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UBC Theses and Dissertations

Zero field level crossing in molecular hydrogen Van der Linde, Jacob 1970-12-31

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ZERO FIELD LEVEL CROSSING IN MOLECULAR HYDROGEN by JACOB  B.Sc,  VAN  DER  LINDE  U n i v e r s i t y o f B r i t i s h Columbia, 1 9 6 7  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of PHYSICS  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 7 0  In p r e s e n t i n g  this  thesis  an a d v a n c e d d e g r e e a t the L i b r a r y I  further  for  agree  scholarly  by h i s of  shall  this  written  the U n i v e r s i t y  make  it  freely  that permission  for  It  financial  of  Columbia,  British for  gain  Columbia  the  requirements  reference copying o f  I agree and this  shall  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  of  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  of  for extensive  permission.  Department  fulfilment  available  p u r p o s e s may be g r a n t e d  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  ABSTRACT  The  lifetimes  have been measured The  transitions  3d E  ->- 2 p E  J  J  o f t h e 3d Z(v=0) J= l, 2 and 3 s t a t e s 1  using  zero-field  observed  level  crossing  w e r e 'the R - b r a n c h members  transition.  between  two c a p a c i t o r p l a t e s t o w h i c h a  frequency  voltage  i sapplied.  MHz  first  rotating the  Polarization  a polaroid  resulting  R(0), to  field  curves  strength against  pressure,  made  was m e a s u r e d by sensitive  detecting  yield  - 1 1  o f 2.37±.12 g a u s s ,  Using lifetimes  *  of the  h a l f w i d t h s , when e x t r a p o l a t e d  cell  the discharge  by p l o t t i n g t h e  thepolarization  in  their  were  u s i n g a ^50  obtained  The h a l f w i d t h s v a r y  1.5xl0  radio-  modulation.  3.25±.25 g a u s s .  roughly  light  i n t h e beam a n d p h a s e  R ( l ) , R(2) l i n e s  zero  and l a t e r  of emitted  The.depolarization magnetic  The measurements  a 180 MHz R . F . s o u r c e  source.  of the  The u p p e r s t a t e i s e x c i t e d i n a  discharge  using  techniques.  yielding  2.60±.15 g a u s s • a n d linearly  collision  with  pressure  cross-sections of  cm . 2  the high  field  Lande  g values  o f these  a r e (2.66±.12)xio~ sec., (3.83±•2)xio" sec., 8  8  and  (3-93±.25)xl0 sec.  f o r J = l , 2 and 3 r e s p e c t i v e l y .  The  discrepancy  the f i r s t  _8  times  states,  between  i s discussed.  and t h e l a t t e r  two l i f e -  TABLE OF CONTENTS CHAPTER I II  PAGE INTRODUCTION  . . . . .  THEORY  8  §2.1  Introduction  §2.2  The M a g n e t i c F i e l d Dependence o f P o l a r i z a t i o n i n C l a s s i c a l Terms, .  §2.3  . . . . .  8 8  Quantum-Mechanical D e s c r i p t i o n o f Level Crossing  III  1  . . . . .  14  §2.4  The 3 d E L e v e l s o f H  §2.5  The E x c i t a t i o n M a t r i x Elements Q ..  x  2  . . . . . .  18  EXPERIMENTAL DETAILS . .  22 28  §3.1  E x p e r i m e n t a l Arrangement . . . . . . .  28  §3.2  The D i s c h a r g e  . . .  30  §3-3  The O p t i c a l System  . . .  34  §3-4  The Vacuum System  §3.5  R.F. S u p p l i e s  §3.6  Helmholtz C o i l s  §3.7.  Current Supplies  §3.8  Lock-in Amplifier  49  §3.9  Rotating Polaroid  50  .  and C o u p l i n g  37 . . . .  40 42  f o r Helmholtz C o i l s  §3.10 The V a r i a b l e Quarter-Wave P l a t e  .  46  50  §3.11 P h o t o m u l t i p l i e r  54  §3.12 X-Y R e c o r d e r .  54  §3.13 L i q u i d Notrogen Bath  55  §3.14 Data P r o c e s s i n g  ........  r . .  56  - i v -  CHAPTER •  IV  PAGE EXPERIMENTAL RESULTS §4.1  Lifetimes  §4.2  Collision  §4.3  Polarization  §4.4  Upper  §4.5  Experimental  ' §4.6  57 57  Cross-sections  State  64 68  Populations Errors  . . . . . .  69  . . . . . . .  71  a)  Discharge  b)  Magnetic  Field  c)  Pressure  I n the Discharge  d)  Temperature  e)  Cascading  f)  Coherence  g)  R.F. B r o a d e n i n g  Helium That  V  . .'  4 D X  Stability  . . . .  72 .  i n the Discharge  Narrowing  73  . . . .  74 75  With  From Other E x p e r i m e n t s  D I S C U S S I O N OF R E S U L T S AND  CONCLUSION  . . .  Introduction  §5.2  Hyperfine  §5.3  E l e c t r o n i c Wave F u n c t i o n V a r i a t i o n with J C o n c l u s i o n and Suggestions f o r F u r t h e r Work  APPENDIX  I  APPENDIX  I I  80  Effects  . .  THE T R A N S I T I O N M A T R I X E L E M E N T S THE  3d  1  S T A T E S OF H Levels  76 80  §5.1  §5.4  73  74  L i f e t i m e Compared  Obtained  71  §A2.1  Energy  §A2.2  The Zeeman E f f e c t  2  80  . . .  . . . . . . .  and E i g e n s t a t e s  85 86 87  .  91  . .  91 97  - v -  PAGE APPENDIX I I I OTHER STATES §A3.1  The 3 d n ~ ( v = 0 ) J = 2 S t a t e  §A3.2  The 3 * K S t a t e  1  REFERENCES AND FOOTNOTES  102 102  .104 105  LIST OF TABLES TABLE I  II III  •  PAGE  R e l a t i v e P o p u l a t i o n s o f t h e F i r s t Few R o t a t i o n a l L e v e l s o f Hydrogen a t Thermal Equilibrium  21 .  P r o p e r t i e s o f E l e c t r o n M o t i o n i n an R.F. E l e c t r i c and D.C. M a g n e t i c F i e l d . . . . .  34  L e v e l C r o s s i n g Curve H a l f w i d t h s  C3d IT and 1  103  3 K) X  IV V VI VII VIII IX X XI  E x t r a p o l a t e d H a l f w i d t h s and L i f e t i m e s . . .  64  P o l a r i z a t i o n Curves' H a l f w i d t h s  65  C o l l i s i o n Cross-sections  68  E x p e r i m e n t a l Upper S t a t e P o p u l a t i o n s  70  L i f e t i m e o f t h e 4*D S t a t e o f Helium . . . .  77  Energies of the 3 d  94  Complex o f H  1  2  Expansion C o e f f i c i e n t s f o r the 3 d  1  +  States  95  R e l a t i v e I n t e n s i t i e s o f P and R T r a n s i t i o n s in  XII  . . .  t h e 3d*E ->• 2 p E J  ( 0 , 0 ) Band.  g-values o f the 3 d ! State 1  .  97  .99  ILLUSTRATIONS AND  FIGURES  FIGURE  PAGE  1  The  S i n g l e t S t a t e s of Hydrogen (H )  .3  2  The  T r a n s i t i o n s Observed  3  Hanle E f f e c t C o - o r d i n a t e  4  P o l a r i z a t i o n f o r (J>=0  5  Polarization for  6  "Three" L e v e l System  7  The  S c a t t e r i n g Angle  26  8  The  Apparatus  29  9  C r o s s - S e c t i o n of Discharge Plates  2  .... System .  9 •". • •  (J>=TT/4  4  .  11 11  ' 14  C e l l and  Capacitor  31  10  The  Vacuum System  39  11  180  MHz  4l  12  T-Matched Resonant C i r c u i t  13  H e l m h o l t z C o i l Power Supply  45  14  D.C.  Power A m p l i f i e r  47  15  -10V  t o +10V  16  P o l a r o i d Rotator  51  17  Quarter-Wave P l a t e  53  18  P h o t o m u l t i p l i e r W i r i n g Schematic  19 20  R.F.  Oscillator  Voltage  . . . ;  Sweep Mechanism  43 .  ....  . . . . . .  48  54  . E x p e r i m e n t a l L e v e l - C r o s s i n g Curve f o r the R(0) L i n e U s i n g 450 MHz E x c i t a t i o n  58  L e a s t Squares F i t t e d Curve f o r the R(0) U s i n g 450 MHz E x c i t a t i o n  59  Line  - viii -  FIGURE  21  .  PAGE  E x p e r i m e n t a l L e v e l - C r o s s i n g Curve f o r t h e R ( 0 ) L i n e U s i n g 1 8 0 MHz E x c i t a t i o n . . . . .  60  22  L e a s t Squares F i t t e d Curve f o r t h e R(o) L i n e U s i n g 1 8 0 MHz E x c i t a t i o n  61  23  L e v e l C r o s s i n g Curve H a l f w i d t h as a F u n c t i o n of Pressure  62  24  L e v e l . C r o s s i n g Curve H a l f w i d t h - a s of Pressure  63  25  Experimental  L e v e l - C r o s s i n g Curve f o r t h e  H e l i u m 4 D -> 2 ? X  a Function  1  Transition  .........  .26  4*D Curve H a l f w i d t h as a F u n c t i o n o f P r e s s u r e  .27  Zeeman E f f e c t i n t h e P r e s e n c e o f S m a l l Hyperfine S p l i t t i n g The F i r s t Few L i n e s o f t h e R Branch o f t h e 3 d Z - ^ 2 p E ( 0 , 0 ) band o f H . . . . . . . . . .  28  1  1  2  78 79 84 101  ACKNOWLEDGEMENTS  I w i s h t o e x p r e s s my g r a t i t u d e t o P r o f e s s o r s u g g e s t i n g t h e problem.  F.W. Dalby f o r '  H i s supervision of the research,  a d v i c e and encouragement were i n v a l u a b l e . I a l s o w i s h t o thank my w i f e of t h i s t h e s i s .  f o r her help i n the preparation  Other c o n t r i b u t i o n s , b o t h t a n g i b l e and  i n t a n g i b l e , have been made by my f e l l o w graduate  students..  T h i s work was s u p p o r t e d by t h e N a t i o n a l Research C o u n c i l o f Canada.  -  1  CHAPTER .1 • INTRODUCTION L i f e t i m e s or o s c i l l a t o r s t r e n g t h s of e x c i t e d  states  o f atoms and m o l e c u l e s have t r a d i t i o n a l l y been measured through a d e t e r m i n a t i o n of t r a n s i t i o n p r o b a b i l i t i e s .  The  most r e l i a b l e o f these t e c h n i q u e s i s p r o b a b l y the Hook  1  method which y i e l d s , under f a v o u r a b l e c o n d i t i o n s , strengths to 10-20% accuracy. the r e q u i r e m e n t  Common t o these methods i s  t h a t the p a r t i a l p r e s s u r e o f the  must be a c c u r a t e l y known.  present  I f , however, the a b s o r p t i o n i s produced  or molecules it  absorber  I f the a b s o r p t i o n o c c u r s from the  ground s t a t e o f a s t a b l e gaseous atom t h i s may problem.  oscillator  no  by atoms  of low vapor p r e s s u r e or by an u n s t a b l e m o l e c u l e  i s v e r y d i f f i c u l t not t o make l a r g e e r r o r s i n d e t e r m i n i n g  the number o f a b s o r b e r s . p r e s e n t .  