above t h a t Q expressed  i n terms of H  f u n c t i o n of H probability  and k l .  &  Q  Thus p l o t s of Q  enable k l to be determined.  a  may  The  Q  be as a  transition  (the E i n s t e i n A c o e f f i c i e n t ) f o r a given  transition  i s r e l a t e d t o the o s c i l l a t o r s t r e n g t h , f , as f o l l o w s : 4  where Z\  a  n  g2  d  a  r  e  t n e  s t a t i s t i c a l weights of the lower and  upper s t a t e s r e s p e c t i v e l y . l i n e s (designated by  I f the k l ' s of two Q  * and ") o r i g i n a t i n g from the same lower  s t a t e are known, then the r e l a t i v e t r a n s i t i o n ( h e r e a f t e r g i v e n by A  -  A  JL  =  absorption  r e l  ) may  be found as f o l l o w s :  g/tt'*^  A" % C. R e l a t i v e A b s o r p t i o n  U J ) '  vAV; (A £)»  N R E L  probability  A  •  0  ...  2.23  The magnetic f i e l d a f f e c t s the number d e n s i t y of atoms i n e x c i t e d s t a t e s i n the d i s c h a r g e tube (even at constant charge c u r r e n t ) making k l dependent upon H Q  and  a  thereby i n f l u e n c i n g the Faraday r o t a t i o n .  (see equation This effect  be measured u s i n g the f o l l o w i n g r e l a t i v e a b s o r p t i o n The a b s o r p t i o n of the source  dis2.16) can  technique.  l i n e by the absorber with  both  d i s c h a r g e tubes i n i d e n t i c a l magnetic f i e l d s i s measured as a f u n c t i o n of the magnetic f i e l d . r" 1  The - A  9 c  t r a n s m i t t a n c e i s given by: 0  £  Ab.(^)  — D O  where I absorber  s a  i s the observed  t r a n s m i t t e d i n t e n s i t y with  d i s c h a r g e turned on,  i s the source p r o f i l e g i v e n  I by:  T  the  with i t turned o f f , and  S(u)  -13S(W)  = / ~  ... 2.25  [~ K S'(co)]  where S'(u) i s the p r o f i l e the source would have i f s e l f a b s o r p t i o n were not p r e s e n t . S*(«)  Because o f the Zeeman e f f e c t  i s g i v e n by: S ' m  = Z,  £-/"-«sS) 7  ... 2 . 2 6  x  and s i m i l a r l y the a b s o r p t i o n p r o f i l e Ab.(w) i s :  where a l l symbols a r e as p r e v i o u s l y d e f i n e d .  Note t h a t the  assumption i s made that the i n d i v i d u a l components a r e Doppler broadened o n l y .  The e x p r e s s i o n g i v e n f o r T accounts  f o r only  one sense o f c i r c u l a r p o l a r i z a t i o n s i n c e the l e f t and r i g h t c i r c u l a r l y p o l a r i z e d r a d i a t i o n does not i n t e r a c t and the a d d i t i o n o f the other sense would not change the r e s u l t i n g value f o r T. If T i s known as a f u n c t i o n o f H as a f u n c t i o n o f H . a  Since k  then a r e l a t i v e change i n k a change i n N. at  Q  Q  then k  a  Q  l can be found  l i s linearly proportional to N  l due t o H  a  can be i n t e r p r e t e d as  Thus the r a t i o o f N a t magnetic f i e l d H  z e r o magnetic f i e l d  M  0  to N  a  i s g i v e n by:  (AJ)  •••  0  2  *  2 8  where the s u b s c r i p t s 0 and H r e f e r t o the value a t z e r o magnetic f i e l d  and a t magnetic f i e l d H  a  respectively.  D. S e l f - A b s o r p t i o n C o e f f i c i e n t The  s e l f - a b s o r p t i o n c o e f f i c i e n t f o r the source, K, may be  estimated by the f o l l o w i n g technique.  The k  Q  l value f o r the  -14absorber i s found q u i t e a c c u r a t e l y u s i n g the Faraday method.  rotation  Using t h i s value and the e x p r e s s i o n , _  1  Jjt  (/-Jt  „  Z  etc .UJ  which i s the t r a n s m i t t a n c e a t z e r o f i e l d on the source and absorber, a value f o r K may be found. Since the s e l f - a b s o r p t i o n c o e f f i c i e n t  i s proportional  to Nf i n the source and s i n c e , f o r the same l i n e , the o s c i l l a t o r s t r e n g t h i s the same i n the source and absorber, then:  '*  =  l»Uo > l  absorber  2  '  3  0  and the m o d i f i e d s e l f - a b s o r p t i o n c o e f f i c i e n t ,  ji, w i l l be the  same f o r a l l l i n e s coming from the same lower  state.  -15-  CHAPTER I I I APPARATUS AND PROCEDURE A. Faraday Rotation Apparatus A block diagram of the experimental arrangement i s shown in Figure 4.  The source and absorber are glow discharge tubes  both f i l l e d with argon gas.  The l i g h t from the source i s l i n -  early polarized by n i c o l prism (NjJ and traverses the absorption tube which i s completely immersed i n the homogeneous, a x i a l magnetic f i e l d of the solenoid (M).  This f i e l d causes the  absorbing plasma to become o p t i c a l l y active and gives r i s e to the  Faraday e f f e c t .  The beam of l i g h t then passes through the  analysing n i c o l (Ng) and into the monochromator where i t s intensity within a certain frequency range i s measured by a photomultiplier tube.  Apart from the monochromator and beam  chopper, a l l components were mounted on a heavy o p t i c a l bench. (1) Source and Absorber A commercial argon Geissler tube (Cenco spectra tube) with a f i l l pressure of about 10 mm.  Hg served as a source.  absorber tube was s p e c i a l l y constructed 5.  The  and i s shown i n Figure  To minimize impurities i n the absorbing gas this tube was  baked f o r 6 hrs. at 400°C. before being f i l l e d to 2 mm.  Hg with  research grade argon obtained from Airco. The discharge was noted to be uniform along the axis and stable except near the points A and B (Figure 5) where some unstable bright spots occurred.  The intensity fluctuations  -16-  s.r L, C  N,  A  Ref fence  M  A  z  A/ L,. z  Siqr-a-l  S.T. - source discharge tube: L^, L 2 •- l e n s e s : Nj., N 2 - n i c o l prisms: C - r o t a t i n g s l o t t e d disk (chopper): A i , A 2 - 4 mm. diameter apertures: M - solenoid electromagnet: S - shunt: V - voltmeter: P.M. - p h o t o i n u l t i p l i e r t u b e . Figure  4  B l o c k Diagram of E x p e r i m e n t a l Arrangement Rotation  for  Faraday  Measurements  44-cm?  constructed of  pyrex  glass  K - kovar e l e c t r o d e s : W - o p t i c a l l y f l a t p y r e x windows: A , B - p o i n t s where f l u c t u a t i n g b r i g h t s p o t s o c c u r r e d i n the d i s c h a r g e Figure  5  A b s o r p t i o n D i s c h a r g e Tube  -17produced by these spots were c o n s i d e r a b l y reduced by running the tube a t o p e r a t i n g c u r r e n t f o r s e v e r a l days.  Moreover,  spots were o u t s i d e the p o r t i o n t r a v e r s e d by the beam. were operated  these  Both tubes  by a c u r r e n t r e g u l a t e d d-c power supply and were  run a t 20 ma. f o r the G e i s s l e r tube and 10 ma. f o r the absorber tube. (2) Magnet The capable  magnet was a home-made, water-cooled of producing  s o l e n o i d which was  a maximum f i e l d of about 2600 gauss a l o n g  the a x i s when s u p p l i e d with about 35 amps, from a d-c c u r r e n t r e g u l a t e d power supply.  The output  v a r i e d manually i n a continuous  from t h i s supply c o u l d be  manner up t o 35 amps.  A water-  cooled shunt was i n s e r t e d i n one of the c u r r e n t leads to the magnet and a s e n s i t i v e voltmeter voltmeter  was placed a c r o s s i t .  was c a l i b r a t e d i n terms of magnetic f i e l d  This  intensity  by a H a l l probe gaussmeter ( B e l l "240" Incremental Gaussmeter) which was placed on the a x i s of the s o l e n o i d .  During  the a c t u a l  experiment the v o l t a g e across the shunt was used as the X-input of an X-Y r e c o r d e r which was used t o c o l l e c t p r e v i o u s l y c a l i b r a t e d voltmeter of a r e c o r d e r The  the data.  The  was used to c a l i b r a t e the X - a x i s  t r a c e i n terms of the magnetic  field.  homogeneity of the f i e l d along the 10 cm. p o r t i o n of  the a x i s where a b s o r p t i o n occurs was w i t h i n £ 1 % .  To h e l p i n  a c h i e v i n g t h i s degree of homogeneity s m a l l kovar c o l l a r s were placed near the ends of the s o l e n o i d core to r e t a r d the d i v e r gence of the f i e l d  lines i n t h i s region.  (3) P o l a r i z e r s Two n i c o l prisms were used as p o l a r i z e r and a n a l y s e r  -18because of their excellent polarization and transmission quali t i e s . The analyser (Ng) was permanently aligned with the monochromator so that the l i g h t output was a maximum.  This was  necessary because the monochromator i t s e l f strongly polarizes incident l i g h t .  The polarizer (Nj) was aligned i n the crossed  position with respect to N  2  by observing when the transmitted  intensity of a He-Ne CW laser beam was at a minimum. The mounting of  was supplied with machined notches so that i t  could also be positioned accurately at 0° with respect to N . 2  A glass plate of proper thickness was mounted at the Brewster angle i n front of and permanently attached to beam would not be displaced when  so that the  was rotated.  (4) Monochromator and Beam Divergence A Spex monochromator (f/10, 10 A/mm.) was used to i s o l a t e i n d i v i d u a l spectral l i n e s .  S l i t widths of about 200 (im. were  chosen, which permitted a l l the l i g h t from one l i n e to be transmitted without any contribution from neighbouring  lines.  The apertures used limited the beam to a diameter of 4 mm. with a maximum divergence of about 0.7°.  This divergence was  small enough to consider the beam p a r a l l e l . (5) Photomultiplier and Detection System A dry i c e cooled P h i l l i p ' s CVP 150 photomultiplier tube (henceforth denoted by P.M.) was used i n order to obtain reasonable response over the range of wavelengths investigated (6965 - 10470 A ) . The dynode voltage divider was as i n Figure 6 and the cathode voltage used ranged between 1000 and 1600 volts depending on the intensity of the s p e c t r a l l i n e being studied. The P.M. was equipped with a mu-metal magnetic s h i e l d because  -19-  S  S,  g  S  S„ a.  9  Amplifier  r C  IM *7K  100 K  47 K  4-7K  4-7K  47K  47K  S6K  -HT  6SK  91K  ApF  .IjnF •If/F j  /POK  k - photocathode: acc - a c c e l e r a t i n g e l e c t r o d e : S -j- dynode number n: a - anode: HT - t o t a l applied voltage. Q  Figure 6  Dynode Voltage D i v i d e r of P h o t o m u l t i p l i e r Tube  of the p r o x i m i t y of the absorber The output  magnet.  s i g n a l from the P.M.  contained  not o n l y the  i n t e n s i t y of the t r a n s m i t t e d beam i n a p a r t i c u l a r  frequency  range but a l s o a noise s i g n a l o r i g i n a t i o n from the absorber which emits the same b a s i c f r e q u e n c i e s s p l i t t i n g s ) as the source.  (apart from s m a l l Zeeman  In order t o d e t e c t the source  s i g n a l , the l i g h t from the source was modulated a t 990 cps. with a chopper ( r o t a t i n g s l o t t e d d i s c (C) i n F i g u r e 4 ) .  L i g h t from  a bulb passing through the chopper and f a l l i n g on a s m a l l photodiode provided a r e f e r e n c e s i g n a l which, along with the P.M.  output  was f e d i n t o a l o c k - i n a m p l i f i e r .  s i g n a l of the l o c k - i n a m p l i f i e r provided X-Y r e c o r d e r .  T h i s output  The d-c  the Y-input  output  of the  was a p p r o p r i a t e l y f i l t e r e d by an RC  c i r c u i t with a time constant  r a n g i n g from 0.1 s e c . t o 1.0 s e c .  depending upon the amount of r e s i d u a l n o i s e .  Thus the i n t e n s i t y  -20-  of the l i g h t transmitted through the system was displayed i n a l i n e a r manner on the X-Y recorder trace as a function of magnetic field. B. Procedure f o r Measuring Faraday Rotation The alignment  of the o p t i c a l system was accomplished  with  the a i d of a He-Ne CW laser temporarily mounted on the o p t i c a l bench.  Further fine adjustments were made so that the signal  from the source transmitted through the system was a maximum. Before any measurements were taken, both discharge tubes and a l l the e l e c t r o n i c s were turned on f o r at least an hour to ensure that steady-state operation had been attained.  Also the P.M.  was cooled with dry i c e for about the same time. The measurement f o r a p a r t i c u l a r l i n e began with selecting the l i n e on the monochromator so that i t s intensity was at a maximum.  Then a graph sheet was put on the X-Y recorder and  aligned to i t s axes. calibrated voltmeter.  The X-axis was calibrated using the With the Y-channel of the recorder at a  fixed s e n s i t i v i t y , the P.M. voltage was adjusted to make the trace cover most of the sheet.  Three traces were taken on each  sheet by increasing the current i n the magnet slowly to maximum and then back to zero. nicol  The f i r s t trace, I , was with T  removed, the second, IQ°, with  and the t h i r d , I 9 0 » with 0  aligned at 0° to N  at 90° to N . 2  2 f  T y p i c a l traces are  shown i n Figure 7. C. Relative Absorption Apparatus and Procedure The following r e l a t i v e absorption experiment was performed to determine the e f f e c t of H  a  on N i n the absorber.  For t h i s  A=  0  fOO  S//S  lOOO  IFOO  A=  A  xooo  zfoo  (a) f o r a strongly absorbing l i n e Figure 7  O  St>0  IO00  7 06 7  Iroo  A  xooo  (b) f o r a weakly absorbing l i n e  Typical Recorder Traces f o r Faraday Rotation (2/3 f u l l size)  iroo  -22determination  i t was required to know how k l changed as a Q  function of H . 0  The k 1 at a p a r t i c u l a r f i e l d could be calcu-  lated using Equation 2.24 knowing both the source and absorber l i n e p r o f i l e s and measuring the transmittance, T, at that f i e l d . Since the Zeeman e f f e c t displaces the absorption l i n e away from )), i t was necessary ^  to place a magnetic f i e l d , H , on the s  source, i n order to obtain any absorption at high f i e l d s .  This  f i e l d was oriented so that the emission l i n e s displayed the same longitudinal Zeeman e f f e c t as the absorption l i n e s . For these measurements of T, c e r t a i n modifications of the apparatus of Figure 4 were required.  The source was placed i n  an iron-cored electromagnet (M^) run by a well regulated power supply and both n i c o l prisms were removed. arrangement i s shown i n Figure 8. set  The r e s u l t i n g  With this setup H  could be  s  at a fixed value and the transmittance could be plotted on  the X-Y recorder as a function of H . a  The was done f o r the l i n e  at 7515 A (displays the normal Zeeman e f f e c t ) f o r several values of H  s  as shown i n Figure 9.  For t h i s experiment the discharge  tubes were run at the same current as i n the Faraday rotation measurements. The transmittances for a l l l i n e s studied i n the r e l a t i v e absorption experiment were measured at the value of H experimental  condition (4r; )  a  where the  = 0 was observed f o r the l i n e at  7515 A (see Figure 9). This value of H  a  was chosen because the  absorption was at a maximum, and the errors involved i n s e t t i n g H  a  and H  s  at accurately known values are minimized.  because, f o r t h i s condition, H  a  i s very close to H  g  This happens and any small  -23-  Magnet  M  K  x  Power Supply  8  Amplifier  Recorder  i r o n - c o r e d electromagnet: M2 - solenoid electromagnet front surfaced m i r r o r . The r e s t of the symbols are as defined i n F i g u r e 4.  Mi  Figure  Lock-in  Y  Block Diagram  of E x p e r i m e n t a l A r r a n g e m e n t f o r Relative Absorption  Measurements  1.0  0.8  O.b  OAX-A*i's calibration mark. 0.X  0.0  JTOO  1000  /roo  H  v  a  Figure 9  Recorder  L xooo  I xroo  taat/ss)  T r a c e of. T r a n s m i t t a n c e V e r s u s H  a  for Line at 7515 3i  -24error i n H _ w i l l cause an error i n A  T of X(H - H ) f 0 ) : i.e. 2  a  s  the error i s quadratic i n the difference between the magnetic f i e l d s H a and H _ .  i s given by the expression: \<AH<L/(J a s  ^  constant \THZJ  \  + \>"*JJk.l  ... 3.1  The f i r s t term allows for the change i n N caused by the magnetic field, H . a  The plots i n Figure 9 show that T as a function of H , at a  constant H , for a normal Zeeman l i n e can be approximated i n the s  neighbourhood of minimum T by the expression:  T = where T  Q  To -  V / 4  + £(H*-H f  ... 3.2  t  i s the transmittance for H  a  =H  g  = 0, 1/ corresponds  to the f i r s t term on the right-hand side of Equation 3.1 and V and  C are independent of H and H . a s a  When sLT = 0 we have  c  therefore:  V  « £ 5 (H±-M )  ... 3.3  s  so that In c a l c u l a t i n g the k ^ l value i n the r e l a t i v e absorption experiment from the measured transmittance, i t was assumed that H  a  = H when f^-\  - 0.  Therefore a systematic error i n the  k l value was expected, corresponding to a deviation i n T of Q  (from Equation 3.2):  ,*  A T ^  1  =  From Figure 9 the values of £  ... 3.5  4-f  = 6.2xl0~^ gauss"*, and  - 2.6x10""® gauss" were obtained so that this error i n T was  about 3.7xl0~  2  4  from Equation 3.5.  Since the experimental  precision i n measuring T i s only ±.005, errors a r i s i n g from the above procedure of assuming £1' = 0 when H = H were n e g l i g i b l e . a  -25Once the f i e l d s were set equal using the above c r i t e r i o n and plots of the l i n e at 7515 A as i n Figure 9, T was measured for  a l l the l i n e s of interest which had detectable source signals  and s i g n i f i c a n t absorption. was disconnected.  F i r s t the X-channel of the recorder  Then a p a r t i c u l a r l i n e was found on the  monochromator and the output signal of the P.M.,  with the  absorber o f f , was adjusted to about the f u l l scale of the Ychanhel by varying the dynode voltage of the P.M.  This signal  was f i l t e r e d to reduce the noise to a reasonable l e v e l .  The  signal l e v e l , which was proportional to the transmitted intensity was measured with the absorber on, and then again with i t o f f . Because of residual noise, the l e v e l was found by moving the signal trace along a small segment of the X-axis manually.  This  was repeated several times and the measurement was only considered s i g n i f i c a n t i f i t was consistent oyer several cycles. transmittance for the l i n e was the quotient of these two  The  levels.  To eliminate coupling e f f e c t s between the discharge tubes when the absorber was  turned on and o f f , they were run by completely  independent current regulated power supplies.  For each l i n e ,  measurements were made at six d i f f e r e n t values of H  s  (=  H) a  between 0 and 2000 gauss. D. Radial Gradient of N i n the Absorber There was a p o s s i b i l i t y of a r a d i a l gradient of N i n the absorber which would a f f e c t the Faraday rotation and make the analysis more complicated. used.  To check for i t , two methods were  The f i r s t u t i l i z e d the arrangement of Figure 8 with the  two 4 mm.  apertures being replaced by 2 mm.  apertures.  With no  - 2 6 -  magnetic f i e l d on the source or absorber, transmittance measurements were made f o r the f a i r l y strong l i n e at 7635 A as the absorption tube was swept across the axis of the beam.  Within  the accuracy of the measurements ( c r = ± . 0 0 5 ) no change i n the T  transmittance was detected i n the region that would be traversed by the 4 mm. diameter beam. The second was to take a Faraday rotation trace of the l i n e at 8115 A as i n Figure 7(a) f o r both 4 mm. and 2mm. apertures. The two traces, although almost i d e n t i c a l i n appearance, were analysed i n d e t a i l i n Chapter IV B to determine i f there were any detectable differences i n the r e s u l t i n g two curves f o r the observed values of Q .  -27-  CHAPTER  IV  ANALYSIS AND RESULTS This chapter describes the techniques used i n reducing the Faraday rotation data and the supplementary  relative  absorption data to the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s . A. Faraday Rotation Data Reduction The data f o r each t r a n s i t i o n studied consisted of three quantities, I ( H ) , I o ( H ) , and l9o°( a'» measured as a function H  T  a  Q  a  of magnetic f i e l d on the absorber.  I  T  i s the lock-in amplifier  output signal due to the intensity of the l i g h t from the l i n e being studied which was transmitted through the absorber with the n i c o l Nj removed (see Figure  4).  I o Q  and  I90  0  a r e  t h e  sig-  nals with N^ i n place, the subscripts 0° and 90° denoting the angle between the axes of the n i c o l s N^ and N 2  The values f o r  these quantities used i n the reduction were the averages from two traces ( l i k e that of Figure 7(a) or (b)) taken on d i f f e r e n t occasions.  Values f o r each quantity were found at 2 mm.  inter-  vals (corresponding to 32 gauss) along the X-axis of the trace up to the point where the trace ended or to where no r o t a t i o n was observed (point Q i n Figure 7(b)). Ig o Q  were each "normalized" to  I  T  The functions IQ° and  (see Equation 2.13): i . e .  Since the r e s u l t i n g curves correspond to the p o l a r i z a t i o n a n a l y s i s of I  , their sum must equal unity everywhere: i . e .  -28-  where A and B would be equal to unity but f o r the fact that the n i c o l  and compensating plate absorb about 20% of the  polarized l i g h t .  A least squares f i t was used to f i n d the  normalization constants A and B which both turned out to have values of about 1.25. They were always s l i g h t l y d i f f e r e n t from each other, however, due to the imperfect compensation f o r beam displacement of the n i c o l  by the glass plate attached to i t .  This causes l i g h t from a d i f f e r e n t part of the source to be i s rotated from the jTL = 90°  directed through the system when  position to the _fL = 0° one, with a subsequent s l i g h t change in intensity. The t h e o r e t i c a l value of Q f o r _fl = 90° i s symmetric about the value 0.5 with the value of Q f o r SL= 0° (note the  cos(2.0. + <f )  factor i n Equation 2.13).  Using t h i s fact the  f i n a l experimental value of Q f o r Jl = 0° was found by "folding' the value of Q f o r Si = 90° about 0.5 and averaging i t with the 0- value: i . e .  Ado-  +(/-B9 -) 9e  2,  _ # —  (sexp  ... 4.3  The standard deviation per point between the folded curves, Cf  Gztp  ,  was used as an indication of experimental r e p r o d u c i b i l i t y and i t s values f o r the various l i n e s are l i s t e d i n Table 3. of ^  Examples  together with the best f i t t h e o r e t i c a l functions are  shown i n Figure 12. B. A Check f o r a Radial Gradient i n N Two rotation traces, one f o r 2 mm. and one f o r 4 mm.  -29+.060 +.0W\1  +• 010  j .  000  ~ 010 <\  -.0*0 -060  J  0  I  I  SOO  1000 Ha.  I  IS00 *-  3,000  (aavss  )  Deviations i n Qex/3 for Line at 8115  Figure 10  7.SOO  A*  Due to a  Radial Gradient of N apertures, were taken to check f o r the presence of a r a d i a l density gradient (see Chapter I I I D).  Each was analysed as  above and the two r e s u l t i n g curves for Ocxp were compared.  The  deviations between the two curves, A. Q = (0 ) — (0 ), i s plotted exf)  for each point i n Figure 10.  exf  The standard deviation per point  between the folded curves that produce (9e* \ i s quite large ( i . e . F  (Ok,, )•>  m  - .015) due to the larger noise signal involved i n the  measurements f o r smaller apertures.  However, i t can be seen  from Figure 10 that the value of A0  i s consistently larger than  (CXa  )„  f o r f i e l d s above ~1400 gauss.  This would indicate  the presence of a noticeable r a d i a l gradient i n N at high values due to an a x i a l r e s t r i c t i o n of the absorber  field  discharge.  This e f f e c t , though, was quite small and since the majority of the rotations measured were sat lower f i e l d values, i t was ignored i n the following analysis. C. Analysis of Relative Absorption The r e l a t i v e absorption data f o r each l i n e consisted of experimental  values of T f o r s i x d i f f e r e n t values of equal  -30magnetic  fields  on t h e  s o u r c e and t h e a b s o r b e r .  complicated function of k l  and u ( s e e  Q  this  point both parameters  values field  of k l  necessary  Q  iterative halving  technique  a monotonic f u n c t i o n of k l ) ,  To g a i n  Then,  a  2.24)  the e f f e c t  on N a v a l u e o f \i was g u e s s e d .  and t h e  Equation  a r e unknown.  to determine  T is  and  at  approximate of  the  magnetic  using Equation  2.24  ( w h i c h works w e l l s i n c e T  values  of k l  were d e t e r m i n e d  a s t o make t h e computed and m e a s u r e d v a l u e s  of T i d e n t i c a l .  Q  necessary  evaluation  of  the  transmittance  was a c c o m p l i s h e d by S i m p s o n ' s r u l e interval  s p a c i n g o f w/10.  w h i c h was  of  the  gas i n  the Doppler w i d t h s )  temperatures  of  the s u r f a c e s  and a s s i g n i n g a s l i g h t l y  glow  K.  for  close  to  of  integration  this  temperature  with  an  I. (necessary  by o b s e r v i n g  the gas  for  These temperature  those a c t u a l l y  2.24)  the  t u b e s when r u n n i n g  to  K.  The  calculation,  were e s t i m a t e d the d i s c h a r g e  so  (Equation  the d i s c h a r g e s  v a l u e o f 320°  the s o u r c e .  the  inside.  absorber  estimates  m e a s u r e d by B u r n e t t  1 0  in  are  similar  discharges. To o b t a i n r e l a t i v e  values  of  k l Q  absorption c o e f f i c i e n t s ,  found f o r each  value.  This reference  effects  o f any d e v i a t i o n s  the f i e l d relative since the  of  higher  produced a temperature  quite  numerical  For d e t a i l s  to c a l c u l a t e  and 3 7 0 °  function  p e r f o r m e d on a c o m p u t e r , s e e A p p e n d i x  The t e m p e r a t u r e s  This  Q  is  values. values,  they  are  line  for  v a l u e was c h o s e n s o a s from l i n e  lines  to l i n e  at  of  the  six  fourth  to minimize the in  lower  levels  of  field the  the  extremes  of  these  c o m i n g f r o m t h e same l o w e r  o n l y d e p e n d e n t on t h e m a g n e t i c  population density  the  were d i v i d e d by t h e  T h e r e s h o u l d be no d e v i a t i o n s G,  G,  effect lines,  state, on N:  -31-  ^  JkJ(^)  J  -fflfr)  -  > ( I ' W )  ... 4.4  Thus a plot of these r e l a t i v e values G(H i) against H f o r a  a  l i n e s with a common lower state should coincide and the v a r i a t i o n of N r e l a t i v e to i t s value at H  a 4  should be able to be determined.  The plots of G ( H ) using the i n i t i a l values of the k l ' s , ai  Q  calculated using the guessed value f o r \ i , had very large deviations however, the e f f e c t of the magnetic f i e l d apparently increasing N f o r some l i n e s and decreasing i t f o r others.  This  inconsistency was p a r t i a l l y removed by varying the value used for | i and r e c a l c u l a t i n g the k l ' s u n t i l the plots of the r e l a t i v e Q  k l values, G ( H ) , most c l o s e l y coincided. Q  ai  The " f i t " of \i was  possible-because G(H ^) was very dependent on u f o r strongly a  absorbing l i n e s . This procedure d i d not work too well (see Chapter V A(2) for a discussion) so that a preliminary determination of the e f f e c t of the magnetic f i e l d on N (required to analyse the Faraday rotation measurements) was determined using only the weaker or s i n g l e t l i n e s .  These showed considerably more consis-  tency than the heavily absorbing l i n e s which probably were strongly self-absorbed i n the source.  Because the value of u,  obtained i n the above " f i t " only lessened, but did not remove, the large inconsistencies i n the plots of G i t was considered only preliminary.  Once values of k l from the Faraday rotation Q  analysis were found (Chapter IV D) using the preliminary determination of the magnetic f i e l d e f f e c t on N and the preliminary value of u, a much more r e l i a b l e determination of p. could be found by a d i f f e r e n t method (Chapter IV E ) .  0-61  IS'A  D-7067A  t\-toisA  CD  zooo  (a) f o r l i n e s with I S 5 lower s t a t e O-73.73 A <,~73frA  a-7rtrA A  zooo (b) f o r l i n e s w i t h l s lower 4  state  U-2UtA A-/O+70A  goo H  Figure  11  /zoo —*-  /60O  (aaussj  (c) f o r l i n e s with l s g lower s t a t e P l o t s of Relative Absorption C o e f f i c i e n t s to Determine Magnetic F i e l d E f f e c t on N  z o o o  -33Using the best value of \i eventually found by the method of Chapter IV E the f i n a l values of k l determined Q  absorption experiment  were calculated.  plots of G(H ^) versus H a  The corresponding  final  f o r each of the lower states studied  a  are given i n Figure 11.  by the r e l a t i v e  Because of inconsistent r e s u l t s , no  l i n e s with a self-absorption c o e f f i c i e n t , K, greater than 8.0 were used. As well as could be determined, decreased l i n e a r l y with magnetic f i e l d .  i t appeared as i f G  Thus N as a function  of magnetic f i e l d becomes:  = N(Hj- Q(o)£•/ -r UJ = N(o) (I-tHj where and  ~"  ana.  y. T a  n  d  G  =  -  Q((L /  .-\  JTf /C(°) x  ••• 4-5 =  ,  z  +  C o n s e n t  ... 4.6  ( ° ) were found from the plots of Figure 11.  The  f i n a l values of y f o r each lower state studied are given i n Table 1.  Notice that the higher the energy of the l e v e l , the  stronger i s the e f f e c t of depopulation by magnetic f i e l d .  cy  level  (x /o~tai>as 1 S  5  number of l i n e s used  1.25  0.40  3  ls  4  1.59  0.20  5  ls  3  2.05  0.20  3  Table 1  F i n a l Values for y  In  -34line (wavelength  Table 2  k  i n A)  absorption  o1 Faraday rotation  6965  1.195  1.246  7067  1.212  1.387  7384  1.080  1.041  7515  1.187  1.070  7635  14.60  10.55  7948  3.61  3.27  8006  0.691  0.682  8015  6.86  4.59  8104  2.87  2.40  8115  22.7  24.6  8425  4.49  4.00  8668  0.600  0.670  9123  9.76  8.50  9658  0.902  0.803  Comparison of Experimental Absorption C o e f f i c i e n t s  addition, Table 2 gives the f i n a l values of the k l ' s found by Q  the absorption method at H rotation method.  a  = 0 and by the more accurate Faraday  The generally consistent values confirm the  legitimacy of the f i n a l values used f o r ji and y. D. Curve F i t t i n g To find values of k l from the Faraday rotation data, a Q  least squares technique was used i n which k l was the variable Q  parameter i n f i t t i n g the theoretical expression (Equation 2.13)  -35to the function Q (Chapter  IV A).  exft  The curve f i t t i n g was done  on a d i g i t a l computer using a standard l i b r a r y programme (UBCLQF). This programme required the evaluation of Equation 2.13 which was accomplished  by using Simpson's rule numerical integration  with an i n t e r v a l spacing of w/10.  Since the two integrals of  Equation 2.13 are symmetric with respect to the o r i g i n , the l i m i t s of integration used were from w = 0.0  to w = 4.6, at  which point the source i n t e n s i t y function was reduced by a factor of 10  8  from i t s value at l i n e centre.  The values of  F(w)  (Dawson's integral) involved in the function were l i n e a r l y interpolated from the values given i n Abramowitz and Stegun . 8  Further d e t a i l s of this c a l c u l a t i o n are given i n Appendix I. The c a l c u l a t i o n was  i n i t i a l l y carried out with the  preliminary values of u and the e f f e c t of the magnetic f i e l d on N found i n Chapter IV C.  The values of k l (the absorption Q  c o e f f i c i e n t s f o r zero magnetic f i e l d on the absorber)  found  using these preliminary parameters, were put into the s e l f absorption calculations (Chapter IV E) i n order to get a better estimate of the value of u.  This could be done because the k l Q  values found from the Faraday rotation experiment were quite i n s e n s i t i v e to self-absorption in the source. These better values for u were then put into the r e l a t i v e absorption c a l c u l a t i o n s (Chapter IV C) to produce more r e l i a b l e values for y , by now  recognized as the relevant parameter for  the e f f e c t of the magnetic f i e l d on N.  These new values for  the parameters were again put into the curve f i t t i n g c a l c u l a t i o n to obtain better values of k l . This i t e r a t i v e procedure 0  was  performed for several cycles, whereupon no further improvement  -36-  O'-'exp^riinentxl Second  12  T y p i c a l Examples of  poini  (every  of Qtxp  doto.)  7\ =  7S/S"A  \  71+9/1  =  IJ*  He,  Figure  point  Q  e  K  /  f  and the T h e o r e t i c a l Best  Fits  -37i n c o n s i s t e n c y was n o t e d .  This is  changes i n the parameters y , k l ,  accuracy  of  the R . M . S .  o f measurement.  the v a l u e error  of k l  d e t e r m i n e d by t h e  Q  a s s i g n e d to the parameter  assigned to k l  cases but one, i n T a b l e 5. k l's  less  Equation  t h a n 1%,  The v a l u e s  thus f o u n d ,  Q  2.23.  of C f  E.  The f i n a l  theorectical  the r e l a t i v e  a reliable of  fits  are  of A  values  k l Q  value  initially  u s i n g the  of  the  listed listed  r e  for  e x p  transmittance  for  found i n the Faraday  Q  i n the R  The v a l u e s  = H  Source = 0 measured  g  to c a l c u l a t e  curve  very  small  (see  values  2.29  K and t h u s u  w i t h a common l o w e r  lines  for  state,  were r e a s o n a b l y  the  method  f£^A,l i n  Table for  consistent  w h i c h s e l f - a b s o r p t i o n was a s m a l l f a c t o r  (from  iterative  rotation for  values  fitting  o f u s o f o u n d , w h i c h s h o u l d be t h e same  lines  obtain  The  c o u l d be done b e c a u s e  Q  (i i s  to  into Equation  t h e k l ' s f o u n d by t h e F a r a d a y  parameter  the  corresponding  T h i s was a g a i n a c c o m p l i s h e d by t h e It  lines  12.  rotation  p a r a m e t e r s was p u t  all  other  the s e l f - a b s o r p t i o n c o e f f i c i e n t .  