More r e c e n t t e c h n i q u e s  as the p h a s e - s h i f t and d e l a y e d c o i n c i d e n c e methods s u f f e r from the above s h o r t c o m i n g s  2  do  such  not  but do r e q u i r e f a s t  "gates"  w i t h r i s e t i m e s s h o r t e r t h a n the e x c i t e d s t a t e l i f e t i m e ' u n d e r c o n s i d e r a t i o n or other s o p h i s t i c a t e d e l e c t r o n i c s . many cases l i f e t i m e s are o f o r d e r 1 0 ~ s e c o n d s  Since i n  or s h o r t e r , i t  8  i s a g a i n easy t o make e r r o r s of c o n s i d e r a b l e magnitude-. use o f z e r o - f i e l d l e v e l c r o s s i n g t o determine  The  lifetimes  s u f f e r s from none o f these l i m i t a t i o n s and has been used t o measure l i f e t i m e s o f many atomic s t a t e s t o measure e x c i t e d s t a t e l i f e t i m e s o f NO*, 1  and more r e c e n t l y OH , 5  and CS . 6  -  An e n e r g y states  2 -  l e v e l diagram  o f molecular hydrogen  f o rthe observed  i s shown i n P I G . l .  shown a r e t h e l o w e s t v i b r a t i o n a l , r o t a t i o n a l electronic  t h e 3d*E  with light  .This t h e s i s  ( v = 0 ) J = l , 2,  observed arises  R(2)  and  state.  and 3 state  o f t h e 3d E l  The  levels  itself  o f each  The  R(0),  -> 2 p E  levels  primarily  lifetimes.  from the t r a n s i t i o n s  o f t h e (0,0)band  corresponding  concerns  singlet  R(l),  system  l  to the transitions  3d" 2  v=0  1  -»• 2 p E -*  J=l J=2 J=3  v=0  l  J=0 J=l J=2  respectively. A more states this  detailed  i s shown i n F I G . 2 .  thesis  states  3D s t a t e s magnitude,  ought  Also and  portion  h e n c e we e x p e c t  states  The H a n l e  those  function  of the atomic  of  Helium  o f t h e same o r d e r o f  seconds .  8  7  o b s e r v e d were  several  a 3 d * n -> 2p"*E t r a n s i t i o n .  these  observed i n  o f t h e wave  lifetimes -  f o r these  arrows.  t o resemble  or roughly 1 0  l e v e l diagram  The t r a n s i t i o n s  a r e marked w i t h  The e l e c t r o n i c these  energy  3K E X  -»• 2p*E  transitions  I n f o r m a t i o n o b t a i n e d on  i s c o n t a i n e d i n Appendix I I I .  Effect o  In of  mercury  1922  excited  Rayleigh  8  discovered  by r e s o n a n c e  that  radiation,  t h e 2537 A  line  was p o l a r i z e d i f  - 3 -  ENERGY (10  3  cm ) -1  levels ns*E  np*E  np !! 1  nd*E  20  10  00  90  The e n e r g i e s o f these s t a t e s  80-  0  were o b t a i n e d from  J  Dieke  1 8  1 F i g u r e 1 - The s i n g l e t s t a t e s o f hydrogen ( H ) 2  with  3d A 1  3d ]! 1  -  viewed a t r i g h t a n g l e s Ellet  1 0  5  -  t o t h e e x c i t i n g beam.  investigated this effect  Wood and 9  f u r t h e r and found t h a t a t  low p r e s s u r e s , i n t h e absence o f a magnetic f i e l d t h e e m i t t e d r a d i a t i o n was almost c o m p l e t e l y  polarized with i t s e l e c t r i c  v e c t o r p a r a l l e l t o that of the e x c i t i n g l i g h t .  Small  magnetic f i e l d s i n c e r t a i n d i r e c t i o n s , were found t o decrease the degree o f p o l a r i z a t i o n . was  The a d d i t i o n o f f o r e i g n gases  a l s o found t o d e c r e a s e t h e p o l a r i z a t i o n .  performing  a more t h o r o u g h i n v e s t i g a t i o n  application  Hanle , 1 1  found t h a t t h e  o f a magnetic f i e l d p e r p e n d i c u l a r t o t h e  e x c i t i n g l i g h t a l o n g t h e d i r e c t i o n o f o b s e r v a t i o n not o n l y decreased  but a l s o r o t a t e d the plane of p o l a r i z a t i o n of  emitted l i g h t .  Breit  1 2  e x p l a i n e d the e f f e c t i n c l a s s i c a l  terms and showed t h a t t h e degree o f p o l a r i z a t i o n , P, i s g i v e n by t h e e x p r e s s i o n P(H) PT6T  1  = a  /1  , rgeHx^ ' ^ mc /  \  K X )  where P = I„ -Ix , L i i s t h e i n t e n s i t y o f l i g h t w i t h e l e c t r i c v e c t o r a l o n g t h e e l e c t r i c v e c t o r o f t h e e x c i t i n g beam and Ij_ t h a t w i t h p e r p e n d i c u l a r p o l a r i z a t i o n , H i s t h e a p p l i e d magnetic f i e l d ,  T i s t h e mean r a d i a t i v e  lifetime,  ge_ i s 2mc  the magnetic moment o f t h e atom. the c l a s s i c a l t h e o r y  A condensed v e r s i o n o f  i s g i v e n i n § 2.2.  By p l o t t i n g P(H) we may then e a s i l y o b t a i n t h e product  gx  }  f r o m . e q . ( 1 ) , and an independent measurement  of g y i e l d s the r a d i a t i v e  l i f e t i m e T.•  - 6 -  A more complete account o f the. e a r l y work on p o l a r i z a t i o n o f resonance r a d i a t i o n i s g i v e n by M i t c h e l l and Z e m a n s k y . 13  The Hanle e f f e c t i s a s p e c i a l example  of " l e v e l -  c r o s s i n g " , the appearance o f i n t e r f e r e n c e e f f e c t s when two s t a t e s a r e degenerate t o w i t h i n t h e i r n a t u r a l l i n e w i d t h . E l e c t r o n Impact P o l a r i z a t i o n Light emitted  from atoms e x c i t e d by low energy  (-20 e.V.) e l e c t r o n s w i l l i n ' g e n e r a l a l s o e x h i b i t p o l a r i z a t i o n r e l a t i v e .to a d i r e c t i o n a l o n g t h e e l e c t r o n beam. Measurements o f p o l a r i z a t i o n were made f o r a number o f atoms by s e v e r a l workers from 1925 t o 1935 *. 11  i z a t i o n i n general  depends on t h e i n c i d e n t  electron  energy i n a c o m p l i c a t e d manner, and no adequate has y e t been d e v i s e d .  The p o l a r -  theory  The d e p o l a r i z a t i o n o f l i g h t  emitted  by h e l i u m e x c i t e d by slow e l e c t r o n s , i n response t o a magnetic f i e l d has been o b s e r v e d by P e b a y - P e y r o u l a e t a l  1  5  The s i g n a l s have t h e same magnetic f i e l d and l i f e t i m e dependence,  g i v e n by e q . ( l ) , as t h o s e produced by o p t i c a l l y  e x c i t e d atoms.  Descourbes  1 6  a l s o r e p o r t s non-zero  l e v e l - c r o s s i n g i n t h e P s t a t e s o f He u s i n g 3  excitation.  P o l a r i z a t i o n of l i g h t emitted  field  electron from H  2  triplet  s t a t e s e x c i t e d by e l e c t r o n impact has been r e p o r t e d by P a t r i c k C a h i l l et a l  1  7  .  I n t h i s work, t h e r a d i a t i v e l i f e t i m e s o f t h e  .  - 73d !  v=0  1  J=l,  2, a n d 3 r o t a t i o n a l . s t a t e s  h y d r o g e n have been m e a s u r e d by z e r o using  electron  similar have  impact  to that  Pebay-Peyroula  not been p r e v i o u s l y  check  on wave  functions  A brief, measurements  Notation  symbols  i s given  only  are those  Conventionally," upper  this  state  convention  primed. nuclear  readable  major used  the  5  .  These  f o rthese  review  lifetimes  provide  a  levels.  of lifetime  Stroke . 9  refers  Lande  from  the  conventional  the various  t o the lower  state  states.  and ' r e f e r s t o  i n a transition.  In this  that  quantum numbers  ground  J i s the total spin  departure  state  thesis  and v i s t h e v i b r a t i o n a l quantum  g-factor  may a l s o  by u  Q  photon  the latter  will  or i t s equivalent,  a r e unof  number.  be e n c o u n t e r e d  o f an emitted  f o ra state;  be a c c o m p a n i e d  we a d d t o  a n g u l a r momentum e x c l u s i v e  the p o l a r i z a t i o n vector  always  1  i n a manner  and s h o u l d  i n denoting  • Some a m b i g u i t y g,  et a l  measured  by  level-crossing  the states  calculated  very  field  and Symbols The  the  to excite  of molecular  between and  g gjj =  nearly -5— .  -  8  -  CHAPTER I I THEORY  §2.1  .  Introduction In  effect  this  analogue  chapter  i n classical  quantum m e c h a n i c a l referred theory  we w i l l  discuss  terms.  will  Following  description of the effect  t o as l e v e l - c r o s s i n g , w i l l  given  first  follow  the Hanle this the  usually  be p r e s e n t e d .  the treatment  of Franken  The quite  1 9  closely.  N e x t we w i l l ground The  states  and t h e 3 d ! 1  transitions I s  ground  state  transition  §2.2  1  of the  l e v e l s o f the hydrogen will  be d i s c u s s e d  and allowed  electric  molecule.  i n terms o f multipole  we w i l l  attempt  to describe  the relevant  o f t h e e x c i t a t i o n mechanism.  "Classical  we r e p l a c e  Field  Dependence  o f P o l a r i z a t i o ni n  Terms In  axis  ! -> 3& Z  the structure  moments.  The M a g n e t i c  sesses  1  populations  Finally portions  consider  order  to discuss  our molecule  an angular  the Hanle  by an e l e c t r i c  effect dipole  momentum L p e r p e n d i c u l a r  and a magnetic  moment y_ = u L .  classically which  pos-  to the dipole  - 9 -  F i g u r e 3 - Hanle e f f e c t c o - o r d i n a t e system F u r t h e r , we suppose t h a t a t some time t i t h e d i p o l e i s s e t i n t o o s c i l l a t i o n w i t h a n g u l a r f r e q u e n c y v. The d i p o l e t h e n e m i t s r a d i a t i o n w i t h i t s e l e c t r i c v e c t o r p a r a l l e l t o t h e d i p o l e a x i s which a t time t i s a t some angle 0 ( t ) t o the x - a x i s .  I n t h e absence o f any t o r q u e s  on t h e d i p o l e , 8 ( t ) = 6 ( 0 ) = 0 .  I f however a magnetic  field  H=H k i s p l a c e d a l o n g t h e z - a x i s , t h e d i p o l e p r e c e s s e s z  about the' z - a x i s w i t h a n g u l a r v e l o c i t y to=uH and 0 ( t ) = =yH(t-ti).  The a m p l i t u d e o f r a d i a t i o n i s r a d i a t i o n damped  w i t h a time c o n s t a n t 2T.  