h a l v i n g method on a c o m p u t e r .  on the  when H  .  obtained using  the  a.nd  in  From  l i s t e d along with $  by  the f i t , 0 ^  i n T a b l e 3.  given in Figure  a l o n g w i t h the a b o v e m e a s u r e d T  dependence o f  was g i v e n  the v a r i o u s  ^ were e a s i l y  are  the  the  a b s o r p t i o n e x p e r i m e n t were u s e d h e r e  preliminary  Equation 2.30).  fit  per p o i n t of  Examples of are  than  by t h e programme,  S e l f - A b s o r p t i o n and R e l a t i v e P o p u l a t i o n s The v a l u e s  for  0 t h  subsequent  programme, (/I A>/^j_ w a s ,  and i s  the v a l u e s  i n f o r m a t i o n i n T a b l e 3. best  by t h e  Q  the  An i n d i c a t i o n o f  t o g e t h e r w i t h the s t a n d a r d d e v i a t i o n The e r r o r  that  a n d u were much l e s s  Q  standard deviations  t o say  (i.e.  if  with K  5).  k l  T r a n s i t i o n Wavelength (Paschen  of  line  Faraday  notation)  {*) ls -2p4 3  ls -2p  7  ls -2p  3  ls -2p  3  A  Q  (from  A el (Wiese2  rel  r  (this  7948. 18  . 006  . 003  errors  experiment)  <20%)  (%)  rotation) 1. 00  3. 269  rel ( P - T and A  errors 20-30%)  1. 00  1. 00  8667.94  . 001  . 002  0. 670  0. 158  8. 5  0. 143  0. 15  . 001  . 002  1. 041  0. 343  5. 3  0. 314  0. 48  1. 67  4. 4  1. 55  2. 8  0. 176  0. 169  0. 29  5  7383. 98 7514. 65  . 001  . 001  ls -2p  6  8006. 16  , 001  . 001  1. 069 0. 682  ls -2p  7  8103. 69 8424. 65  . 002  . 006  2. 405  1. 00  8. 2 —  1. 00  1. 00  . 021  4. 001  0. 888  2.9  0. 842  0. 58  9657. 78  . 003 . 004  . 002  0. 803  0. 197  9. 3  0. 215  0. 14  1. 246  4. 3  0. 699 0.410  0. 98  2. 84  2. 1  1. 00  i : 00  3. 79 2. 20  2. 2  4  4  4  4  ls -2p 4  8  ls -2p 4  1 0  ls -2p  2  6965. 43  . 003  . 002  ls -2p  3  7067.22  . 002  . 003  1. 387  0. 689 0. 440  6  7635. 11  . 005  . 028  10, 55  2. 66  8014. 79  . 00 3  . 007  4. 592  1. 00  8115. 31  . 004  . 061  24. 56  3. 68  3. 9 — 6. 2  2. 09  2. 7  5  5  ls -2p 5  t;- P8 ls5-2pg 2  l s  ls -2p 5  1 0  9122.97 Og, is v  . 006  . 023  8. 497  the d e v i a t i o n b e t w e e n the 'folded* e x p e r i m e n t a l  Ch^  i s the d e v i a t i o n b e t w e e n the e x p e r i m e n t a l  Ar.i  i s the e x p e c t e d  a  ^"1  4. 0  curves  and t h e o r e t i c a l  for  curves  e r r o r i n the v a l u e o f A\ ( s e e C h a p t e r V A f o r an e x p l a n a t i o n of how  Table, 3  0 for Q  re  Experimental  Values  of  A  r e  ^  obtained)  0. 59  1. 07  J C  -39l e s s than 5.0)  were i g n o r e d .  The  f i n a l v a l u e s are l i s t e d i n  Table 4 a l o n g with t h e i r standard d e v i a t i o n s ,  .  They  correspond, w i t h i n a f a c t o r of 2, to the p r e l i m i n a r y v a l u e s of p. chosen to make the r e l a t i v e a b s o r p t i o n experiment consistent  (see Chapter  IV C ) . number of used  level  l s  5  l s  4  ls  1.7  0.3  4  5.7  0.7  4  1.7  3  results  Table 4  -  lines  1  F i n a l Values f o r y.  It can be seen t h a t the u's f o r the l s  5  and  were about e q u a l while the value f o r the l s  4  l e v e l was  by a f a c t o r of 3.4.  l s ^ levels larger  Since u i s p r o p o r t i o n a l to the r a t i o of N  i n the source t o N i n the absorber  then:  Mi.  ... 'sovrcc  where the s u b s c r i p t s l s energy  levels.  4  and  4.7  h /abierker t  l s ^ r e f e r to the c o r r e s p o n d i n g  T h i s i s i n agreement with the f i n d i n g s of I r w i n , 9  s i n c e the e l e c t r o n i c s t r u c t u r e of neon and argon  i s similar.  In  neon glow d i s c h a r g e s he found t h a t the p o p u l a t i o n s of the c o r r e s p o n d i n g metastable  s t a t e s were d i s p r o p o r t i o n a t e l y l a r g e  a t low c u r r e n t d e n s i t i e s and approached a Boltzmann d i s t r i b u t i o n at high current d e n s i t i e s .  In the present case the c u r r e n t  d e n s i t y i n the source i s more than 500 the absorber. ls  2  times g r e a t e r than i n  The r e l a t i v e l y low p o p u l a t i o n s of the I S 4 and  l e v e l s seems to be the major reason Why  the r o t a t i o n s f o r  -40-  lines originating from them are either small or unobservable on t h i s apparatus.  -41-  CHAPTER  V  DISCUSSION AND CONCLUSION A. Experimental Errors (1) Faraday Rotation In t h i s section a l l sources of error involved i n the measurement and analysis of the Faraday rotation which possibly would contribute to uncertainties i n the A 's are discussed rel and evaluated.  The symbol,  , used throughout Chapter IV and  in the following analysis represents estimated value of the absolute  either the measured or the  standard deviation i n the  parameter p. (a) Random Errors The errors discussed here are those errors of measurement which occur randomly from trace to trace.  Traces (see Figure 7)  for the same l i n e taken of d i f f e r e n t days were indistinguishable within the noise signal on an i n d i v i d u a l trace (when the intens i t i e s were adjusted  to be equal),  This demonstrated the o v e r a l l  s t a b i l i t y of the system, and p a r t i c u l a r l y the s t a b i l i t y of the discharge tubes.  For t h i s reason the various parameters  describing the discharges ( i . e . k l , y , j i T , and T ) were Q  t  s  a  thought to remain constant when the discharges were run at constant currents. However, the following deviations, not e a s i l y seen i n a v i s u a l comparison, could occur i n d i f f e r e n t traces.  When  c a l i b r a t i n g the X-axis against the calibrated voltmeter f o r  -42individual traces a random error,  »  e a  *  u a  l  to about 1 gauss  cm. would occur i n the c a l i b r a t i o n constant, M, of 160 gauss -1 -1  cm. For the weaker l i n e s , where the rotations are small (see f o r example Figure 12 f o r X = 7515 A), another error occurs due to the low s e n s i t i v i t y of Q always close to unity. determined  to the value of k l when Q Q  With such l i n e s the value of k l i s Q  predominantly by  there i s maximum net rotation. £) (j  remains  ~ ~  -L  around the values of H Here,  <P  +. — -r  a  where  • •• = °1•  1  assuming the absorption i s small and the rotation does not change very much over the range of frequencies containing the majority of the source intensity (see Equation 2.13).  Due to  the noise l e v e l and errors i n the zero l e v e l , the accuracy of  0  &xf>  Q  i s limited to about jr.005.  Knowing t h i s , and the value of  at maximum rotation, the accuracy of (p may be determined  using a trigonometric table.  Since k l i s proportional to (p Q  (Equation 2.21) the f r a c t i o n a l standard deviation i n k l,faAQ, Q  due to errors from t h i s source was also estimated and i t s values l i s t e d i n Table 5.  For stronger t r a n s i t i o n s , where 0,^0.5  at  some point, the error due to t h i s small uncertainty i n Q ^ i s e  negligible. (b) Systematic Errors The systematic errors are those uncertainties i n the parameters and analysis which are the same f o r a l l traces taken. The l i n e a r i t y of the intensity measuring system was confirmed by the excellent f i t of the "folded" rotation data, as demons t r a t e d by the small values of (%. (Table 3). There was  -43c o n s i d e r e d t o be no e r r o r The e v a l u a t i o n IV D)  of  was w e l l w i t h i n  data.  Even f o r  in  Simpson's r u l e  the  from t h i s  the  theoretical  the a c c u r a c y  s m a l l magnetic  to account  for  fractional  deviation  source. expression for  obtainable  f i e l d values  the  5.  scatter  t h e most r a p i d c h a n g e s i n i n the values  F o r weak l i n e s in  the d a t a  to o r i g i n a t e  the  (e.g.  this  the  accurately The  a s s i g n e d by  fit,  is  listed  deviation  strong lines  inadequacies of  chosen  integrand.  0  the cause o f  experimental  interval  o f k l , (A^)^,  p o i n t s and f o r  w i t h the  the  i n t e g r a t i o n was s m a l l enough  t h e c o m p u t e r programme, due t o an i m p e r f e c t Table  for  (Chapter  Q  it  was  was  theoretical  the  thought model  c o n s i d e r i n g o n l y a D o p p l e r b r o a d e n i n g mechanism i n  s o u r c e or not c o n s i d e r i n g a s m a l l r a d i a l account  for errors  a random e r r o r  in  expected i n the The l i n e  A  gradient  from both these s o u r c e s , the a n a l y s i s r e  of  the s t a n d a r d  the  of N).  „ was  in  To  included  as  deviations  ^'Si  profile  of  i n d i v i d u a l Zeeman c o m p o n e n t s i n  the  a b s o r b e r was c o n s i d e r e d t o be p u r e l y D o p p l e r b r o a d e n e d w i t h a width of  the  about  c o r r e s p o n d i n g to a temperature temperature  25°  discharge  K.  above  of  which  neutral  the e x t e r n a l  the  surface to  20° TtK  broadening.  little  on t h e d i s p e r s i o n , e s p e c i a l l y  Doppler half-width  K.  However,  in  the  discharge of  the  little  was  the the  pressure pressure  a s m a l l amount w o u l d have at  more t h a n  away f r o m t h e a b s o r p t i o n peak  The temperature  estimate  T h i s was At  .  s h o u l d be v e r y  (collisional) effect  This  temperature  standard d e v i a t i o n , 0  t u b e was r u n t h e r e  K.  g a s atoms i n t h e  t u b e and s h o u l d be a c c u r a t e  v a l u e a s s i g n e d to the at  the  of 320°  (see  one  Appendix  s o u r c e was c o n s i d e r e d t o be 3 7 0 °  K.  II).  -44w h i c h was a b o u t 3 0 ° of  the  K.  higher  t u b e when r u n n i n g .  than the  temperature  The t e m p e r a t u r e  s o u r c e and a b s o r b e r c o r r e s p o n d c l o s e l y by B u r n e t t  1 0  i n a study of  neon glow d i s c h a r g e s . broadening half-width limit  of  He a l s o  used here  to the  the d e t a i l e d  of  line  the for  surface the  temperatures shapes i n  found the r a t i o  of  to the D o p p l e r h a l f - w i d t h  found  similar  the  collisional  t o have a n u p p e r  ~ 0 . 0 7 i n an a b s o r p t i o n t u b e f i l l e d w i t h neon  to  2 mm. H g . The e r r o r temperature  i n t r o d u c e d by t h e e s t i m a t e  had a n e g l i g i b l e e f f e c t ,  virtually  independent of  profile.  There e x i s t e d , however,  a large  degree of  the  fine  of  the  line  since  the  the A  source r e  ^'s  details  of  for  transitions  self-absorption in  c a s e s made t h e w i d t h  of  at  the the  are  the s o u r c e  studied,  s o u r c e , which  half  line  in  peak e m i t t e d  intensity  more t h a n d o u b l e t h e c o r r e s p o n d i n g D o p p l e r h a l f - w i d t h . analytic  self-absorption profile  for  gross e f f e c t  this  The e f f e c t  of  2.8)  made t h e a s s u m p t i o n o f  g e o m e t r y and h o m o g e n i e t y i n case.  (Equation  this  the s o u r c e , which  is  accounted  parallel  not q u i t e  a s s u m p t i o n on t h e d e v i a t i o n  compared t o  that of n e g l e c t i n g c o l l i s i o n a l broadening in  T h i s b r o a d e n i n g mechanism h a s i t s  the wings o f a l i n e  For  field  in  the c e n t r e  absorbing lines  on t h e a b s o r b e r t h e  contained only these cases,  those  greatest  when t h e r e  light  frequencies  collisional  of  the  was  the  in  truncated.  