Thus an o b s e r v e r l o o k i n g a l o n g t h e  z - a x i s , h a v i n g a r e f e r e n c e system £,ri,z where £ and n a r e i n c l i n e d a t a n g l e f t o x and y r e s p e c t i v e l y , w i l l a time dependent e l e c t r i c  f i e l d w i t h components:  observe  E (t)=A cos[a3(t-ti)-cf)]e?  e  1 ( v t +6 )  0  2 x  t-ti  E (t)=A sin[a)(t-t )-(!)]en  0  1  or the i n t e n s i t i e s the K and  1 ( v t + 6 )  of radiation  e  2 T  with, e l e c t r i c v e c t o r along  axes:  n  t-ti  I ^ ( t ) = I c o s [w(t-ti.)-<f>]e 2  T  0  t-ti  I  (  2  )  ( t ) = I s i n - [ u ( t - t i )-4>]e 2  0  Now, i f o u r o b s e r v e r does n o t d i f f e r e n t i a t e emitted i n the i n t e r v a l  light  where T > > T , t h e i n t e n s i t i e s  (ti,ti+T)  observed a r e 11  V  +T  V  Ur  t ) d t  85  /t V  t ) d t  x  ti+T I = /, I (t)dt = /" I (t)dt n  ;  t i  n  •'tin  The p o l a r i z a t i o n  P(<j),H) i s then ^ g ~ ^ r i , o r ?  '  n  t-ti  / " { c o s [ w ( t - t i )-<|)]-sin [ u ( t - t i )-<J>] }e P(cj),H)=—^ t-t oo f dt / ,e 2  2  T  dt  11  performing the i n t e g r a t i o n , P(cJ>,H)=  1  +  ( 2COT ) a Ccos2d> -2wTsin2<j>]  and s u b s t i t u t i n g P((j),H)=  OJ=UH  (2 HT) [ 2  1 +  u  cos2(  t>-2yHTSin2(j)]  (3)  - 11 -  <J> = 0 t h i s reduces t o  For' the s p e c i a l case o f P  (  H  )  1+(2UHT)  =  2  a 'sketch o f t h i s shape i s shown i n F I G . 4.  iP  H  F i g u r e 4 - P o l a r i z a t i o n f o r <}>=0 I t s h o u l d be n o t e d t h a t when  P(H) . ploy •  x o f t h e o s c i l l a t o r i s g i v e n by T  :  1/2 t h e " l i f e t i m e " 1_ 2yHi /2  For  t h e case tt = TT/4  F { R )  -  l+(2yHx)  2  y i e l d i n g t h e curve shown I n FIG 5-  I  A  I  I  1  '  1 1  1  \ \  Hl/2  1  -H  1/2  F i g u r e 5 - P o l a r i z a t i o n f o r tt = TT/4  - 12 -  This of  curve  not only  the magnetic  yields  moment  the l i f e t i m e but also  of the oscillator.  F r o m e q . ( 3 ) we may of. p o l a r i z a t i o n r o t a t e s for  P takes  for  an a n g l e  with  <{> •= t a n  set  4UHTCOS2<})  along  have N o s c i l l a t o r s  magnetic  field,  at a given magnetic  field  =  0  ,  or •  (2yHx)  - 1  we h a v e  s y s t e m was c o m p o s e d  into motion  the direction  = 0 , i.e.  Up t o t h i s p o i n t radiating  see that  increasing  dP ^  that  -2sin2<J) -  •  also  o n i t s maximum v a l u e <j> s u c h  the sign  of only  the x-axis  excited  tacitly  assumed t h a t  a single  at time  t i .  a t random t i m e s  our  oscillator Actually  we  t . and having t)  their  initial  motions  The oscillators decaying  on axes  i n t e n s i t i e s s e e n by t h e o b s e r v e r  incoherently  during  time n  I (t)= ?  I n  excited  T » T i s  I I ( t ?  We x-y  plane  leads  the x-axis  and  just  j  S  t ) d t * n/~ I ( t c  t ) d t  t o equation(3)  now c o n s i d e r  with  due t o n  j  5  t ) d t  T  V *.^ I V V again  along  the x-axis.  T  10  which  d i s t r i b u t e d about  i t s axis  88 n  't.W  f o rthe polarization,  an o s c i l l a t o r  a t an  t ) d t  angle  6  0  excited  i nthe  to the x-axis.  The ' p o l a r i z a t i o n  s e e n by  P(4> e ,H)= ^ j  o  1 +  the observer  y H T ) 2  i s then  [cos2((i) -6 ) Q  -  Summing now  over n o s c i l l a t o r s w i t h i n i t i a l  distributed  a b o u t t h e x - a x i s i n t h e x-y n I  P((J) H)= >  P((J>,e  3=1 =  where P = D  ±  I j=l  Thus a l t h o u g h polarization  plane  but  no  cos26  plane  H) =  ,  c o s 2 (  ^  "  2  l  j H T s l n 2 (  H  |P lll 0  J  the p o l a r i z a t i o n shows t h e  has  same f i e l d  i t c a n be  been decreased,  tne  dependence.  shown, a l t h o u g h more t e d i o u s  t h e maximum p o l a r i z a t i o n  e x c i t e d the  s h o u l d be  time  clear  orientation  s h o r t i n d u r a t i o n compare  collision  t h i s s h o u l d be w e l l  The  occurs.  l i f e t i m e f o r the above d e r i v a t i o n  i n v o l v e d i n the  seconds thus  depencence  decreased:  t h a t the p e r t u r b a t i o n which  o s c i l l a t o r s h o u l d be  the r a d i a t i v e  i s of order  or i n t e r r u p t i o n  by  the p r e c e e d i n g  10  t o appl. 1  satisfied.  o s c i l l a t o r s h o u l d a l s o n o t be  o r a change i n t h e p o l a r i z a t i o n apply  i s again only  change i n i t s m a g n e t i c f i e l d It  The  a x e s synuretric,=  f o r a symmetric d i s t r i b u t i o n o f o s c i l l a t o r s i n the  x-y  to  0  3  P l+(2yHT)^  Similarly that  2yHTsin2(#-6 )]i  s u b j e c t to r e -  c o l l i s i o n w i t h i t s neighbor, will  again r e s u l t .  t h e o r y t o an a t o m i c  or  To  molecular-  - 14 -  system we need m e r e l y e v a l u a t e t h e magnetic moment y=gy L e Thus f i n a l l y : where y i s t h e Bohr magneton 2mc 0  Q  P(<t>,H) = ^ T J r[cos2cj> - £^siri2<|)] ^ |-geHx^ mc K r >  §2.3  Y  Y  Quantum-Mechanical D e s c r i p t i o n o f L e v e l I n 1933 B r e i t  2 0  Crossing  d e r i v e d a quantum m e c h a n i c a l  •expression, the " B r e i t formula",  f o r the r a d i a t i o n r a t e  from c o h e r e n t l y e x c i t e d , n e a r l y degenerate s t a t e s o f a system.  T h i s was l a r g e l y i g n o r e d u n t i l  Pranken e t a l  2  1  1958 when  r e p o r t e d t h e measurement o f t h e f i n e  s t r u c t u r e i n some H e l i u m l e v e l s u s i n g l e v e l - c r o s s i n g . They r e - d e r i v e d t h e " B r e i t f o r m u l a " , under c o n d i t i o n s o f p u l s e e x c i t a t i o n s as w e l l as e x c i t a t i o n by white' l i g h t . Consider state of  a " t h r e e " l e v e l system w i t h  ground  |a> and e x c i t e d s t a t e s |b> and |c> w i t h t h e energy  Ia> t a k e n f o r c o n v e n i e n c e t o be z e r o .  v  F i g u r e 6 - "Three" l e v e l  system  -  They  have  i.e.  We  magnetic have  15 -  sublevels  m  the unperturbed  |a,J,m;t>  =  |a,J,m>  |b,J',y;t>  =  |b,J' y>e  c , J " ,,vv;;tt >  =  || c , J "",vv > e  y, and v,  3  respectively,  states: m=-J,-J+l,...,J +rv/2)t  -(ia v  y=-J',-J'+1,.. ., J '  D  5  -(iu,, + r „ / 2 ) t v  v=-J"  c  5  E a u =4::—, E i s the energy a In ' a  ,-J." + l , . . . , J " 1  J  where  D  expresses  the r a d i a t i o n  At its  ground  time  states  perturbation |bjj' y>.  t=0,  The s t a t e  5  perturbation  theory  of state  damping  t h e atom  |a,Jjm>  Q which  J  may  when  a, and r  of the state.  i s assumed  t o be i n one o f  i t i s subjected  excite  = — a x  i t t o some  to a  of the  vector  of the system  using  i s then  at subsequent  times:  pulse  states  first  order  •.  -(iw  ; t > = |a,J,m> + I  |X  |b,J«,y><b,J',y|Q|a,J,m>e  +r  /2)t  y  y Now  the states  |b,J',y>  s t a t e s |c,J",v> The  rate  R  Where the as  by e m i s s i o n  at which  m,b,c  ( Q  ' ' g  t )  this  =  well  photons R b c  states  to obtain  m  t >  o f a photon  occurs  moment  could  have  the t o t a l  ) = II mv  i s given  of decaying  to the  o f p o l a r i z a t i o n g. by  2  m  of polarization  (Q>E  capable  l<X ;t|g-r|c,J",v>|  I  g*r i s the.dipole  ground  a r e assumed  operator. been  Since  excited  instantaneous  g a t time t .  I <x ; m  t.| g - r j q, J " ,v> j 2  we  any one o f sum o v e r  emission  rate  m of  - 1 6 . -  • Substituting  f o r l y , t > we ( i  R (Q,g,t) = H I !  obtain:  - r  w  /2)t <b,J' y|Q|a,J,m>-  e  b c  mv  5  y •<b J' y|g-r|c J",v>| J  and  expanding  the square,  this  b c  (Q g,t)  =  3  -where  e = <a | Q | 3>  S g  =  Now  i f we  a  a  <  a t random  observe  the system  •  b c  R  have  times  N systems within  (  Q  > ^  = N/  =  o  T R  b c  each  T>>^-  (Q,g_,t)dt Q  HI I  N  ym  ^  v  y  subjected  t o an  (0,T) and  , the radiation  N  r  i s the "Breit  Q  - N/~ ,g  y  rate  I H I  R  b c  (Q,g,t)dt  , g  y  r'-i(w  myy'v  1)  V  the i n t e r v a l  f o r a time  myy'v  easily  V  becomes:  =  This  V  »  a|g*rIB>  impulse  observed  becomes b  v  Q ^  2  3  -co , ) - r ] t  [i(w R  i  y  -UJ^,)  Q 0 .g . K y y v v yy 1 - i x ( a ) -w . ) m  1  m  formula".  Some  (6)  of i t s implications  are  investigated. I f the various |T(U  -w  (eq.(6))  |y>  |y'>  are well  ,)|>>1),  t h e sum  i n the Breit  reduces  to just  those  "boCa-.e)  -  HI myv  this  and  i s just  l« H  l lg 2  p m  p v  l  2  terms  resolved ( i . e . formula f o r which  y=y'  - Ro  K  fluorescence with  no  interference  terms  -  2)  |x(co -CO , ) ! < 1 we  I f , however.  ference  terms  as w e l l ,  17-  - R  y  y'  VV V A ( m , y ,\i' , v ) A* (m, y , y ' , v ) ^ £ , l - i x ( c o -co , ) . 1+ixCco - 0 ) , ) mvy>y ' y y' y y'  ,  "°  ......  +  + H I mvy>y  D  -CO , ) 2 y y'  1+ 2  f  T  1  '  •{A+A* + 1T(O> -a) It  will  ..will  be shown i n A p p e n d i x  also  A=A e  2 i  0  ^  be shown t h a t where  3  inter-  A=Q Q ,g , g = A ( m , y , y ' , v ) t . . - ym my y v vp >^»^ >  mvy^y' P  some  setting  >  -  obtain  I that  i n this  cj> i s t h e a n g l e  u n l e s s y-y'=2, i t  experiment  A has t h e form  g makes w i t h  Then i f t h e l e v e l s  exhibit  magnetic  field  se x = co + f ^ - H y  and  co -co , y y'  M  A=0  a linear  (7)  , ) ( A - A * ) }  the  Zeeman e f f e c t  x-axis. i n a  b  ge —H mc .  =  e  Hence = Ho + n I mvy>y'  R ,(Q,g,<iO h  2Ao 1+  ]  •{cos2cJ) - £ ^ s i n 2 < j > } mc The  polarization  _ \  C  P, w i t h  ( Q , & A )  R (Q,g (()) b c  3  -  +  R  respect  b c  to the axis  (Q g,<!)+TT/2) 5  R (Q,g,cj)+^/2) b c  E, i s t h e n  (8)  - 18 -  Substituting  p  the expressions  Lu , 2 * I , ,,fgeHx^i R  =  1 mc '  D  eq.(8),  from  I I I" . A ( m , y , y ' , v )  2  T  1+1-  for  Q  ° mvy>y' p  K  •{cos2cj> - £^^sin2(f>} mc  Po eHii ^ mc  {cos2cf> - ^ ^ s i n 2 ( } ) }  2  (9)  mc  y  J  2 p  o  = 5°  II I A ( m , y , y ' ,v) mvy>y' 0  Comparing derived a  eq.(9) w i t h  classically  the expression  (eq.(5),  §2.2)  difference i n the definition  are  The 3d E  Levels  J  In as H  of H  the highly  expressions  2  excited states  r a p i d a t even  the Born-Oppenheimer  vibrational  eigenfunction and r o t a t i o n a l describe  small  into  between  the internuclear axis  the coupling  The  rotational  molecules  a product  the states  by H e r z b e r g  types 2 3  numbers  of electronic, i s no  i n terms  longer  of the coupling  and t h e e l e c t r o n i c motion,  o f e l e c t r o n i c motion with  various  quantum  the resolution  eigenfunctions,  We m u s t  described  of light  approximation,  2 2  valid.  case  Q  2  -of t h e t o t a l  and  o f P , t h e two  except f o r  a n d H e , t h e r o t a t i o n o f t h e n u c l e i may b e  2  sufficiently that  we s e e t h a t ,  t h e same.  §2.4  such  f o r the polarization  of coupling  the axis  of rotation,  i n a molecule are  a n d u s u a l l y r e f e r r e d t o as Hund's  a, b, c, d, and e i n t h e i r  extreme  limits.  The 3d E x  - . 19  states  o f H 2 c a n be w e l l  between was  Hund's  given  who  2  derived  as h a v i n g  .A t r e a t m e n t ' o f  "* w h e n h e  discussion  b and d has been also  described  b a n d d.  by D a v i d s o n  more r e c e n t case  case  -  analyzed  of coupling given  of  H  splittings  In  may  have  values  zero,  between  As nuclear I  that  2  expanded  here  and which  even  J states  theory  a n d A.  Budo  i n a  t o t h e 3d  I i + I  values  2  2  liquid  transition  Ruling  out this J'  have  state  2 5  magnetic  levels  1  will  composed  probability  transition  we  = J,J±2,J±4,  state.  o f atoms  a resultant nuclear - I  2  | .  For H  f o r the have  having  of order have  spin this  2  states, 1 = 1 .  I = 0 t o one w i t h  event  with  levels alternately  and odd J s t a t e s  improbable  moments  J  rotational  a state with  moments.  expectation  1  have  excitation  of these  l ' s ^ a n d t h e 3d E  111  The  the  multipole  non-zero  a molecule  - 1,  1 = 0  from  an e x c e e d i n g l y  see t h a t  of I; i n particular  have  transition  will  i n electric  will  the ground  I = 1 o r 0.  these  between  '  t o e s t a b l i s h which  I i and I  ,  have  The  s e c t i o n we  i s w e l l known  spins  = I i + I  means  is  be  i s convenient  will  A  i n Appendix I I .  the next  perturbation It  application of this  coupling  states.  expected •  i s found  2  these  intermediate  field.  The  coupling  this  by Von I . Kovacs  t h e Zeeman  a  then  10  1 = 1  i n the pure -  7  sec.  1  .  the s e l e c t i o n rule  - 20 -  The  parity  multipole  moments  note  £  that  2&-pole see  unless of  are zero  states  moment  that  Q  <a, J |  ^ | 3, J±2n>  harmonics  and that  i  (-1)  .  we  have  In assumed  that  the  I t i s then  from  molecules  electric  at thermal  temperature  moment  quadrupole  of level  state  to  i s zero  the  properties  i s usually  so.  equilibrium  populated  with  t h e i r populations  Boltzmann  d i s t r i b u t i o n law, i . e .  we  f o r the  between  moment  crossing  | a , J > was  n o t one, b u t s e v e r a l  J i s given  easy  that:  multipole  the discussion  this  electric  = 0  the e l e c t r i c  problems  We  n i s an i n t e g e r ,  o n e c a n show  the ground  of the  J and J ' .  furthermore,  I 3,J±2n>  nonvanishing  states  state  (-1)  where  integer;  state  2n<&, a n d i n t h e c a s e n=0, £>2J, s o t h a t  unless  atomic  parity  has p a r i t y  <a,J  first  have  s p e c i f i e s which  between  (£)  H i s an even  spherical  of the states  Q  .  a  04 p  we  always"  s p e c i f i e d and f o r  When find  considering that  a t room  rotational states given  o  (2)  are  by t h e M a x w e l l -  the population  Nj o f the  by E  _J N  where  j  =  N  f j  e  f j i s the degeneracy Ej  i s the energy  kT of the state  of the state  =  (2J+1)(2I+1)  J  _ k  i s Boltzmann  constant  T  i s the absolute  N  i s a normalizing  =.1.38... 10  temperature factor  x  1 6  ergs/C°  21  -  The temperature the  -  r e l a t i v e ground  and those  state  when l o w e r e d  p o p u l a t i o n s a t room t o 80°K  ortho-para conversion restriction  Table  remembering  are tabulated i n  I.  J  '  at  292°K  at  80°K  I  0  . 132  .2491  0  1  .663  .7492  1  2  .115  .0017  0  3  .086  7  4  .004  Table  I - Relative  xl0~  6  1  0  Populations of the F i r s t  Rotational  L e v e l s of Hydrogen  at  Few Thermal  Equilibrium..  It state having the  will,  under  either  ground  seen  that  a t room  exists state.  temperature  the previous selection  t h e J = 0 and J=2 s t a t e s  temperature  ambiguity as  c a n be  i s lowered any more  t o 80°K and o n l y  rules,  as g r o u n d  the  be  excitedi  state.  h o w e v e r , no the J=0 state  J'=2  When  such will  serve  -  §2.5  The E x c i t a t i o n In  an  Even non  though  t h e laws  26  and has only  so long  scattering  cross  been attempted  of  Therefore,  a  be s a i d  f o r t h esimplest atomic  t o eq.(9),  §2.3,  i t should  of P  D  value  i sirrelevant  consider  i nt h e absence  field.  o f a magnetic  excitation  and decay  a r e now c o n s i d e r e d  ent  processes.  transition  We c o n s i d e r  be s p e c i f i c  first  l e t us c o n s i d e r  J = l ->• J = 0 o b s e r v e d  along  the axis  a r e A m = ± l a n d Am=0 h a v i n g  parallel  t o thez-axis  emitted  light  should  was a r b i t r a r i l y '  as two  independ-  The t r a n s i t i o n s  polarizations  level  thus  perpendicular  I fa l l three  areequally  populated  since the  I f .however, t h e e x c i t a t i o n  mechanism populates  t h e s t a t e m=0 d i f f e r e n t l y  states  t h e A m = ± l o r t h e Am=0  m=±l, e i t h e r  Instead,  perpendicular  show n o p o l a r i z a t i o n  chosen.  The  a hypothetical  an axis  respectively.  m=0, +1 a n d - 1 o f t h e J = l  for this;  what  t h e decay.  theaxis of quantization (z-axis).  observed  noted  d i g r e s s i n g s o m e w h a t , we w i l l  the  To  be  t o the determination  i s not convenient  states  •  o f t h e magnetic  formulism  and  collisions at  known, t h e c a l c u l -  level-crossing  to  find  sections i sextremely .  o  i t s numerical  can  a r ecompletely  we m u s t  elements Q g-  electron-atom  as P ^0 and i sindependent  field, T.  formula  .  Referring that  theBreit  governing  energies  o f low energy  complex cases  t o apply  f o r t h ep e r t u r b a t i o n matrix  relativistic  ation  M a t r i x Elements', Q  order  expression  22 -  from t h e  transitions  z-  -  dominate light  as  will  the be  source  23  of  polarized  -  emitted  along  or  When t h e ' p o p u l a t i o n s the  molecules  radiation to  the  only the  show, some p o l a r i z a t i o n w i t h  a  that  =  the Pk,  distance  the  electron  show t h a t  momentum P  the  quantization.  obtain  has  sufficiently  electron be  t r a v e l along  the  z-axis  with  scattered  inelastically  £_ =  of  the  small  P R,  the  by  the  molecule  electron  in particular £  X  assume t h a t that  If  occurs.  i s then  sufficiently  molecules. i t is  s c a t t e r i n g centre  is  i t is  o  energy  momentum o f  c o l l i s i o n we  respect  low  angular  the  the  a P ^0  e x c i t a t i o n a l i g n the  alignment  and  to  z-axis.  unequal  i n general  The  after  R.  Thus  and  are  will  of  aligned  |m|  emitted  Let  at  different  the  the  to  colliding to  of  to  said  necessary  easy  perpendicular  hence  are  axis  be  r a d i a t i o n , and  the  energy  of  the  about =  z  0  electron  |.£'| = |P'xR|<<h, i . e . t h e e n e r g y p. 2 2 E' = << > then £ ' =0. 2m 2mR z =0 and c o n s e r v a t i o n o f a n g u l a r H  scattered  electron  n  p 2  5  Thus  for  the  c o l l i s i o n A£ z  momentum f o r t h e Thus state that  i f the  whole  collision  l e v e l s occurs. the  to  system  then  increases  J,  I f AJ<_0 we  s c a t t e r i n g cross  implies  that  alignment  need  sections  as  an  of  AJ  =  z  the  extra  for different  0.  uppercondition  |m|  be  non-equal. The  condition  on  the  scattered  electron  O  (above) i s q u i t e instance  „  ^ 1  stringent,  h 2mR^xio 2  2  s  for R  =  5A  , , , , so 0.15e.V.  n  energy  u.  and  ^'^JQ  that  for  even i n  ;  cases  -  where  the  ground  polarization to  be  can  realized  excesses  P  the  field.  completeness,  to  the  various and  imation  are  be  of  used.  matrix  the  be  Is  zero  and  i t ' i s not  much  the  relative  y'  valid  so  be  only  at of  better  the  expected  greater  These  my  energy  matrix  discussion of  such  order  let p  and  be  elements  of  Born  highly  where the  No  as  for  a  few  hundred and  absolute  are  used  the  effect  •  theory  approx-  approximation  least  natural  excitation  Born  erroneous,  at  sake  adequate  the  the  of  f o r the  simplicity  although  , will  i n §5.2  the  be  function as  field,  crossings,  elements  i t s relative  my'  a  considered.  that  magnetic  level  well  matrix  theory, the  , Q  the  e n e r g i e s above  obtained w i l l Q  as  as  approximations  because  elements  splittings  reason  will  of  field  computed  this  I t i s expected  correct. in  where  independent  For  y,  a much  sections  predicted  f o r non-zero must  c  various  Nevertheless, lack  i s not  0  A  of  exists  uniquely  case  elements  magnetic  a n g u l a r momentum  occur.  