through the  the wings o f  the  the  was o n l y a s m a l l  transmitted in  line  small  effect  and, with s e l f - a b s o r p t i o n present  was e n h a n c e d b e c a u s e  strongly  probably  the  the  profile  effect  p r o f i l e was  of  theoretical  source.  from the r e a l  The  which  plane  some  line  b r o a d e n i n g c o u l d have been  magnetic system and  very  -45important. To determine  the e f f e c t  s o u r c e on t h e v a l u e s used i n curve  place of  fitting  of  k l,  lines  at  profile  broadened h a l f - w i d t h  involved  the V o i g t  the d e v i a t i o n ations ,  best  fit,  of k l Q  profile  t o be 0 . 0 8 .  and t h e r e l a t i v e  the h i g h e r v a l u e  compensation f o r However,  the  because of  broadening,  the  lack the  of  this  value  50% f r o m t h o s e  values  the  the o t h e r  that  devi-  on t h e  least  squares  2% w i t h t h e u s e The  it  t o measure  better  than  the value  i n measuring the  was u s e d i n  was e q u a l  that  this  pressure  (Equation  to that  axially  line  shape,  transmittance,  to  line,  2.8)  was  for  together  are  the  t h e measurement o f u .  t h e measurement o f u. f o r  was a l s o c o n s i d e r e d t o be t h e The f a c t  which  analysis.  t h e d e v i a t i o n s , Cfa , f o u n d i n  o n l y one l i n e  that  partial  c o u l d change f r o m l i n e  The above c o n s i d e r a t i o n s on s o u r c e with inaccuracies  is  of  b r o a d e n i n g mechanism.  absorbed Doppler source p r o f i l e final  the  obtained  much l e s s .  of u found there of  the  half-width  o f \i w h i c h was 4 0 % l e s s  inability  and t h e f a c t  only a self-  for  in  the case w i t h o n l y D o p p l e r b r o a d e n i n g c o n s i d e r e d ,  indicates  used i n  At  c h a n g e d o n l y by a b o u t  was a c h i e v e d w i t h a v a l u e  used f o r  The r a t i o  the D o p p l e r  self-absorbed Doppler p r o f i l e  The v a l u e s  the V o i g t fit  the  2.8  was  was c h o s e n , by r o u g h l y m i n i m i z i n g  , were r e d u c e d by a b o u t  u s i n g the fit.  of  profile  to  the  p r o f i le3  g i v e n by E q u a t i o n  7635 A a n d 8115 A.  collisional in  c o l l i s i o n a l broadening i n  a self-absorbed Voigt  Q  the s o u r c e  for  of  the  ls^ level,  the  so that  reasons Although  IS3  level,  t h e cfa.  same.  , the homogeniety o f  the  absorbing  -46plasma was i n doubt, causes no concern since only the integrated k l i s important.  Any change i n the a x i a l dimensions of the  Q  plasma column caused by the magnetic f i e l d are, accounted f o r i n the measurement of y.  As indicated i n Chapter IV B, there i s a  p o s s i b i l i t y of a r a d i a l gradient i n N which increases with magnetic f i e l d .  An increase i n a r a d i a l gradient would tend to  decrease the absorption and rotation that might be observed at higher f i e l d s .  This i s the same e f f e c t produced by a decrease  i n N with magnetic f i e l d .  Thus the measurement of y also partly  accounts f o r the e f f e c t s of such a r a d i a l gradient. A discussion of the errors involved i n measuring the magnetic f i e l d e f f e c t parameter, y, which lead to the deviations,  C?f, i s given i n Chapter V A(2). In c a l i b r a t i n g the voltmeter with respect to the magnetic f i e l d i n t e n s i t y i n the solenoid a d e v i a t i o n , ^ , of about 4 gauss cm. * could possibly occur i n the X-axis c a l i b r a t i o n constant, M. The small inhomogeneity of the f i e l d w i l l broaden the absorption l i n e s l i g h t l y but should have l i t t l e e f f e c t on the r e s u l t s since only the integrated r o t a t i o n i s of importance, (c) Accuracy of the A ^ i ' s Now the extent to which th above errors produce possible uncertainties i n the values of the  A r e  i ' s i s evaluated.  Equation 2.23 i t can be seen that the error i n A  r e  From  i w i l l result  solely from the errors i n the corresponding k l ' s since the Q  other terms are very accurately known or cancel. f r a c t i o n a l standard  error i n A  r e  i i s given by:  Thus the  -47 •  w h e r e 0[-^FJ is the the  various  error  would produce  p,  the  [A ) nl  These  errors  exactly 1  the  a  following  t e n d to c a n c e l  same  in A  errors  r e  ^  value in k  due  i n the  y  (i.e.  1' and  k  from  the  o  1".  i n the  w i l l be  to t h i s u n c e r t a i n t y .  By  combination,  i s the  r  the  error  errors  may  be  calculated.  0*,  Also, deviation  ~ -fir  ~Xr  ^p.  of the  This  k l's.  could If the  0  M due  error  fractional standard  in A  systematic . due rel  to  the  i . e.  combining to the  +  °r  for a particular parameter,  d e v i a t i o n Cy>, Lthen the  By  have  k j "  +  ment  of  5.3  standard  r  systematic  value  must  c o m b i n i n g the the  a value,  ...  systematic,  If Ax> •  For  mean  n^r)*p-  t h e n f~^—j  worst possible  M).  a  A p , being  of  D  fractional deviation  since  o  k l is a function  , ju , T ,  deviation  {-cr  -  p  Each  "J*^" •  systematic  of that p a r t i c u l a r p a r a m e t e r , error  ratio  measured parameters  particular parameter, A p ,  i n the  in value  deviation  error  of p  introduced  p,  there  from  errors  i n the  could  be  into A  usual  a  measurement  is characterized  - [mr + (  standard  random  vary  5. 4  r g  ^  because  random to  by  a  measurestandard  of t h i s i s :  W J * . . .  manner  the  error  in  & Y Te  errors is:  pal frl  Sr&n&om  +  ...  5.6  -48w h e r e the f r a c t i o n a l f i t , fa hf).^, and  due  standard to the  d e v i a t i o n s due  accuracy  of Q ,  to the l e a s t  , (AJLJ) ,  &X/  squares  are  as  curve  explained  above. The  partial  i n v o l v e d by  derivatives  about the  l i n e a r i t i e s ) and  The  values  standard value  leaving  c a l c u l a t e d by  e r r o r , c£>,  standard  then,  c a l c u l a t i n g the v a l u e  are  of the k l b y Q  deviations estimated fractional  d e m o n s t r a t e s the a c c u r a c y  depend are  r e l a t e d to.  The  on  the p a r t i c u l a r  standard  produce  o v e r a l l the l e a s t e r r o r  5. 4 a n d  5. 5 and  other  the  i n the A  of the t r a n s i t i o n s  as  the  5,  squares  was  >  3.  That  non-  "best" values, curve fit. 5.  Using  the  the  final  chapter,  calculated  /Systematic  of the A line  as f o l l o w s :  J  they  are  r e  ^'s  here  ^'s.  found  a l l under  be  above  probabilities  were  However,  the e r r o r s  standard may  r e  the t r a n s i t i o n  transitions used  data i n Table  at t h e i r  from  experiment.  i n the m e a s u r e d v a l u e s  quite m a r k e d l y  ^*rti  in Table  of the  errors  o r i n this  \ Artl  10%  parameter  given i n Table  IV  AreX /random  l i s t e d f o r the i n d i v i d u a l l i n e s  errors  are  deviation,  and  The  the u s u a l l e a s t  in Chapter  standard  V  Arel  (to m i n i m i z e  a l l other p a r a m e t e r s  of t h e s e p a r t i a l d e r i v a t i v e s  of the  v a r y i n g the  chosen using  to Equations  involved in using readily  any  calculated.  (2) R e l a t i v e A b s o r p t i o n In t h i s field  effect  of k l due Q  determined  s e c t i o n the on  N  w i l l be  accuracy analysed.  to u n c e r t a i n t i e s as f o l l o w s :  of the The  assessment  of the  magnetic  p o s s i b l e d e v i a t i o n s i n the  i n the m e a s u r e m e n t s  values  of the t r a n s m i t t a n c e  was  Wavelength of l i n e  k 1 o (from Table  (A  k.l)  )UAu.  AJ  e  (%  3)  Ad'  gauss  - 1  (%xl0  A.A  4  (%)  (%)  0. 8  0. 3  . 0. 52  8  0. 7  0. 31  9. 5 5. 9  + . 40  8668  3. 269 0. 670  7384  1. 041  5  0. 5  0. 30  5. 4  + . 70  7. 7  7515  1.069 0. 682  4  0. 4  0. 33  4. 5  8  0. 6  4. 5  + . 19 + . 24  7. 9 7. 6  8104 842 5  2. 405  0. 4  4. 001  1. 3 0. 5  0. 31 0. 35  0. 8  0. 53  9. 5  + . 31 + . 05  9658  0. 803  9  1. 1  0. 32  2. 9  + . 09  9. 1 11. 5 8. 0  6965  1.246  2. 3  0. 4  0. 35  4. 1  + . 04  8. 4  + . 30  8. 5 15. 6  <*)  7948  8006  7067 7635  1. 387 10. 55  0. 37  0. 5 0. 3  0. 83 0. 57  0. 7 0. 5  0. 87  17. 6  -. 53  15. 5  0. 77  10. 0  -. 79  15. 2  4. 592  8115  24. 56  0  8. 497  0  a n e x p l a n a t i o n of  Table  6. 8  5  and ^ . ^  8  e  e  4. 9 11. 8  10. 5 7. 5  0. 4  8015  for  -. 10  2  2. 2 0 0  9123  (%xl0~  gauss.)  cm. )  -. 85 + . 46  8. 6  Chapter  A n a l y s i s of the S o u r c e s  V  A  of E r r o r  (1)  12. 6  K"  1  40  30  20  3i i  ro  0 L-i  o.osr  o.io  oXo  i.oo  0.50  kJL  .9sr  Figure 13  J  .90  I  .sjr  I  .so  i  10.0  .to  I J  .ro  I  .40.30  L  .7.0  I  .to  J  .or  I .03  T  Plot of -^MJlJ Versus k l and Transmittance 0  ~Tj-\ The  ZO.O  *I  no  f.00  zoo  fractional derivative,  5. 8  ^ ^ , was calculated using Equation 2. 29,  with the self-absorbed source being replaced by a purely Doppler ened source profile.  broad-  The temperature ratio, R, was set at 1. 16 as in .  the rest of the calculations.  A plot of —^^p" calculated as a function of  k l and T is given in Figure 13. Q  Since the measured transmittances had a standard deviation of about . 005, the minimum error in k l (using Equation 5. 8) was about Q  -512%  f o r k l b e t w e e n 0. 5 and  values the  outside  this  range.  c a l c u l a t i o n of /* w e r e The  magnetic  present. which  This  4jf = 0 i n the  caused, b y  assuming  Chapter  negligible H_  - H„  absolute to the  I I I C, i n the  above v a l u e lines  that H  Also,  kj.  and  of G  seems  to r e q u i r e  K,  c h a n g e d the  not was  be  = 0. - H  a  relative  population  same  verified.  d i s t o r t e d by  T  i n the  G  systematic  very  the  large  11).  In a d d i t i o n , magnetic  as  about  i t was field,  being  the compared G,  From  for  the G  for  of F i g u r e  i n the  coefficient,  field.  source  source  f o r c e d to one  This  is  changed i n  absorber.  that the  11  In c a l c u l a t i n g  self-absorption  s t a t e s i n the  noticed  when  3%.  plots  inaccuracy.  those  in  deviation for  Q  lower  slowly  error  that k l c h a n g e d w i t h m a g n e t i c of the  are  i n c o n s i s t e n c i e s i n the  v e r s u s H_  that the  at  error  coefficients,  Figure  standard  were  point  a negligible effect  (see  made  shown , however,  varies  have been  rather  relative manner  the  magnets  gauss  absorption  4. 4 the  should  24  small,  than m e a s u r e m e n t  s a m e way  o n l y be  of t h i s m a g n i t u d e  is very  made  could  It was  s  Q  of the  a s s u m p t i o n was  that the  e x a c t l y the  more  the  of k l s i n c e  ^"  for different lines  0  4jr  in H  Equation  Q  range.  absorber  fields would have  u s e d to m e a s u r e  k l ' s , the  and  at  i n the  of k l u s e d i n  f i e l d w h e r e no  at a p a r t i c u l a r f i e l d v a l u e  values  say  = H  error  rapidly for  u n c e r t a i n t y "in f i n d i n g the  9  since  deviations  f o r ^4  the  the  quite  a l l values  and  at z e r o  p l o t s of F i g u r e  T h u s the e x p l a n a t i o n  to  except  determination  observed  rises  nearly  the s o u r c e  arises from  c a l i b r a t i o n of the  different lines  most  on  that d e v i a t i o n s  is small.  error  inside this m i n i m u m  gauss,  error  The  However,  fields  e q u a l to about - 5 0  in  2. 0.  Q  side  This  can-  discharge of  the  -52c a p i l l a r y bore i n the G e i s s l e r tube.  This would have a definite effect  on the optical thickness of the source dent of magnetic field.  so that jm  would not be  indepen-  This could certainly be the cause of some of  the i n c o n s i s t e n c i e s for heavily self-absorbed lines ( i . e . those with k l Q  greater than 3.5 for  the A  for the l s ^ and  l s ^ lower l e v e l s , k l greater than Q  1.0  l s ^ lower level). test of Voigt source  p r o f i l e s produced no improvement i n the  consistency nor much o v e r a l l change i n the plots of F i g u r e  11.  Since  only the l e s s strongly self-absorbing lines were used i n the m e a s u r e m e n t of ^  , the dependence of this p a r a m e t e r on the value The  best conclusion that could be  relative absorption experiment was number densities of the a f i r s t order %T s for the l  I S 4 and  The  l i n e a r l y with magnetic field,  standard  deviation, C^- , i n the  readily estimated  at the range of slopes of a straight line which would be 11 (b) and  degree of inconsistency i n the values state ( F i g u r e 11 (a)).  small.  that in the f i r s t 2000. gauss the  ls-^ lower states was  to the data plotted in F i g u r e  was  reached f r o m the-.results of the  Is states decreased  approximation only.  of JJ.  (c).  by looking  a good fit to  However, there i s a large  of G i n the plot for the  In this case the values  l s ^ lower  of G for the line at  6965 A* were considered m o r e reliable than the results f r o m the two  lines.  other  This is because the results for n o r m a l Z e e m a n lines show  a high degree of consistency ( F i g u r e 11 (c) contains only results f r o m n o r m a l Zeeman lines).  The  line at 6965 A  is not a n o r m a l line,  but  the splitting factors for its three p a i r s of components are quite close  -53to  each  other  in value  Zeeman pattern. t h a n the l i n e factors  for this level,  .B.  0.40x10  , gauss" ,  E v a l u a t i o n of The  twice  1  compares  only method b a s e d the  components.  above.  the v a l u e  dispersion  The  The  other  standard  account,  f o r the  on  presented  f a v o u r a b l y with  especially  anomalous  was  o t h e r two  f o r the  for strongly absorbing  spectrograph  above  line splitting  line  the  used  factors  deviation in estimated  to  levels.  others  ( f o r example  but  using a s i m i l a r  absorbing  see  of l i n e p r o f i l e s , lines,  Faraday  for obtaining relative  stronger transitions.  The medium  Pery-Thorne  and  produces  r e q u i r e s an  to o b t a i n e v e n m o d e s t  Absorption methods,  i n d i c a t e s that the  d i s p e r s i o n w i t h i n an  It i s i n d e p e n d e n t  1  results  normal  Zeeman  a d v e r s e l y affected by  R o s c h e s t w e n s k y hook technique  Chamberlain '''). best  the  to a n o r m a l  d i s c u s s i o n into  evidence  transition probabilities,  is  resembles  Technique  experimental  rotation method  closer  probably  taking this  line  l a r g e d i f f e r e n c e s b e t w e e n the  of Z e e m a n was  split  much  ot s e l f - a b s o r p t i o n m e n t i o n e d  -4 be  K)  8015  the  which has  of i t s f o u r p a i r s  related  thus  It is i n fact  at 7067 &  i n the p l o t ( a t  y  and  and  the  extremely  high  accuracy.  source  and  absorber,  (for  12 example on  see  the w o r k  detailed line  done  shape.  wings  of the  while  l a c k of k n o w l e d g e  can  on  For  s p e c t r a l line  c o n t r i b u t e to the  Line  emission  I by  Robinson  strong transitions,  are  of the  also  Ne  t r a n s m i t t e d , this  ) depend where  quite h e a v i l y  often only  the  dependency i s i n c r e a s e d ,  extent of s e l f - a b s o r p t i o n i n the  source  errors.  intensity i s another  widely  used  method  for deter-  -54mining  transition probilities  complicated  experiment  to  or  measure  The k l's  major  obtained  o  absorbing  These  rotation.  low  Another  in  fairly order  and  the  1s  many more  of the  the  not  2  argon  lower  the  caused  For  optimized  to o b t a i n a m a n a g a b l e  by  three  by  absorbing  the  ratio.  the  source  shape v e r y  uncertain.  possibility  of a r a d i a l  d e n s i t y g r a d i e n t a f f e c t i n g the  the n u m b e r The  s e l f - a b s o r p t i o n i n the  of d e t e r m i n g  d e n s i t y of l o w e r  rotations.  the  effect  of  error.  intensity  is a large source  the  the  source  source  i n • ordo-p-toproduces,  which  makes  l e a d s to the  measurements.  of the m a g n e t i c  absorber  enough  f a c t that the  This  large aperture  s t a t e s i n the  uncertainty introduced here  showing large  The  small  column.  a high  high  no  c u r r e n t and  absorption, d i s c h a r g e  s i g n a l to n o i s e  low  weakly  things:  the  v a r y i n g the  strong transitions,  the m e t h o d  possible  there was  chief source  for  Thirdly,  parameters  generally  r o t a t i o n was  d r a w b a c k i s the n e c e s s i t y of h a v i n g through  and  many  level,  glow d i s c h a r g e ,  s h o r t l e n g t h of the  large aperture  more  necessary  i s the  ~ 4).  l i n e s the  coefficients were  e x c i t a t i o n was  fill pressure,  and  For  characteristics  discharge  experiment  of i t s m e a s u r e m e n t w a s  absorption  a much  errors.  a b s o r p t i o n tube (below  p a r t i c u l a r l y with  accuracy  inherent  also introduce  i n the  However,  Uncertain background intensity  d r a w b a c k to the p r e s e n t  lines,  observable that the  can  ).  i s involved, with m a n y m o r e  calculate.  self-absorption  (see W i e s e  i s not v e r y  field  on  accurate.  of e r r o r f o r the  lines  -55C.  Conclusion It h a s  the  been demonstrated  strong transitions  the t h e o r y . ranging  The  from  i n the A r  effect was  zero  up  line centre.  determining can  complement  transitions, upper The  and  s t u d i e d and  be  which  combined with  the v a l u e s The  obtained  are  listed  given i n a recent  effect  s t a t e s i n the ~2000  here  glow  discharge  g a u s s i n the  containing  used  relative  on  has  the n u m b e r  number  densities with  increased magnetic  greater  f o r the h i g h e r  energy  decrease on  are  detailed  listed  line  in Table  shapes,  and,  2.  The  lines.  The  results  probabilities.  densities  with  up The  the  the  than  of the  results,  of although  i n the  effect  by  a  being linear dependence  anomalous  Z e e m a n splitting. F u r t h e r w o r k to i m p r o v e technique  w o u l d n a t u r a l l y be  the  accuracy  i n the  area  and  a p p l i c a b i l i t y of the  of r e d u c i n g the  10%).  4s  to a f i e l d  of t h i s  is hindered  for non-singlet lines,  well with  decrease  coefficients  method  of the  (deviation less  a linear field,  The  for weak  compare  absorption experiment.  states.  of  been determined  a l a r g e p o s s i b l e d e v i a t i o n , show  from  way  measurements  Wiese  of the  accurate  suitable  3 and  with  fields  components  f o r the t r a n s i t i o n  in Table  field  and  are m o r e  c o m p i l a t i o n by  of the m a g n e t i c  excellently  half-widths  for these  lifetime  s t a t e s to o b t a i n a b s o l u t e v a l u e s  values  Zeeman  simple  probabilities  other techniques can  a  effect for  for magnetic  about f o u r D o p p l e r  method provides transition  the  Faraday  agrees  analysed  at w h i c h  d i s p l a c e d by  The  relative  I 4s-4p m u l t i p l e t  to a v a l u e  absorption line were the  e x p e r i m e n t a l l y that the  effects  of the  -56above m e n t i o n e d d r a w b a c k s .  The  reliably increased  a longer  would  require  Increasing be v e r y  by m a k i n g  a longer  the a c c u r a c y  useful.  absorption  solenoid with  coefficient  absorption a higher  of the r e l a t i v e  c o u l d be  column.  This,  capacity power  absorption  most however,  supply.  experiment would  also  -57BIBLIOGRAPHY III W.  Seka and F. L. C a r z o n , J. Quant. 8, 1147 ( 1968).  12/ W.  I_. Wiese, Eighth International Conference on Phenomena in Ionized Gases. Vienna., I A E A ( 1967).  /3/ B.  L. Stansfield, M a s t e r s ' T h e s i s , P h y s i c s Dept., B r i t i s h C o l u m b i a ( 1967).  /4/ A.  C.  G. M i t c h e l l and M. W. Zemansky, Resonance Radiation and E x c i t e d Atoms, Ch. I I I . Cambridge U n i v e r s i t y P r e s s , New York, ( 1934).  /5/ R.  L.  F o r k and L. C.  /6/ C.  E . Moore, A t o m i c E n e r g y L e v e l s . V o l . I, Washington, U. Department of C o m m e r c e , NBS C i r c u l a r 467 (1949).  11/ E . V.  181 M.  /10/ J. C.  A.  /12/ A.  3_, 137 (1964). S.  Condon and G. H. Shortley, The T h e o r y of A t o m i c Spectra, Ch. III. Cambridge U n i v e r s i t y P r e s s , Cambridge (1963).  Irwin, Ph. D. T h e s i s , P h y s i c s Dept. , U n i v e r s i t y of B r i t i s h C o l u m b i a ( 1965). Burnett, Unpublished M a s t e r s ' T h e s i s , P h y s i c s U n i v e r s i t y of B r i t i s h Columbia, (1969).  Dept.,  P e r y - T h o r n e and J. E. C h a m b e r l a i n , P r o c . Phys. Soc. 133 (1963). M.  /13/ L. D.  /14/ C.  Opt.  U n i v e r s i t y of  Abramowitz and I. A. Stegun (ed. ), Handbook of M a t h e m a t i c a l Functions. New York, Dover Publications, Inc. ( 1965).  /9/ J. C.  /II/  B r a d l e y , Appl.  Spectrosc. Radiat . T r a n s f e r .  Robinson, Ph. D. T h e s i s , P h y s i c s Dept., B r i t i s h C o l u m b i a (1966).  82,  U n i v e r s i t y of  Landau and E . M. L i f s h i t z , E l e c t r o d y n a m i c s of Continuous Media, Ch. IX, Reading, M a s s . , A d d i s o n - W e s l e y P u b l i s h i n g Co. , Inc. (I960).  Young, J. Quant.  Spectrosc. Radiat.  Transfer.  5, 549 (1965).  -58APPENDIX CALCULATIONS INVOLVING THE  A.  OF  A  DIGITAL  COMPUTER  C a l c u l a t i o n of k l by C u r v e F i t t i n g to F a r a d a y Rotation Data p  The  large number of calculations involved in fitting the theoretical  e x p r e s s i o n for Q value, Oexp, An  USE  I  as a function of H  r e q u i r e d the use  iterative technique,  U B C L Q F , was calculations.  a  (Equation 2. 13) to the  experimental  of a high speed digital computer (IBM  as i n c o r p o r a t e d into the l i b r a r y  7044).  programme,  used i n the F o r t r a n I V p r o g r a m m e which p e r f o r m e d the T h i s p r o g r a m m e i s given below as i s a chart of the  general flow of the calculation (see F i g u r e A l ) . The of Q  subprogramme U B C L Q F  r e q u i r e d not only the theoretical value  at each point i n order to fit to the experimental  the p a r t i a l derivative, -,Y-T • iteration converge quickly.  The The  use  iteration proceeded until the Q  accuracy The  integration was  while keeping  programme  plotted i n F i g u r e The  l e s s than 0. 02%.  