will.be  matrix  be  -  experimentally  When as  state  24  e.V.  for will  crossrelative  qualitatively . H  explicitly of  only  hyperfine  linewidth,  will  be  considered. We electron  b e f o r e and  probability  p'  after  amptitude,  Q  R  be the  the  momenta  of  collision.  , from  a  state  the  Then a  incident for  to I ,  transition  whose  p Q*  energies Born  differ  by  approximation  E 2 7  0  ;  ,  we  take  i n accordance  with  the  - 25.-q.r  p'-p where  q =  r  g  U  /2mE.-  £  ,  |q| = /  i s the incident  £2  electron's  position  i s the interaction potential; electrostatic  taken  n  5  ,R  with  R  },r ) = e e 5  n  g  2  I i r -R^= " i r -R%—  L  e,n|  i  e  n|  = the position  {n} = t h e s e t o f R  | e  vector  i  e|  of nucleus  vector  of electron  {e} = t h e s e t o f m o l e c u l e ' s  Carrying  of  <3|  a 3 a  e  electrons  electron  out the i n t e g r a t i o n  Q  n  nuclei  = the position  m i s t h e mass  t o be t h e  interaction 2  U({R  vector  .1  .e ie{e n}  over  r  we  g  obtain  - i q • R, |a> x  3  We  expand the e x p o n e n t i a l  Q  = QD ^ 1  D  i n a power  + Q ^  +• Q o ^  2  + Q ^  3  D  series  D  Where  q  {I)  or the  the e l e c t r i c  M  states  3 and a.  pole  ...  (10)  1  R. i  2-1  obtain  ~2T  ~2~  (iq)  +  and  i  —i  a>  m o m e n t i n t h e q_ d i r e c t i o n  between  - 26 -  (Qp°^  = <3|l[a>  = 5 g  from  a  orthogonality  F o r E•= l 4 e . V . a n d R. —1 so  that  imately  this  series  a s pj-p .  We  seen  Qp"^ = Qo"^ = 0 , - h e n c e we may  For but i.e. ion.  unfortunately  2m We  = E  ,  a3 n  approximately The  1  converges  that  approximate  that  along  so long -p;  Q  completes  our problem, p'  are easily  "K  seen  =0,  approximatq will l i e  p' , k' = — - , q = k ' - k  "ft  from  (FIG.7).  Figure  by:  R  as p' i s s m a l l ,  p setting k =—  approx-  f o rour states  we m u s t m a k e a f u r t h e r  d i r e c t i o n s q takes  diagram  8  q i s not s p e c i f i e d unless  so t h a t  expect  x  i n §2.4  a specified q this  Iq^'R^l^l '— — l  5 10~ cm  e x p a n s i o n , eq.(10), have  o f |a> a n d |3> )  7 - T h e S c a t t e r i n g A n g l e 0,  the following  -  By  conservation  of  energy  27  k.'  -  is restricted  i n magnitude  to  '2mE k'  I -  =  I k|  -  '  a  h  p  2  -ik'  The  maximum a n g l e  Thus  16  for a  energy  of  q makes w i t h  e.V.  Incident  2 e.V.,  |fi| <_  scattered  so  k'  tributed,  <q>  that = -k  -k  i s then  fl=sin  ^  .  electron scattered with =  20°;  i f the  is spherically i . e . <Q>  =  0,  an  electron is  symmetrically  dis-  and  TT  2  <sin fi> 2  =  k' sin gdg ' (k-cosg) 2  —  2  2  o 1  f  \'  2  sin gdg 2  n  /  f  k S \  2TT J o  hence, using  the  same k  <sin ft> 2  leading  as  above,  -  /<fi > < 2  shall  k'  .06  <  to  Although  and  this  assume  14°.  i s not  an  impressively  i n d e r i v i n g the  q k» t h a t — = - — = q k p r o p o r t i o n a l to  „ i and  hence  Breit the  small  formula  matrix  dispersion matrix  we  elements  elements  Q  are  R a  those  of  the  ele-ctric  formulae  and  the  quadrupole  moment  (2) Q  .  Specific  formulae  to  the  transitions  a p p l i c a t i o n of  observed  are  found  these in  Appendix  -  28  -  CHAPTER I I I EXPERIMENTAL  §3.1  Experimental In  the  view  essentials  charge the  Arrangement of  of  e x c i t e d by  the  d i s c u s s i o n i n the  this  experiment  15  20  -  e.V.  and  region, a  a device  basic  was  A capacitor  filter  discharge  p l a t e s each  cell  0  direction  the  discharge  the  vertical  other  of  into  the  of  of  field by  a monochromator  interest.  signal  on  field given  a  dis-  along  in  the  transition,  polarization.  attempt  at  between  The  scaling  the  y-z  electric  field  or  energy.  lenses  spectrally the  onto  exit  slit  of  a  lock-in  form  in  the about  to  cancel  field, i n the the  r e s o l v e the  a p h o t o m u l t i p l i e r whose  channel  the  one  Light emitted of  at  of  Centered  magnetic  = Hk.  of  Between  oscillate  earth's  a pair to  plane.  coils;  the H  a pair  Helmholtz  Light appearing falls  to  appropriate  component a  no  free electrons to  2 pairs  i s focussed  monochromator fed  the  are  to produce  direction slit  with  a  measuring  parallel  E  causes  magnetic  i s placed  plates a radio-frequency  (E cosa)t)i  chapter  sketch.  these =  c o n s i s t of  for resolving  i s shown i n F I G . 8 ;  made i n t h i s  preceeding  electrons travelling  f o r d e t e c t i n g and  apparatus  realism  of  should  x - a x i s , a homogeneous v a r i a b l e  discharge  ±x  DETAILS  of  the z  entrance transition the  output  amplifier.  is  CAPACITOR P L A T E S MONOCHROMATOR  Z-AXIS X-Y RECORDER Figure  8 - The Apparatus  - 30 -  The through  component  between t h e two l e n s e s  i t s plane  i s r o t a t e d - cfoout  of light  frequency.  i s thus  A small  light  perpendicular  the x-axis.  modulated  at twice  and p h o t o d i o d e  f o r the lock-in  amplifier.  a m p l i f i e r i s connected  recorder. across coils  The x - c h a n n e l  Helmholtz  coils  a  at the r i m reference  The o u t p u t  to the y-channel  of the lock-  o f t h e x-y  i s c o n t r o l l e d by t h e v o l t a g e  producing  i s slowly  the f i e l d  swept  t o produce  f o r m H = H + H i t where H ~-10 g a u s s c  to the  rotational  placed  H = Hk.  a r e d r i v e n by a p a i r o f power s u p p l i e s  e r whose o u t p u t  i s passed  The p o l a r i z e d  t h e p o l a r o i d , one on e i t h e r s i d e p r o v i d e  signal in  passing  a polaroid, with  beam, w h i c h  of  light  and H  o  These  and an  fields ~+10  amplifi-  of the gauss.  ITlclX  A quarter its  fast  before the  axis  light  slit  preferentially  t h e R.P.  frequencies ponding  §3-2  to the x-axis  ensure fields,  i n t h e x-y plane  with just  transmitting y polarized  x polarized  light.  the e f f e c t s observed  the experiment  f = l 8 0 MHz  changes  The  over that  i n t h e beam  o f t h e monochromator t o e l i m i n a t e  o f the monochromator  To to  a t 45°  the entrance  effect  wave p l a t e i s p l a c e d  and l a t e r  i n electric  were n o t due  was p e r f o r m e d  with  f = 4 5 0 MHz  using  and c o r r e s -  fields.  Discharge •The d i s c h a r g e  .pyrex d i s c h a r g e  cell  apparatus  consists of a  o f 5 cm d i a m e t e r  a n d 2.5-3  cylindrical cm  length  -  connected  sho\\rn i n F I G . 1 0 .  as  slightly against diameter 1  t o t h e vacuum  cm  convex  radio  in  a manner  The below.. the  A  cell  cm  O.D.  end  brass  The  i s coupled  planes  output  to these  plates  of  plates  §3.5-  and p l a t e s  a r e shown  the co-ordinate  full  scale  axes  used  shown:  »  9 -  about  of the  y  Figure  are  cm  i t extends  The  tubes  strength of 7  plate  so t h a t  pyrex cell  mechanical  of observation.  i n  are also  1  of the discharge  of the c e l l .  For convenience  discussion  two  circular  transmitter  described  by  to provide  to the axis  frequency  ends  at either  the edges  are. p a r a l l e l a  outwards  i s placed  beyond  system  The  a i r pressure.  31 -  Cross-Section Cell  and  of  Capacitor  X  Discharge Plates  i n FIG. 9 throughout  - 32 -  The d i s c h a r g e gas  of neutral molecules  electrons.  The m o t i o n  electric  field,  and  magnetic  zero  mx  the kinetic  smaller  of the electrons  field  =  to consist  a n d a much  assuming  a complete i s governed  of a  number  i n a  absence  dilute of  free  radio-frequency of  collisions  by:  eE coso)t 0  x .= and  i s assumed  ^f-coswt  muK  energy  K.E.:  1*? K.E.  =  -|mx  2  1  2m  =  The  assuming  solving  other  ratio  =  eE coswt-eHy  my  =  eHx  y(cot)  = -x(ut+TT/2) a)  words,  o f major  1)  field  H = Hk  yields,  equations: = -Hiifl.  discharge  t  D  x(oot)  In  U  o r damping,  mx  these  2  magnetic  no c o l l i s i o n s  J  the  2  OJ  application of a small  s t i l l  in  (iEp_) sin  ,  x  , cos(tot) ,  v=— m  t h e e l e c t r o n moves t o minor  order  elastic  whose  diameter i s  f o r t h e above  cell,  on an e l l i p s e  3 conditions  collisions  t o meet t h e c o n d i t i o n s must be  of electrons  satisfied: with  neutral  of  - 33 -  molecules This the  may  discharge merely  of  3)  ought  ionize  lists in  of  discharge. third  condition  some o f t h e p r o p e r t i e s field  dition  between molecules density  p:  molecule  of the  This discharge  the amplitude  we  i n order  energy  that  of the motion  and s u s t a i n  give  Table  f =  2TTOJ .  satisfied  for a l l  2 i n a discharge  c o m p u t e t h e mean f r e e o,  our  of the electrons  t o s e e how w e l l  of cross-section  t o make  20 e . V .  at frequency  i n Table  to  we m u s t  or order  2 (above) i s c l e a r l y  I n order  energetic  by r e c o m b i n a t i o n  implies  oscillating  tabulated  i s satisfied  of the  events.  considerably  lost  a maximum k i n e t i c  used.  be r a r e  must be s u f f i c i e n t l y  the  the size  of i t s  motion.  f o relectrons  frequencies  the walls  any d i m e n s i o n  the occasional  Condition the  path of  the amplitude  with  also  t o exceed  up  an e l e c t r i c  infrequently.  t h e mean f r e e  exceed  must  means t h a t  electron  The  as:  of electrons cell  electrons  electrons  should  sufficiently  considerably.  collisions  cell  t o occur  be r e s t a t e d  electron  •.motion  2)  ought  cell  the f i r s t path,  distributed _  con-  L with  a  2  E required to produce K.E. =20e.V. max  frequency (xiO Hz)  amplitude of electron motion (major diameter)  Q  6  minor axis to major axis r a t i o i n 10 g a u s s magnetic f i e l d  10.0  100  V/cm  4  mm  .07  200  200  V/cm  2  mm  .035  500  500  V/cm  0 .8  Table  II -  Properties R.F.  