done n u m e r i c a l l y by Simpson's rule.  used i n the n u m e r i c a l  three times  The  subsequent The integration  interval  spacing  chosen s m a l l enough to obtain good  the computing time down to a reasonable  also found  0^  and  amount.  gave the best fit t h e o r e t i c a l curve  as  12.  data at points where the displacement  components was  but also  of this d e r i v a t i v e made the  change i n the calculated value of k l was involved was  curve,  of the absorber  beyond the half-width of the source  m o r e heavily as those  of the greater a c c u r a c y  at lower fields.  Zeeman  line were weighted T h i s was  of the data for the higher fields and  done because the  lesser  -59re.li.abili.ty of the theoretical m o d e l for the lower fields.  Farther  information about the p r o g r a m m e i s given i n the comment statements (C).  MAIN P R O G R A M M E Reads i n F(tu),  T ,  ARFARA T ,  a  g  and  M  starts on individual line data Reads i n A , <*s , fi , &etp , f and an i n i t i a l guess for k l s  I I  Calculates  >  starts on new  j*  line  Q  4^,  Weights data points m o r e heavily when Zeeman components split beyond source half-width  Uses U B C L Q F to iterate to least value  of k _ l ; evaluates  Calculates  Ol  0F  squares  = (AK-I)I  &,  ***  Calculates value . o< A  P r i n t s k l and theoretical best fit values of 0 Q  to subprogramme  Figure A 1  f r o m subprogramme  ARAUX  Flow C h a r t  of C u r v e F i t t i n g C a l c u l a t i o n (continued  on page  60)  ARAUX  -60-  r e t u r n to m a i n p r o g r a m m e ARFARA  from main programme ARFARA  SUBPROGRAMME I Calculates  SM  self-absorbed  - 0 =  ARAUX  (/ - €  Calculates;  - K£  I  source p r o f i l e ;  )  J S(u» L*.  C^TJ  oCou  }  4.6 S(a>)  and  LZ  - h i  Jdcv  t  integrals involved i n value of  I  Calculates Q also £ £ -  for present  SUBPROGRAMME  A  k l and Q  INTFUN  Calculates the v a r i o u s expressions under a l l the i n t e g r a l signs and combines them with S[u>) (from A R A U X ) to f o r m the integrands at each i n t e r v a l of the Simpson's rule integration  Figure  A1  Flow C h a r t of C u r v e Fitting C a l c u l a t i o n (continued  f r o m page 59)  ¥T8TTC ARFARA COMMON AS(5)» BS(5>» F A ( l O l ) » FB<51)» NS* DDS» DDA> YEX(81)» NN» 1CMAG » POPFACt GAMMA* SAB* SX(51)> LA» IT* LN, DRAT* D P F ( l O l ) DIMENSION P ( 1 ) » D ( l ) » E l l l l t E 2 U ) t Y F ( 8 1 ) t XX(81)» I D ( 1 0 ) . XT(81) 1* I D I ( 1 0 ) » W(81) DATA I END/3HEND/» B O L T Z K / 1 . 3 8 0 5 4 E - 1 6 / » A T W T / 6 6 . 3 3 1 G E - 2 4 / » 1 CCrGHT7"2T9"9"7'9"2'5"ET07" EXTERNAL AUX C READS IN VALUES FOR DAWSONS INTEGRAL . R'EA'D f5T2"J (~FA~OT» 3 ^ 2~» TOT) ~ 2 FORMAT (8F10.8) READ (5*2) ( F B ( J ) • J = 1. 51) F ' A T D =~~0T0 PRINT 20 20 FORMAT (/68H DETERMINATION OF K.OL BY LEAST SQUARES FIT TO FARADAY R rOT"A'T"mN~CUR\/E771 : READ (5*1) IDI PRINT 7* IDI C ^TS"rS-S-OURCE~TEMP"rr-TA~rS"AB"S"ORB"ER~TEMP"i READ ( 5 » 8 ) TS» TA* SAB* CMAG» NIT* EP 8 FORMAT (4F10.4* 15* F10.5) PRINT 2T* T"A"» T"S» SA'Bl CM'AG"* NTT"* EP ~ 21 FORMAT(5H TA = * F 6 . 1 » 1 5 H DEG.K.* TS = » F 6 . 1 » 1 9 H DEG»K•» KAPPA = 1»F6.3»20H#KOL*POPFAC* HA = » F 6 . 2 » 1 1 H GAUSS/2MM./30H MAXIMUM NO. ~" 2 O F ITERATIONS = » ITT4"Wi MTNTMUM CHAN'GE OF^KUC rN~L7A"ST rT"£RA"TlON" 3 =»F7.4/> DRAT = SQRT(TS/TA) _ 'C ^C7\"CCarATE'S A"ND STOR'E'S~D"OPPrER rrN"E~PR"0"FT'L"E ' DO 31 LD = 1. 101 Z.D = FLOAT (LP - 1 ) /20.0 3T~DPT"(LD) = EXFC^ZD^ZD) LINE = 0 C STARTS ON INDIVIDUAL LINE DATA • rOT0'O~R'E'A'D~"r5Tr)~"TB 1 FORMAT (10A6) IF( ID(1).EQ.IEND) GO TO 1002 C WLT~TS THE WAVELTEN'GTfll PT T~) PS A~N I'NTT IA L GUESS OF KOL C GVAL IS THE MULTIPLICITY OF THE UPPER LEVEL READ (5*3) WL»CVAC» NS* NN» P(l)» GVAL, POPFAC* GAMMA 3~FORMA-T rFTOT3T~F'5T3T~2"I"5T~3fFTOTF5 C READS IN ZEEMAN SPLITTING FACTORS AND INTENSITIES READ (5*4) (AS(JJ)» B S ( J J ) , J J = 1, NS) —  —  -  _  —  -  —  —  -  —  -  -  FORMAT ( 5 1 2 F 7 . 3 ) ) WVAC = WL + CVAC CALCULATES WAVENUMBER  l | J  4  C  VA  C  C  ,  j  (1.0E+8J/WVAC  =  CALCULATES DOPPLER H A L F - W I D T H D D A = 2 . 0 * V A » S Q R T ( 2 « 0 * A L Q G ( 2 . 0 ) * B Q L T Z K * T A / <ATWT*CLIGHT»CLIGHT ) ) DDS = DRAT*DDA READS I N THETA(EXPERIMENTAL) , READ ( 5 * 5 ) ( Y E X ( I ) » I = 1 , NN ) 5 FORMAT ( 1 0 F 8 . 4 ) PRINT 6 6  FORMAT(//52H*********************************^  [ i I  )  i j  PRINT 7 » ID 7 FORMAT (10A6/) PRINT 2 5 * PDA» DDS » PQPFAC. G A M M A » NN 2 5 F O R M A T ( / 6 H DDA = . F 9 . 6 » 1 5 H CM.-l. D D S= » F 9 . 6 » 6 H C M . - 1 / / 9 H POPFAC 1 = * F 7 . 3 » 6 X » 7 H G A M M A = » F 5 . 2 * 1 I H * 1 0 - 4 / G A U S S » 6 X > 1 6 H N 0 . O F POINTS = » I 2 /  j  2 )  =0 IT = 1 WEIGHTS DATA POINTS MORE HEAVILY PAST HALF-WIDTH O F SOURCE S'A'B'OTP = P (1 )*SAB*POPFAC HFWD = S Q R T ( A L O G ( S A B K A P / A L O G ( 2 . 0 / ( E X P ( - S A B K A P ) + 1 . 0 ) ) ) ) DELFAC = CMAG»AS( 1 ) » 7 . 7 7 3 7 1 E - 5 / D P S , DO 16 J N = 1. NN X X ( J N ) = F L O A T ( J N - 1) P I S = X X ( J N ) »DE'LFAC HTJN) = 170 IF(PIS.LT.HFWD) W ( J N ) = 1 . 0 16 CONTINUE C'A'LTtrS OB'C'ITQ'F" T"0 FTN'D K O T T B Y IT'ERATTNG T O A LEAST SQUARES CURVE F I T CALL L Q F ( X X , Y E X » Y F * W » El» E 2 » P t 1 . 0 . N N » 1« N l T , N D » EP» A U X ) IF (ND.EQ.l) G O TO 1 7 P'RTNT-9 : ' 9 FORMAT 1 1 8 H SYSTEM N O T SOLVED ) GO TO 1000 CArULn*\TES V-ArUE-pRaPORT 1 7 AREL = P(1)*DDA*1.0E+10/(WVAC*WVAC*GVAL) PE IS VALUE O F SIGMA(LQF) LA  C  -  -  —  -  1  "C C  ' ,  i .  | | | I I I J j I  _  j  PE-H—rooTo-*-ET("rr/pi"r) 10  PRINT 1 0 . P ( l ) . FORMAT(7H K O L =  E 2 1 1 ) . . F 1 0 . 5 /  PE. AREL 5X.16HRMS.TOT.ERROR  = .F10.5.6X22HPERC  I  '  i  2ENT TOTAL ERROR = .F10.5/8H AREL = "»F10.5/> PRINT 12» (YEX(J)» J = 1, NN) 12 FORMAT ( 27X, 13HY(EXPERIMENT)/(10F8.3)) C TF I~S THE THEORETICAL BEST F I T PRINT 13i(YF(J)« J = l.NN) 13 FORMAT (/27X» 23HY(THEORETICAL BEST F I T ) / ( 1 0 F 8 . 3)) I'FTFE.GE.rO.) GO TO 1002 GO TO 1000 1002 CONTINUE .  E  .  N  .  D  ,  SIBFTC ARAUX C USED BY UBCLQF TO CALCULATE THEORETICAL THETA (EQN. 2.13) FUNCTION AUXTP. D» XJA» JA) COMMON AS<5>» BS(5)» FA(101)» FBU51)* NS. DDS• DDA» Y E X 1 8 1 ) . NN» 1CMAG * POPFAC. GAMMA» SAB. SX(51) » LA» IT* LN» DRAT* D P F ( l O l ) DTMEWS1W"~PTTTT~ D T D IF (JA.GE.2) GO TO 301 C AKOL IS VALUE OF KQL FOR A PARTICULAR ITERATION AKOL" =~~PTD C CALCULATES SOURCE LINE PROFILE (EQN. 2.8) AND STORES DO 21 K - 1»51 21~S~XTR) = 1.0DO - DEXP ( DBLE <~SAB*POPFAC*AKOL*DPF ( 2*K - 1 ) ) ) ) SQ = 0.0 301 HAN = XJA*CMAG bKQUi  IF < LA.EQ.(NN + 1)> GO TO 303 GO TO 302 jOra-CA"^"! • IT = IT + 1 C CALCULATES INTEGRALS IN EQN. 2.13 USING SIMPSONS RULE 3~0~2™STW2T^~~0T0 ~~ SUM2B = 0.0 SUM2D = 0.0 SUM2 G~H-~0T0 SUM4T = 0.0 SUM4B = 0.0 —SU'MZTD ^~0T0 SUM4G = 0.0 DO 13 N = 1» 4-7 • 2 x-2-^~Fuo"A"T-CNTrr/roTo— " X4 = FLOAT(N)/10.0 LN = N + 2 r  -  ~  C C  C C  TOPX IS THE VALUE OF THE I NTEGRANP~ IN THE NUMERATOR INTEGRAL SUM2"!" = SUM2T + T0PXIX2* HAN* AKOL* BOTX* DIFF* DBTX) BOTX IS THE VALUE OF THE INTEGRAND IN THE DENOMINATOR INTEGRAL SUM2B = SUM2B + BOTX SUM2D = SUM2D + DIFF SUM2G = SUM2G + DBTX ' \ LN = N + 1 SUM4-T = SUM4T + T0PX(X4» HAN» AKOL* BOI'X* DIFF* DBIX) SUM4B = SUM4B + BOTX SUM4G = SUM4G + DBTX 13 SUM4D = SUM4D + DIFF LN = 1 _ TPO = TOPX(0.0» HAN* AKOL» BOTX* DIFF* DBTX) BTO = BOTX DFO = DIFF DBO = DBTX LN = 51 TP46 = TQPX(5.Q» HAN* AKOL* BQTX* DIFF* DBTX) BT46 = BOTX DF46 = DIFF DB46 = DBTX L~N = 50 ~ TP45 = TOPX(4.9» HAN* AKOL* BOTXT DIFF* DBTX) BT45 = BOTX DT"4"5"^T~DTFF DB45 = DBTX TOP = TP0 + TP46 + 4.0»(SUM4T + T P 4 5 ) + 2.0»SUM2T BOT = BT'O + BF46 + 4.0*(SUM4B + BT45) + 2.0*SUM2B DFF = OFO + 0F46 + 4.0*<SUM4D + DF45) + 2.0*SUM2D DBT = DBO + DB46 + 4.0#(SUM4G + DB45) + 2«0*SUM2G A'UX I"S TTiE VTAXUE OF T"HET7\ A~T MAliNETIX Fl'ECD VALUE HAN ~ AUX = 0.5 + TOP/BOT D H ) IS THE DERIVATIVE OF THETA WITH R E S P E O 10 KOL D-(T)—^-PFF'/BOT—PB'T-*-T0"P7TB"0T-*-B*0'T-) OYEX = YEX(JA) - AUX SQ = SQ + PYEX*PYEX -  TC C  -  —  —  -  -  IT ("JA" ^ N'N~) 1~0T» TO~Z* T0"2 -  102  -  -  -  XNF = FLOAT(NN) SQ = SORT(SQ/XNF) S'Q I'S THE RM'S~P'EVTA"T TO"N P"ER~P'0TN'T 0'F~Tfl'E TrTEURTTTCAL^R"VE FROM THETA(EXPERIMENTAL) PRINT 103* SQ -  -  -  —  _  I  5  '  * Q I 8 | 6 #^ J"* 131 I 9  L  o l  7  103 FORMAT (~72~2HRM~S~TJEV. P~ER~POTNT"3 » F"8T57~) 101 RETURN END SIBFTC INTFUN C CALULATES INTEGRANDS FOR THE INTEGRALS OF EON. 2.13 FUNCTION TOPX(XX» HAN* AKOL* BQTX » DIFF* DBTX) COMMON AS"(5)» B"ST5 ) » F A d O D * FB(51)» NS* DDS» DDA* YEX(81)» 1CMAG* POPFAC* GAMMA* SAB * SX<51)» LA* IT* LN* DRAT. D P F ( l O l ) DI MENS I ON SUMP < 51» 8 1 ) * SUMN(51* 8 1 ) * SUMC(51* 81) D~E"L = HAr^*7T77T7lE-!J/D~DB~" C HFAC ACCOUNTS FOR THE MAGNETIC FIELD EFFECT ON N HFAC = 1.0 - GAMMA*1.QE-4»HAN RK'OL = A"K"0"L*HFAC -  C C  C  "C C  —  IF(IT.GT.1)  GO TO 20  CALCULATES AND STORES LINE PROFILES FOR LEFT AND RIGHT POLARIZED LIGHT* AND THE DISPERSION SUMP(LN* LA) = 0.0 SUMN(LN» LA) = 0.0 SUMClLN* LA) = 0.0 SUMS OVER ZEEMAN COMPONENTS (EQNS. 2.20 A N D 2.21) DO 10 J J = 1 * NS D~EO~S~^D~Er*A~S"(HJT XP = (XX-DELAS)*DRAT XN = (XX + DELAS)*DRAT LTN E A R I N T E R~PO L~A T TON~TO~~FTNTr~V A'CLTE^OT^WO R PTTOTT"Rl5M"S"T 0 R E D DOPPLER LINE SHAPE IFD = 0 X'D~^~XN 13 IFD = IFD + 1 IF(XD.LT.O.O) XD = -XD rFTXDTL~ET4V95T~G~0~~ T'CTTl YP = 0.0 GO TO 12 n.—rp-^~rF"rx-(-xD#-20T0-)~+ 1 PYP = XD*20.0 - F L O A T ( I P - 1) YP a (1.0 - P Y P ) * D P F ( I P ) + PY'P*DPF(IP + 1) 12"rF"(TFDTG"ET2") G0 T0~r4 YN = YP XD = XP G'0~T"01"3 "~— 14 S U M P ( L N » L A ) = SUMP(LN» LA) + B S ( J J ) * Y P SUMN(LN» LA) = SUMN(LN» LA) + B S ( J J ) * Y N -  —  —  —  _  !  NN *  c  FINDS DISPERSION BY LINEAR INTERPOLATION FROM DAWSONS INTEGRAL DATA IFX = 0 FAC = 1*0 XF = XN 1 IFX = IFX + 1 I F ( X F ) 2» 3» 3 2 XF = -XF FAC = -1.0 3 IF (XF.GT.2.0) GO TO 4 IX = I F I X ( X F * 5 0 . 0 + 1.0) PX = XF*50.0 - F L O A T ( I X - 1) FP = ( ( 1 . 0 - P X ) * F A U X ) + PX»FA('IX + 1>)*FAC GO TO 6 h RX = 1.0/(XF*XF) IX = I F I X ( R X * 2 0 0 . 0 + 1.0) PX = RX#200.0 - F L O A K I X - 1) FP = ( ( 1 . 0 - P X ) * F B ( I X ) + P X * F B ( I X + 1 ) ) * F A C / X F 6 I F ( I F X . G E . 2 ) GO TO 5 FAC = 1.0 FN = FP XF = XP GXTTTj-i ~ 5 SUMClLN* LA) = SUMC(LN» LA) + B S ( J J ) * ( F N - F P ) 10 CONTINUE 2~0~~A'BST = EXP(-HK"OL*0.5*1"SUMP(LN» LA) + SUMN(LN» L A ) ) ) BTP = EXP(-HKOL*SUMP(LN» L A ) ) BTN = EXP(-HKOL*SUMN(LN» L A ) ) 1  !  SHCP1"E~^SDMC("CN~I^T*T5W1"8 9"5"8 1  C  CARG IS VALUE OF EQN. 2.21 CARG = HKOL*SMCPIE R-OT-5—CO'S-rCAR-G") DROT = SIN<CARG)*SMCPIE*HFAC K = I F I X ( X X * 1 0 . 0 + 1.001) •Q rNTEG'R"A"N'D~F'O'RN'0'M'E'R"A"TUR ITtfTEUR'AU" TOPX = SX( K)*ABST*RQT. C INTEGRAND FOR DENOMINATOR INTERGAL . BOTX = SX CKpHTiTP +~BTN") DIFF = - S X ( K ) * A B S T * ( (SUMP(LN». LA ) +SUMN ( LN » LA ) ) *. 5*HFAC*R0T + DR0T •) DBTX = -SX(K)»HFAC»(SUMP(LN> LA)*BTP + SUMN(LN * LA)*BTN ) r0T~R"ETOR'N END SENTRY -  -  9  I  i t .  r  f  i  1  •  -61B.  C a l c u l a t i o n of k l F r o m 0  The  use  Relative Absorption  of the digital computer was  of the relative absorption data.  The  Data  also n e c e s s a r y  in the analysis  k l ' s for a p a r t i c u l a r line at each 0  magnetic field were found by the iterative halving technique. value of k l was  chosen and the theoretical transmittance,  Q  calculated f r o m Equation  2. 24.  compared with the observed obtain  Depending on how  value of T,  k l was Q  better agreement i n the next calculation.  until the difference between the calculated and less than 0.0004.  