Then at  a  of  Electric  density  of  .014  mm  Electron  Motion  and  Magnetic  D.C.  1.77 10 Vera x  1  (.05  3  in  an Field  mm  Hg  at  o  room t e m p e r a t u r e ) we  f i n d the  the  be  §3.3  focal at  its focal  its  focal that  light  traverses and  is  length  the  aperture  the of  of  the  Thus  at  electrons'  ~10  of  these  motion  A densities  can  at  qualitative.  F  8,  i n FIG. 20  =  cm  length from  space  from  the  the  a pair  each  the  the the  are  discharge,  the  other  is  large  lenses placed, at  slit  of  the  monochromator,  centre  of  the  discharge  slit  a rotating polaroid  polaroid  plano-convex F/6,  lenses  entrance  of  aperture  and  entrance  between  onto  lenses the  cross-section  - 6 mm.  L  o r i g i n a t i n g at  focussed  Between  be  a  'System  shown  length  one  so  to  Optical As  of  path  description  expected  The  assuming  mean f r e e  foregoing  best  and  in a parallel of  the  monochromator.  i s placed;  enough not  beam  to  the constitute  - 35 a  "stop"  i n the o p t i c a l system.  i n F I G . 16.  shown  The  At the wavelengths  rotating polaroid i s this  experiment  was  o  (4930  performed'  -instrument,  A)  t h e monochromator, an F/8  h a s , when u s e d  i n second  order,  a  o  ~6A/mm, a n d a r e s o l u t i o n  widths  m  dispersion  o of  1  Spex  o f 0.3  mm  f o r both  of better  entrance  0.1A.  than  and e x i t  Slit-  slits  were  used  o throughout, was  found  yielding a resolution  to polarize  incident  o f ~2A.  light  The  monochromator  i n a manner  that  o varied  with  wavelength.  approximately  80%  ing  lines).  As  at  the e x i t  slit  rotating to  polarization.  y-axis.  described it  with  etical  the  This  makes  z-axis.  was  E with  wavelength produced  obvious  of light  intensity on t h e  be made i n s e n s i t i v e  by p l a c i n g a t 45°  a  quarter-  t o t h e x-  quarter-wave  for this  how  grat-  i s incident  must  axis  to the  plate  purpose.  Because  the quarter-wave  plate  polarization a brief  theor-  consider  an e l e c t r o m a g n e t i c  wave whose  electric  an a n g l e  6' w i t h  travelling  along  The  system.  on  x  the x-axis,  a p o l a r i z e r r o t a t i n g at  incident  components.E  reference  (or slow)  light  follows.  .It falls  to.  light  accomplished  f o r the monochromator  treatment  frequency  to modulation  was  i t sfast  i n §3.10  (parallel  the monochromator  A variable  We vector  the y-axis  leads  i s not immediately  corrects  A i t polarized  when u n p o l a r i z e d  polaroid,  wave p l a t e and  along  this  4930  At  and E^  light  has  referred  then  angular  an e l e c t r i c  to the usual  vector  x, y,  z  -  and  E„ x  =  A cos0e  E  =  A sin6e °  y  The  =  electric  wt  to  the  k  z  -  V  t  )  i(kz-vt)  0  The  light  TT/4  to  cos 6  field  =  p  calling  and  the  A  passing and  E^  A/4  along  E  q  E  L  falls  (  k  z  and  a  A/4  V  t  electric  axis  (at  an  angle  with  i t s fast  perpendicular field  along  along  to  axis the  i t s fast  i t s retarding  at  axis  axis:  p  the  A/4  a  phase  components are =  E j  /I  e  e  the ( E  plate  E  shift  of  the  |(  suffers  field  1 6  1 ( 6 + i r / 2 , )  x  and  'v- V) E  y  a  phase  shift  .  6+7T/4.  then:  E  of =  is  )  plate  i t s plane  field  "  i(kz-vt) E cos(3iT/4-a)t)e"  E| =  x  1  =  1  axes  polaroid  i(kz-vt) E p C o s (7T/4-cot) e"  plate  E  the  =  suffers  i n terms  on  E „ the  through  E  these  2  o  electric  The the  E^  to  is  x-axis  z-axis E  sin i  A cos(wt-0)e  then  the  -  2  x-axis) E  or  (  n  -  i t s polarization referred  P  6  i  36  components:  =^(E,-iE ) A  after passing  through  - 37 -  The  monochromator  a -6 2  a  2 g +  at  passes  aE' + 3E' where y x  i t spolarization  =  2  along the y-axis.  2  the exit I  The i n t e n s i t y  s l i t i s then: cc ( E ' ) + ( 3 E ' ) 2  of light  appearing  2  a  y .=  a  a  2  x  ^ (E +E )  2  —  2  2  + 3  2  ~  T 7  Cj  2  2  2  p  = ^!lAlcos (a)t-0)  (11)  2  Thus  we o b t a i n  a signal  of exactly  have  o b t a i n e d i ft h e monochromator a +3 2  only  by t h e f a c t o r  produces  a signal  depending  §3.4  .  conventional  reduced  where  2  efficiency  and t h e i n t e n s i t y  then  C i s a constant and g a i n , t h e of radiation  the discharge.  The Vacuum The  not there,  The p h o t o m u l t i p l i e r  V ( t ) = C*cos (wt-6)  of the optics,  from  were  would  2  on t h e p h o t o m u l t i p l i e r  efficiency emitted  —  t h e same f o r m we  System  vacuum glass  system  system  i s shown i n F I G . 1 0 .  pumped by a  I t i s a  "Cenco-Hyvac"  m e c h a n i c a l pump w h i c h . i s c a p a b l e o f r e d u c i n g t h e p r e s s u r e in  the system  nitrogen in  place.  needle  cold  trap  Hydrogen  valve;  pressure  to less  than 5 10 X  - 1  *  mm  Hg when t h e l i q u i d  t o p r e v e n t b a c k s t r e a m i n g o f pump o i l i s gas i s leaked  the leakage rate  o f t h e system.  Into  the system  establishing  by a  the equilibrium  When t h e d i s c h a r g e w a s . s u b m e r s e d  -  in  liquid  freeze  that  n i t r o g e n any i m p u r i t i e s  on t h e w a l l s  obtained  from  within  quite  a high pressure bottle  a few hours  bottle,  hydrogen through  halfway  downstream  to  a flexible  from  halfway  downstream  rubber  t h e second  walls  impure  become  installed  cold  Pirani  before  t o a cold  trap  cold  v i a glass  gauge  cold  i t was a t t a c h e d t o t h e vacuum  was m e a s u r e d  McLeod  gauge  a t about  2 hour  gauge.  are attached Proceeding trap i s  inside  glass  intervals  swivel  system.  the pressure  and c o n t i n u o u s l y monitored  Pirani  cell,  i n the position of  was p r o v i d e d by t h e g r o u n d  system  Proceeding  o u t by t h e m e c h a n i c a l  t u b i n g has an  taken  trap  v a l v e and  trap.  stopcocks.  another  Some f l e x i b i l i t y  filled  to the discharge  and a McLeod  When d a t a w a s b e i n g  pressure  trap  t o the needle  t h e g a s i s pumped  discharge cell  atmospheric  From t h e c h a r c o a l  the discharge cell  o f ~7 mm.  calibrated  hose  t o a second  and connected  by w h i c h  above  2  charcoal.  A l l the connecting glass  diameter  joints  i s sufficiently  t r a p s were  ~ 2 0 cm o f h o s e  along^a  from  encountered  the  a t ~2 p o u n d s / i n  through  t h e system  pump.  hydrogen  t h e r e d u c t i o n v a l v e on t h e p r e s s u r e  there a t low pressure  about  The  the discharge cell  from  with activated  proceeds  from  tend t o  upstream.  Starting  passes  i n t h e hydrogen  of the discharge c e l l .  opaque, t h e r e f o r e two c o l d  further  it  38 -  i n the  with the  with the roughly  MCLEOD  GAUGE  NEEDLE VALVE  Figure  10 - T h e V a c u u m  System  HO  -  §3.5  R.F. S u p p l i e s The  180 MHz  a 928 B t u b e . plates the  take  cathode  supply for  I t sschematic  i s f e d from  and a t t a c h e d  loop.  b y a 30  high  circuits.  attached  at contact  room,  sheet at the  end.  transformers  Feedback t o  power  periods  repeated  were  o f 300ft  The w i r e  at points, equidistant transfer  t h e maximum p o w e r  about  output  was e s t i m a t e d  8 cm f r o m  This  output  of order  was e s transfer  about oscillator  rather  radiated into  at intervals  from  was q u i t e u n -  10 m i n u t e s ,  had a tendency  was  the centre.  t o be  o f 300V.  The power  o f R.F. p o w e r w e r e  or special  by a l e n g t h  points  exceeding  voltage  capacitor plates  lead wire.  In practice,  and t h e o s c i l l a t o r  bursts  grounded  at either  circuit  strip  s e v e r a l drawbacks.  fractions  copper  circuit  between t h e p l a t e  at a cathode p o t e n t i a l  over  of thin  p o i n t , w h e r e maximum p o w e r  The  stable  The " t a n k "  i s laid  impedance) twin  to occur.  occurred  t e r m i n a l o f a power  o f 16 g u a g e i n -  loop  "tank"  while  cm l e n g t h  to obviate  t o the copper  grounded  timated  ground p o t e n t i a l  cm l o n g ,  The d i s c h a r g e  to the plate  (characteristic  5 - 10W  whose  using  The  A t t h e p l a t e s t h e i m p e d a n c e a n d R.F.  sufficiently  coupled  a n d 30  t o one anode  copper wire  coupling  in  about  the negative  1 cm w i d e  oscillator  i s shown i n F I G . 1 1 .  o f -80V t o -600V.  grids i s provided  strip  had  was. a s i m p l e  the plates consists of a strip  sulated  the  source  at a potential  centre  is  Coupling  on a n R.F. v o l t a g e  approximately  the  and  -  large  the surrounding  to pulse o f 10  6  i t s output seconds.  Figure  11 - 180 MHz R . F .  Oscillator  - 42 -  Later, transmitter  the source  was r e p l a c e d  c o n s i s t i n g o f a n R.C.A. MI-17436-1  used  as " d r i v e r "  high  frequency  power  amplifier  50ft a n d o u t p u t  power  of approximately  1:1 s t a n d i n g is  and a Canadian Marconi  wave r a t i o  we n e e d  capacitor  a field  by  t h e "T m a t c h e d "  length By  o f RG  approximate  or  Helmholtz  50 t u r n s  placed  circuit  by moving while  and  a  50W 50V.  a t 3 cm,  order  were  obtained  shown i n F I G . 1 2 .  t h e power t o t h i s  point  the resonant  circuit.  circuit  The c i r c u i t i s  the crossbars  keeping  A  A ( s e e FIG.12) an  may b e o b t a i n e d .  closer  together  them e q u i d i s t a n t f r o m t h e  Coils earth's  magnetic  earth's  field  o f #24 c o p p e r w i r e .  5 gauss/A. field, to less  f a c t o r o f 50.  cancelling coils  and a r e spaced  i n series the f i e l d  approximately the  I f we h a v e  400V/cm b e t w e e n t h e d i s c h a r g e  of this  carried  1 9 . 5 cm m e a n d i a m e t e r ,  has  impedance o f  plates.  The  a  resonant  line  further apart,  §3.6  50V/.  163-107  a v a i l a b l e i s only  impedance match between  t o resonance  capacitor  Voltages  of the contact  transmission  tuned  an output  MHz  transmitter  (S.W.R.) o n a 50ft l i n e  of order  8-U c a b l e  adjustment  and  Model  p l a t e s and t h e p l a t e s a r e spaced  1200V i s r e q u i r e d .  a  with  t r a n s m i t t e d , t h e R.P. v o l t a g e  Since  b y a 450  than  cm a p a r t ;  each  When t h e 2 c o i l s a r e  produced These  9.7  have  at the centre i s  coils  are expected  .01 g a u s s ;  The i n h o g e n e i t i e s n e a r  t o reduce  i . e . by a t l e a s t  the centre  are of  _  i |  3  RG8-U TRANSMISSION LINE  -  '  U.H.F CONNECTOR  CAPACLTOR PLATES  5  CM -SLIDING CROSSBAR  Figure  12 - r - M a t c h e d  Resonant  Circuit  - u n -  it  order  (^-)  where  r i s the displacement  from  the centre  and  n  • r  R i s the coil or  about  1 part The  must have total  of  100  <10  4><10  over  3  - 3  .  i . e . they  Therefore  these  and i n t h i s  centre  i t i s only  no n e e d  necessary  was  field  w i t h i n the Helmholtz  coil  component  with  was  later  effect  with  . 0 1 gauss  to calibrate  The f i e l d  was  magnetic  one  part  each had  used i n produced at  along field.  until  coils a  zero  of the earth's  zeroed  be d e t e c t e d  a d i p needle  using  a  rotat-  the horizontal Fields this  c a n be u s e d  down t o  way. to  I t  this  accuracy.  Helmholtz  across  diameter  the smaller  component  coils  could  a  inhomogene-  coils  the current  of rotation  t h e same  larger  of voltage  within.  that  about  The terms  i t s axis  found  with  were  i n a  f o ri n -  The  These  field  gauss/ampere.  The v e r t i c a l  of the earth's  approximately  2.4  time  be r o u g h l y  The c o i l s  to adjust  field  ing  reached.  coils  used.  should  region.  provide  configuration the f i e l d  i s approximately was  larger  coils  but this  must  o f 1 8 . 5 cm w e r e  o f #18 copper w i r e .  turns  (^-) ~ (if)  the a p p l i e d magnetic  homogeneity  the discharge  There as  provide  absolute  incurred with  parallel the  which  3 7 cm a n d s p a c i n g  ities in  coils  o f 10 g a u s s ;  homogeneities  cell,  per 250.  t h e same  field  F o r o u r 5 cm d i s c h a r g e  radius.  1  h  coils  the coils  measurements  were  calibrated i n  as a f u n c t i o n o f  w e r e made  first  with  field a  15  FIL.TI  2N1132  AA/V^r470  2N37022N3702 2 5 0 :1K U1  ^ 2 5 0  -o ALL  R E S I S T O R S 1 WATT C A P A C I T O R S 10 V O L T Figure  13 - H e l m h o l t z  Coil  Power  Supply  - 46 -  " B e l l 240"  H a l l probe gaussmeter.  e r r a t i c behavior  Because of the somewhat  o f t h i s gaussmeter the measurements were  checked w i t h a "Magnion FFC-4" r o t a t i n g c o i l  magnetometer.  I n o r d e r t o save c a l i b r a t i n g the x - c h a n n e l o f the r e c o r d e r , the p o s i t i o n o f the pen was of f i e l d .  x-y  measured as a f u n c t i o n  W i t h the r e c o r d e r on the 2 V / i n s c a l e (which  always used i n i t s c a l i b r a t e d mode) the f i e l d charge r e g i o n was  found t o be  i n the  was  dis-  3.60±.08 gauss per i n c h o f  pen movement from the c e n t r e .  The  c o i l s were u s u a l l y run  w i t h a 1ft r e s i s t o r i n s e r i e s .  The  f i e l d produced was  then  2.00±.04 g a u s s / i n .  §3.7  Current The  Supplies f o r Helmholtz C o i l s c u r r e n t f o r the e a r t h ' s f i e l d c a n c e l l i n g  c o i l s i s p r o v i d e d by a s m a l l power s u p p l y I V and FIG.13. lmV  and  150mA i n t o the c o i l s .  The  drift  approximately  i s l e s s t h a n lOmV.  c u r r e n t d e l i v e r e d t o the l a r g e r H e l m h o l t z  c o i l s i s s u p p l i e d by a p a i r o f "EICO 1064" and r e g u l a t e d by a D.C. i s shown i n FIG.14.  power a m p l i f i e r  The  2 8  po;«;er s u p p l i e s whose s c h e m a t i c  a m p l i f i e r d e l i v e r s up t o 7A  e i t h e r p o l a r i t y i n t o a 1ft l o a d w i t h r e a s o n a b l e has  to  I t s s c h e m a t i c i s shown i n  At 0.65V C-llOmA) the r i p p l e i s the D.C.  d e l i v e r i n g up  a v o l t a g e g a i n of a p p r o x i m a t e l y  impedance o f a p p r o x i m a t e l y  lOKft.  linearity;  u n i t y and an The  of  input  amplifier i s controlled  by the v o l t a g e s u p p l i e d by a m e c h a n i c a l l y  swept -10V  to  All  •27K 2N3792 •2N278^  resistors  1 Watt  ^=-12V  220 LOAD  22  22  WsAiVWMVsAAA  —5  220' 12V 3 :  2N3792  2N278/1 27K<  o INPUT o ~ Figure- L'l - D . O . P©w§r A m p l i f i e r  O  0  IK  AMPLIFIER CONTROL V O L T A G E  OUT  w w  4^1000/ 40  REVERSIBLE 10 r p m MOTOR  A / W V IK  'O-RING  -10V  t o +10V V o l t a g e  Sweep  Mechanism  -  .+10V  source  shown  in  -  1J9  FIG.15.  The  sweep  time  is  about  7.minutes.  .- §3.8  Lock-in  Amplifier  The amplifier The  of  a  10 *.  The  1  signal  of  about  used,  triggered  used  Model  output  was  this  Q  amplifier  amplifier  Research  mode.it  In  with  lock-in  Applied  lock-in  10 in  120,  experiment  a  a  this a  ±10V  i t supplies an  and  has  i s D.C.  by  consists  phase  a  tuned  experiment,  at  full  i t s own  time  of  phase  detector.  Princeton  of  1%  scale.  and In  sinusoidal  supplied  second  pre-  sensitive  linearity  externally  3  of  was  gain  the  reference  reference  constant  a  used  signal. through-  out .  The easily V(t)  understood.  i t produces  s  where cy, ly  to  function  cj> i s  and  x  large,  our  a  a is  the  Essentially,  a  „  T/2TTV  I n=0  an  V(t)cos(vt-c)))d(vt)  selected  phase  the  constant.  time  signals  We  see  Choosing  S  «  now  derived  v = 2w  /  and  signal  2mr  in  angle,  what  v  So  is  the  long  as  cos  2  cj> = 0,  tuned T  frequen-  is sufficient-  will  average  to  the  lock-in  amplifier  eq.(ll) §  3-3-  2TT •  input  is  2(n+l)7r  /  incoherent  signal  i f given  detector  signal  ^  can  sensative  (cot-6)cos(vt-(J))d(vt)  zero.  does  - 50 -  S <* /  2TT  c o s 2 ( u ) t - e )cos2cot  d(2wt)  o  •  <* c o s 2 0 In  other  just  words,  §3-9  t o the. x - a x i s .  The polaroid  laid The  2"  r.p.m.  diameter  pipe  ball-bearing.  light  CA3GRH,  Universal TJ-Q H.P.  groove  the polaroid  which  The•  Is fitted  A sewing maching  and t h e motor p u l l e y  and a b e l t  rotates  Electric  the polaroid.  Co., runs a t •  The. m o t o r p u l l e y  of lj-^- inch i s rotated  belt  has a  diameter.  With  at approximately  cycles/second.  The centre the  Light  front  Two  from  black  t h e lamp  of the polaroid  The  quarter  the polaroid  The V a r i a b l e  described  t o be. " c h o p p e d " p a s s e s  are painted  lamp b e h i n d  turns.  §3.10  light  of the pipe.  polaroid  small  in  signal i s  i s shown i n P I G . 1 6 .  rotator  and d e l i v e r s  arrangement  output  •  t o a 2" I . D . b r a s s  a large  motor, model  this 14  i s glued  over the pipe  1050  = P  2  Polaroid  polaroid  inside  - sin 6  to the p o l a r i z a t i o n of the emitted  The R o t a t i n g  tightly  2  the lock-in amplifier  proportional  referred  = cos 6  the•  of the rim of  to interrupt  light  from  falls  on a p h o t o d i o d e ,  Quarter-Wave  placed  Plate  plate  and Saloman  a  p e r i o d i c a l l y as t h e p o l a r o i d  disk.  quarter-wave  by H a p p e r  segments  through  2  used 9  .  i s based  on one  Figure  16  - Polaroid  Rotator  - 52 -  -  When  fused  quartz  fringent  with  its optical  strain.  The r e l a t i v e  vector, p a r a l l e l dicular  This  variable  1"  - a n d one w i t h  effect  wavelength  The  major  o n x l " x ^-g. g r o u n d  edges,  a brass  axis  case,  along  retardation  to the d i r e c t i o n  applied.  i s s t r e s s e d i t becomes  a wave  i t s electric  i s made  of  with  vector  increases with  electric  perpenthe stress  use o f t o c o n s t r u c t the  quarter-wave  components flat  the d i r e c t i o n  between  of strain  bire-  plate  shown  are a piece  and p a r a l l e l  and a s t e e l  i n FIG.17.  of fused  on two  pressure  quartz  opposing  plate  to dis-  1" tribute ground the  the force face  quartz  a  N.F. t h r e a d  of the quartz  pressure  contact  from  plate.  inhomogeneities  spaced  inspection tightened corners  by  and t h e i n s i d e  2 pieces  with  crossed  to strain  a fairly  I n an e f f o r t  o f the steel  bottom  1 " are  over the to  reduce  due t o i r r e g u l a r i t i e s  surfaces; the surfaces plate,  screw  o f ^-g  surface  plate, the of the  case  3 " x  x  1" t e f l o n .  polaroids while  the quartz  homogeneous  of the  shows  Eye  the screw i s  that  quarter-wave  except plate  at the i s pro-  duced. The the  screw  produces  quarter-wave  until a  zero  unpolarized output  plate  i s adjusted  light  signal  from  from  by t i g h t e n i n g  the discharge  the lock-in  amplifier.  FULL SCALE  Figure  17  - Quarter-Wave  Plate  -  §3.11  -  Photomultiplier The  is  54  an E.M.I.  photomultiplier.used  9558QB.  for this  experiment  I t h a s an S-20 (NaKSbCs)  surface.  o  The  quantum e f f i c i e n c y  turer  t o be -23%.  