The  Fortran IV programme  calculations i s given below. in F i g u r e A2.  The  The  An  T,  initial  was  this calculated value r a i s e d of lowered to The  observed  iteration proceeded values of T  used to p e r f o r m  was  these  general flow of the. calculations i s given  symbols used there are as defined for the body of  the thesis. The  integrals involved were calculated n u m e r i c a l l y as i n the  fitting p r o g r a m m e  of Appendix I A.  The  curve  l i m i t s of integration calculated  in the p r o g r a m m e were set to include only the significant intensity contributions from the source The  programme  line i n order to not waste computer  also calculated the relative absorption values,  F u r t h e r information about the p r o g r a m m e i s given i n the statements (C).  time.  G(H ^).  comment  a  -62-  MALN P R O G R A M M E Reads i n H  R E LABS  's, T_, T  s  starts on individual line data Reads i n /\ , ju. , a , j9 , m e a s u r e d T's, and an i n i t i a l guess for k l  starts on new line  s  s  Q  Calculates  AV  0  , A\>  t  starts on f i r s t value of T using i n i t i a l guess f o r k l Q  Begins calculation of theoretical T using the present value for k ^ l  better  value of k l  Calculates l i m i t s of integration r e q u i r e d for integrals of Equation 2. 24  Calculates both integrals  of Equation 2. 24  Calculates theoretical transmittance = P  C o m p a r e s P to m e a s u r e d T to determine what change i n k l would make values c l o s e r (iterative halving) Q  If  |P - T J ^ 0.0004 then value of k 1 presently i n use i s chosen as c o r r e c t one  I  A f t e r calculating k l for a l l six values of T, then calculates values f o r G (Equation 4.4) Q  T S U B P R O G R A M M E SUB Calculates the value of the integrands for each i n t e g r a l at each i n t e r v a l of the Simpson's rule integration F i g u r e A2  F l o w C h a r t of Relative  Absorption  Calculation  o  J I B F T C~R E C A B S K O L D E T E R M I N A T I O N BY I T E R A T I O N TO R E A C H T H E E X P E R I M E N T A L V A L U E OF c THE TRANSMITTANCE c C O M M O N A S 1 5 > * B S « 5 ) » S A B * D R A r» A X l l O l ) . N S * D E L * N Q » I T DATA B O L T Z K / 1 . 3 8 0 5 4 E - 1 6 / * ATWT/66.3310E-24/* 2 C L I G H T / 2 . 9 9 7 9 2 5 E 1 0 / . D F A C / 7 . 7 73 7 1 E - 5 / BOLTZK IS BOLTZMANNS CONSTANT IN ERGS PER DEGREE K E L V I N c A T W T I S T H E M A S S O F AR I N G R A M S c C L I G H T IS THE S P E E D OF L I G H T IN C M . ( S E C > - 1 c DFAC IS THE ZEEMAN S P L I T T I N G CONSTANT * 2 * S Q R T ( L N ( 2 ) ) c DIMENSION T(6)» HA<6>» HKOL<6>» -KOL'6) R E A D ( 5 » 1 0 ) ( H A d ) » I = 1* 6 ) H A IS THE MAGNETIC F I E L D IN GAUSS c 10 FORMAT (6F10.1) TS = 370.0 TS IS THE SOURCE TEMPERATURE IN DEGREES K E L V I N c TA = 3 2 0 . 0 TA IS THE ABSORBER T E M P E R A T U R E IN D E G R E E S K E L V I N c C A L C U L A T E S AND S T O R E S D O P P L E R L I N E P R O F I L E c D O 2 0 L = 1* 1 0 1 XL = FLOAT IL - D / 2 0 . 0 20 A X I L ) = E X P ( - X L * X L ) S T A R T S C A L C U L A T I O N S FOR I N D I V I D U A L L I N E S c 1 5 0 R E A D < 5 » 6 ) W. N S » C V A O P K O L * GAMMA* S A B 6 FORMA 1 I F l O . i * 15* 4 F 1 0 . 5 ) W IS THE WAVELENGTH IN ANGSTROMS c N S I S T H E NUMBER OF Z E E M A N COMPONENTS c C V A C C O R R E C T S T H E WAVE L E N G T H T O V A C C U U M c I F ( W . L E . 0 . 1 ) GO T O 1 0 0 0 R E A D ( 5 * 4 1 ) ( T ( I ) » I = 1, 6 ) c 1 IS 1 HE OBSERVED T R A N S M I S S I O N 41 FORMAT (6F10.6) READ < 5 » 4 ) ( A S ( J J ) » B S ( J J ) » J J = 1*5) A"S rS T H E Z'EEMA'N R E L A T I V E 5'P'L I 1 I T N G F A C T O R " WTI H" I NT E N S I T Y B S c 4 FORMAT (5(2F7.3)) WVAC = W + C V A C VA = ( 1 . 0 E + 8 ) / W V A C V A I S T H E WAVE N U M B E R I N R E C I P R O C A L C M . c C A L C U L A T E S D O P P L E R H A L F - W I D T H S F O R A B S O R B E R ( D D A ) AND S O U R C E ( D D S ) c D D A = 2 « 0 * V A * ' i U K r ( 2 . U * A L O G 1 2 . U J * B O L T Z K * I A / ( A T W T * C L I GH T * C L I G H T ) ) DRAT = S Q R T ( T S / T A ) DDS = DRAT*DDA  c  c c  WRITE ( 6 * 3 ) W 3 FORMAT (5H W = » F8.1/) LK = 0 S l A R I S C A L C U L A T I N G KOL FOR EACH MAGNETIC F I E L D VALUE DO 99 I = 1 » 6 TRANS = T( I ) HA I = H A ( I ) DEL = <HAI#DFAC)/DDS ERR = 0.01 AKOL = P K 0 L * ( 1 . 0 - GAMMA*1.0E-4*+AI> PKOL I S A I N I T A L GUESS FOR KOL S T A R T S METHOD OF I T E R A T I V E H A L V I N G T Q FIND S U C C E S S I V E V A L U E S TT = 0 ILO = 0 IHI = 0 NK = 0 121 I F ( I L O . G T . O . A N D . I H I . G T . O ) GO TO 127 GO TO 120 127 GO TO (36» 3 7 ) » MK 36 2 = AKOL Y = AKOL*1.20 GO TO 124 37 Y = AKOL 2 = AKOL/1.20 124 AKOL = CZ + Y ) * 0 . 5 NK = NK + 1 120 C O N T I N U E I  C  C  . __ T  r T  __  1  I F U T . G T . 3 0 ) GO TO 122 C A L C U L A T E S I N T E G R A L S IN EQN. 2.24 USING SIMPSONS S~D~~= A~S~n.T*"D"E L ED = A S ( N S ) * D E L F I N D S L I M I T S OF I N T E G R A T I O N * EM IN AND EMAX BMTN = S'D ="4~5 ~" EMAX = ED + 4.5 N I H = I F I X U E M A X - E M I N ) * 5 . 0 + 0C5 ) NT - 2"*"NIH FN I = F L O A T ( N I ) FNIM = F L O A T ( NI - 1) NT'I —NT-="~3 SUM4T = 0.0 SUM4B = 0.0 -  -  —  -  RULE  OF  KOL  SUM2T SUM2B  = 0.0 = 0.0  DO 145 N = 1 » N i l *  2  X2 = FLOAT (N + D / 1 0 . 0 + EMI N X4 = F L O A T ( N ) / 1 0 . 0 + EMI N ' NQ = N + 1 SU"M2T = SUM2T + F H T I X 2 * AKOL» SOURCE) C FHT IS THE VALUE OF THE INTEGRAND IN THE NUMERATOR INTEGRAL SUM2B - SUM2B + SOURCE C SUQRCE IS~TH'E VALUE OF THE INTEGRAND IN THE DENOMINATOR INTEGRAL NQ = N SUM4T = 5UM4T + F H T ( X 4 * AKOL» SOURCE) 145 SUM4B = SUM4B + SOURCE NQ = 1 FHO = FHT(EMIN» AKOL* SOURCE)  sw~^~~SOWCE  C  EMAX = F N I / 1 0 . 0 + EMIN NQ = NI + 1 FR"4'0 = FHTTEMAX* AKOL* SOURCE) S040 = SOURCE EMAM =• FNIM/10.0 + EMIN NQ = NI ; ~ FH39 = FHTIEMAM» AKOL* SOURCE) SQ39 = SOURCE TOP = (FH"0 + FH40 + 4 . 0 X S U M 4 T + FH39) + 2.0*SUM2T) BOT = (SOO + S040 + 4.0#(SUM4B + S 0 3 9 ) + 2.0*SUM2B) P I S THE THEORETICAL VALUE OF T CALCULATED BY EQN* 2.24 p—T'OP"/"BO"T IF(NK.GT.O) GO TO 39 IF < P - TRANS) 30* 123* 31 3~0~&Kd£~~Z~&KQir/1.20 IHI = IHI + 1 MK = 1 GO~TO 1"21 31 AKOL = A K 0 L * 1 . 2 0 ILO = ILO + 1 MK~^r"2 GO TO 121 39 D I F F = ABSCP - TRANS) GHE"C"K"S—IT lTE'R"AT"I"aN~CA'R'Rl'E D~00 T ^ 0~S"0 F F i r ITN"T~ATC 0 RATY IF ( D I F F . L E . 0 . 0 0 0 4 ) GO To 123 IF <(P - (TRANS + E R R ) ) . G E . 0 . 0 ) I = AKOL —  'C  -  IF H P - < TRANS-ERR) ) .LE.0.0 ) Y # AKOL IF (DIFF.GT.ERR) GO TO 124 ERR = ERR/10.0 GO TO 124 122 WRITE (6.7) 7 FORMAT (15H DID NOT CLOSE ) LK = LK + 1 GO TO 99 123 QKOL(I) = AKOL C AKOL HERE 13 FINAL VALUE OF KOL AT A PARTICULAR FIELD VALUE WRITE (6.8) I-» TRANS. AKOL 8 FORMAT ( H H FOR TRANS(, I2» 5H ) = » F6.3* 10H » KOL = .F10.6) 99 CONTINUE DO 58 I = 1.6 C FINDS VALUE OF G(HA I) FROM EON• 4.4 58 HKOL(I) = QKOL(I)/QKOL(4) IF (LK.GT.O) GO TO 98 WRITE (6.42) ( H K O L ( I ) . I = 1. 6) 6(F10.6» 3X) ) 42 FORMAT I / 6X» 12H KOL(H) 98 WRITE (6»9) 9 FORMAT (//) GO TO 150 1000 CONTINUE STOP SIBFTC SUB C THIS SUBPROGRAM CALCULATES THE VALUE OF THE INTEGRANDS FUNCTION FHTTXX. AKOL » SOURCE) COMMON AS<5)» BS(5)» SAB» DRAT. A X ( l O l ) . NS. DEL* NQ» IT DIMENSION SUMA1400). SUMS ( 400 ) ~rF"(TTTGTTT) GO T0~~2"0 C SUMS OVER ZEEMZN COMPONENTS (EQN• 2.26 AND EQN. 2.27) SUMA(NQ) = 0.0 —SUfSTSlNQ) = OTO ~ DO 10 JJ= 1. NS DELAS = D E L * A S ( J J ) XS-^-XX^^DELAS IF(XS.LT.O.O) XS = -XS X = DRAT*XS "C FT'ND-S-VAXUE OF~EiMl"S"Sl-0'N~A"ND~S"00'R"CE~PR'0"FT'L"E B'Y-L 11^ A'R^TNTET^PWATTON C FROM STORED DOPPLER PROFILE DATA I F ( X S . L E . 4 . 9 ) GO TO 3 -  -  -  _  Ol  CTl  Nj  CO  P  UD  fG •«-  C13.1 i w n NIVUD *n y -  o z tO  s:  tO  * _l  o  <  <  < *  i  +  < >-• r-  <  *  o CM *  o  OIL  — o  O CM  tO O  *  <r • •  M r H UJ I • v o • •* XU_ C LO  «-« X —  O  o  II  II  1— II  >-0  UI O  ifl  J  cn  —I •  X  O  UJ — Q  O II  O  I—  u. x II  tO oO  O CD Q < O —I *  i-  • o  r-l  00 -0  i-i  i- ' II  I  —3 < to LO  ^  < _l *  —  C3 CD II  to to co a  to to < < LD _ l Q. < o o in  UJ  Z < U to tO Qi  o o  n < to i -  >- u.>- o < < .> LO •—>  CD CO  < + 'r+ <  X I- I — o X Kl O  tO o  ~-» p o  to iO  * -  •  • o a cc— z — a a a a UJ x U J  + !<c p x o q —U < z z  r-  cn  UJ  U. CD  a  +  to — I o  —  _ l CM  < -**£  CL  X  LU < in  X  tO  I  CD <  +  +  LU U  CC  • o z to  3 S tO  II  r-  UJ  to K-lco t-u . or zO < o tO  a z UJ  — *«  O U. CM  UJ  B — y  ^  - 6 3 -  EFFECT  ON  ANOMALOUS OF  APPENDIX  II  DISPERSION  OF  THE  In a d i e l e c t r i c medium  ABSORPTION  The  dependent.  =  sew) £_  ecou)  =  e  € Ccv)  and  'ccv)  ^  c  +  £ (a)) which i s complex and  e'too)  • • .  A .  . . .  A .  i 2  £ (u>) are related to the usually defined  absorption coefficient k and the index  £  D is related to the  i . e.  Q  r e a l quantities  BROADENING  LINE  the e l e c t r i c displacement  e l e c t r i c field E by a d i e l e c t r i c p e r m e a b i l i t y frequency  COLLISIONAL  of r e f r a c t i o n n as follows  13 :  2 ^ g "(CJU) • • .  A .  3  A .  4  which assumes n « 1 and,  €'(&)-/ which assumes little absorption over one wavelength.  K r a m e r s - K r o n i g relations give the relation between s'Cco) and  The £>;  as  1 3  :  t(WJ  i  _  ^  cu'*--CO**  r  ...  where P indicates the p r i n c i p a l value i s to be taken. expressions  for k, n, and £&>) we  =  ZTT*-'  Defining (ju and u ) ' by Equation  Combining the  obtain a relationship between the  absorption coefficient and the index  n(w) — /  A .5  of r e f r a c t i o n :  j  CV -U," , Z  ^  ... A . 6  2. 9 changes this equation to:  -64-  4-7T If k(aO may  1  Oil  J  be given by:  A(UJ')  =  2 . 18) as:  However, processes.  "  =  F«o)  the absorption line may  In this case  A.S  only, the e x p r e s s i o n for n i s given  and B r a d l e y ^ (see Equation  * "'  ...  A,*"  which assumes Doppler broadening by F o r k  CO  —  . . .  A . 9  also be broadened by c o l l i s i o n a l  the absorption coefficient i s given by a Voigt  4 profile : "  n>  J  a  i +  (ou-p*-  ...  where a i s a ratio of the half-width due to the c o l l i s i o n a l to the Doppler half-wi dth, AV  A . 10  broadening,  : i.e.  p  CL —  /ZJ  Thus, i n this m o r e general case,  . . .  A .  11  A .  12  n i s given by:  where  ,  „*•  + 0 0  ~ r  = The  r\  value of the i n t e g r a l  -^y-Hj  Jr^o.) was  d  "  ... A . 13  calculated n u m e r i c a l l y on a  computer for v a r i o u s values of OJ and (X. .  The values of the Voigt  p r o f i l e i n t e g r a l r e q u i r e d i n the calculation were computed using a F o r t r a n I V subprogramme The  developed  at the U n i v e r s i t y bf M i c h i g a n  results of this calculation are given i n F i g u r e A 3 .  14  -66F o r (X = 0. 0 the r e s u l t i s identical to the values of — Z/lFT{cO) which comes f r o m the relation which  assumes no c o l l i s i o n a l broadening.  For  (X =-0. 05 the difference of the index of r e f r a c t i o n f r o m unity i s about 5% l e s s that for the pure Doppler case within one Doppler half-width f r o m the line centre.  However, for l a r g e r values of UJ the d i s p e r s i o n  becomes i n c r e a s i n g l y independent of the absorption line shape, for values of (X ^ 1. 0 and  becoming  0U>7. 0 i n v e r s e l y p r o p o r t i o n a l to uJ: i . e . A. 14  

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