potential  tained  t h ecathode  a t -150V  to  ground  is  shown b e l o w  Ai ssaid  I t was o p e r a t e d  of -1280V.  33KQ w h i l e  a t 4900  The dynode to first  by a z e n e r  chain  a cathode  t o anode  resistors  were a l l  dynode p o t e n t i a l  i s main-  diode.  b y a lOOKft r e s i s t o r .  with  by t h e manufac-  The anode  i s connected  The a b b r e v i a t e d  circuit  i n PIG.18.  ANODE CATHODE  D  Di  D3  2  D 11 100K  100K  150V  A A A A A ^ A A / V X A ^ V -• 33K  - v V A A A M  33K  33K  AAAA/V 1  100K  6  D. i s t h e j ' t h d y n o d e  .  J All  V=-1280V  Figure  §3.12  18 - P h o t o m u l t i p l i e r W i r i n g  1 Watt  Schematic  X-Y R e c o r d e r The  having into  resistors  x-y r e c o r d e r  alinearity  each  channel.  used  was a V a r i a n  o f 1% a n d i n p u t  impedance  model F100 o f lOOKft  - 55 -  §3-13  Liquid Nitrogen The  band  of  the  R(0),  Bath  R ( l ) , and.R(2) l i n e s  3d E x  of  the  (0,0)  2p TI t r a n s i t i o n  are  resolved  monochromator  1  separated  by  about  o-  3A  which  fairly  i s easily  wide  vibrational line  and  cooling  slits.  about  discharge surprising  from  the  =  is  Thus  the  line  proportional level. J  =  5 level  from  an  liquid  To  lower  cell 2  and  the  nitrogen the  nitrogen  level  hour  and  was  cell  only  of  level  J  =  population  0 and  J  =  temperatures  observed 2  to  the  on  with at  the be  we  see  at  room  level  state. to  that  the  fitted  nitrogen._  negligible.  with  ^ of  i n s i d e exceeded  a  side,  The  roughly surface  arises),  discharge  front outside  the  3  temperature  is  of  =  whereas  i t s population  the  be  state J  R(l)  a rate  pressure  the  expected  discharge,  at  arises  ground  ground  that  completely.  2 from which  liquid  boil  found  Under  l a r g e vacuum Dewar  window  decreased  H  of  R(l)  section this  the  §2.4,  the  almost  state.  would  I,  on  transition  of  of  with  electronic,  I t was  line  theory  R(4)  Table  filled  when t h e  3  same  almost  R(4)  upper  population  flat  Dewar I s  the  =  inside a  diameter  the  J  temperature  i s placed  inch  the  the  falls  the  i n the  appreciable  (compared w i t h at  of  the  Referring to  3 has  since  strength to  of  same i n t e n s i t y .  rule derived  populated  line  e l i m i n a t e d the  i s not  selection  the  however  the  This  J'  R(4)  transition  has the  The  by  liquid  inch the  200u.  per discharge  - 56 -  §3-14  Data  Processing  The  graphs  plotted  FIGs.19  a n d 21) w e r e  extract  t h e h a l f w i d t h , Hi/ .  by t h e x-y r e c o r d e r ( s e e  subjected  were r e a d 39  from  the graphs  vals  to provide  used  t o f i t a f u n c t i o n of the  P = Ci +  data  processing to  Relative values  2  polarization  to numerical  points.  = 7 - ^ 1+^(H-H )  of the  a t 0.2  inch  The p o i n t s w e r e  inter-  then  form:  [cos26  - =^(H-H  0  )sin26]  2  0  n  where  C i , C  computer  2  , Hi/ ,. H 2  "least  D  1/2  and 0 a r e p a r a m e t e r s  squares"  The  fitted  curves  the  hydrogen pressures  f i t t i n g routine  a r e shown used,  6 t o 8 (depending  noise)  graphs were produced  by  (12).  The  pressure, ated Hi/  2  plotted  to zero  sections  2  then  ( s e e FIGs.23  graphs  on  be  of  signal fitted  computed f o r t h a t  a n d 2k).  the radiative  a r e c o m p u t e d as w i l l  a  At each  and i n d e p e n d e n t l y  as a f u n c t i o n o f p r e s s u r e  pressure  v.s. pressure  H i / was  by  (U.B.C. L . Q . F . ) .  i n F I G s . 2 0 and 22.  to  average  fitted  seen  and  extrapol-  From  lifetime  these  and c r o s s -  i n the next  chapter.  - 57 -  CHAPTER  IV  EXPERIMENTAL §4.1  Life-times A typical  curve  obtained,  A similar  excited  b y t h e 180  squares"  fitted  polarization  graph MHz  curves  figures  R.F.  a n d 22.  and n o r m a l i z e d  bars  bars  t o 0.9  —  on curves  halfwidths, Hy , 2  2  represent  i n pressure  i n the statistical  represent  i n r e a d i n g t h e McLeod guage.  to the upper-state's  T  The  are t a b u l a t e d i n Table I I I .  i n H]/  The h a l f w i d t h o f a l i n e  <J  "least  The h a l f w i d t h s o f t h e  a n d 24 t h e a v e r a g e  the error  maximum e r r o r  T  f o r the discharge  Their respective  a r e s h o w n i n FIG.20  The e r r o r  only, while  converted  i s shown i n  a r e shown as a f u n c t i o n o f p r e s s u r e  vacuum s y s t e m .  be  discharge,  field.  plots.  23  of the p o l a r i z a t i o n  i s shown I n FIG.21  at the various pressures  the curves  errors  plot  MHz  scale i s arbitrary  In  the  t h e 450  computer generated  obtained  of  experimental  using  FIG.19.  the  RESULTS  ~  ^ — F j —  ,  u  0  at zero pressure lifetime,  may  using:  i s the Bohr magneton.  2gjU h]/2 0  The e x t r a p o l a t e d h a l f w i d t h s and l i f e t i m e s of  t h e J = l , .2, a n d 3 o f t h e  in  t a b l e IV.  The g f a c t o r s  v=0 used  levels  state are tabulated  are those  g i v e n by  Dieke  3 0  .  POLARIZATION  MAGNETIC J  -8.0  |-i  1  -6.0  -4.0 Figure  19  :  (  -2.0  :  1  1  I  2.0  - Experimental Level-Crossing U s i n g 450 MHz E x c i t a t i o n  1  4.0 : ;  Curve  6.0  f o r t h e R(0)  FIELD  1  8.0 Line  GAUSS  - 59 -  Figure  20  - Least Squares F i t t e d Curve f o r the R ( 0 ) L i n e U s i n g 450 MHz Excitation  Figure  21  - Experimental Level-Crossing U s i n g 180 MHz Excitation  Curve  f o r the  R(0)  Line  - 61 -  Figure  23-  Level  Crossing  Curve H a l f w i d t h  as  a Function  of  Pressure  24  -  Level  Crossing  Curve H a l f w i d t h  as  a Function  of  Pressure  - 64 -  8  J  g  (gauss) 3 0 0 ° K 180MHz  Hl/2  J  1  .901  2  .571  .3  .445  2. 47±.l  3. 4l±.l -  3.10+.2  Collision  reasonable  the discharge.  we  compared  assume  that  ground  state.  N*,  f o r molecules o f t h e two  ~ *  of the  V.  individual  .  o f t h e H1/2  .•  v.s. pressure  graphs,  cross-sections  may  about  that  t h e number  of excited  o f ground  molecule  state  equation  competing  decay  = -r N*-avN*N  molecules  molecules;  be conditions  t = 0 , may  molecules,  then  state  then  i n the  i s small  be w i t h  f o r the upper  at time  electrons  of neutral  suffers will  excited  0  of free  t o t h e number  t h e number  The  a  '  compared  an e x c i t e d  the  terms  Lifetimes  :  t o t h e number  collision  f o r each  and  some p l a u s i b l e a s s u m p t i o n s  '  i s small  assume  Halfwidths  of the c o l l i s i o n  i f one makes  discharge and  the slope  estimate  We  4.12  Cross-sections  From  in  2.78  3 . 74  obtained  i n Table  sec) 450MHz  3.83  IV - Extrapolated  i s contained  obtained  80°K  2 . 55  2.27±.l  The h a l f w i d t h s  §4.2  10  T(  2.60±.08  Table  runs  _8  x ( 10 ' s e c ) Hi/2 (gauss) 8 0 ° K 450 MHz' 3 0 0 ° K 180MHz  any  one i n population,  be w r i t t e n i n  processes,  (13)  - 65 -  R ( 0 ) 450 MHz  average  R(l)  450  • 90y  70y '  50y  35y  3.43194 3.40270 • 3.43443 3.48059  3.14845 3..256143.15160 3.20694 3.'33747 3.23944  2.71100 3.17868 2.92867 2.98884 3.02103 2.82810  2.75137 2.69689 2.74707 2.68692 2.74468 2.75787  3.2.23±.031  2.943±•07  2.731±.0l4  4.41363 4.26093 4.38411 4.39620 4.12871  3.93656 3.76604 3.74870 3.45032 4 .00709 3.96007  3.44569 3.41708 3.32601 3.38945 • 3.53151 3.39896  4.317±.o6  3.811+.09  3.4l8±.03  6.10883 5.54281 6.97308 5.98210 6.74916 5.86058  5.41178 5.91381 5.31858 5.85825 5.61430 5.9H32  5.18763 4.72888 4.94178 4.80084  4.43328 4.545974.12689 4.17776 4.26244 4.34724  6.203±.25  5.676+.13  4.915±.11  4.3l6±.07  3.437±.l8 MHz  4.61027 4.91537 4.62872 4.52513 • 4.84251 4.81395 average  -  4.723±.07  R ( 2 ) 450 MHz  average  Table V - P o l a r i z a t i o n  Curves'  Halfwidths  -  'R(O)  180  66  -  MHz 33y  50y  2.60260  2.85038  2.96818  2.87502  2.88300  3 .03098  2.51146  2.7057-2  2.90821  2.64745  2.79014  2.95136  2.57688  2.83666  2.98517  2.8l3±.035  2.969±.022  3-74852  4.33185  4.33183  3.87775  4.08811.  4.35565  3.64140  4.12431  4.43897  3.57200  4.22226  4.31595  • I5y  -  2.67594 2.648±.056  average  R(2)  180  MHz  •  4.65773  3.96459 average  Table  3 .76l±.1  V(continued)  4.192±.065  -  Polarization  4.420±.07  Curves'  Halfwidths  - 67 -  where  T  i s the radiative  0  collision  cross-section,  molecules,  thus '  v i s the relative  e q . ( 1 3 ) we  * ( t )= N e- ^r  N  probability,  and N i s t h e d e n s i t y o f n e u t r a l  Solving •.  transition  0  v  N  =  t  0  the inverse • ^ = r(N)  =  o i s the  velocity  of  colliding  molecules.  obtain N e-  r  (  N  )  t  0  lifetime r +avN Q  •then  dr  = av  -Txr  dN  dr/dN  or,  J  from  ,  a = — .v  5  o u r g r a p h s we h a v e  n  , . v  (14) /  TJ  ^  2  , which  may  be r e l a t e d  to  df by  the following  substitutions  r = 2 u H g j  N  D  N PT T  K  0  0  where  N  i s Avogadro's  D  number/Molar  P i s the pressure  T  i n standard  volume atmospheres  = 29.7°K  0  T i s t h e a b s o l u t e t e m p e r a t u r e o f t h e g a s i n °K then dT_ dN  2 =  s  J ° N T y  0  T  dHi/ dp 2  0  o r more c o n v e n i e n t l y  dN  =  n Q8xlQ-  7  c  m  3  y  gauss  1 sec T  d H E D  &  J  ^ dp 2  -  finally,  substituting  68  this  -  into  eq.(l4)  = 4. 8xiO- J5l ^  —  7  A  9  T v  dp  D  where Hi/ (=  i s measured  2  and P i s measured  3  80°K  values  i nthe liquid  Boltzmann assumed  obtained  T = 300°K-in  assuming  velocities,  cooled  are listed  i ncalculating  the discharge  for the collision cross-section,  t h e room t e m p e r a t u r e  nitrogen  cell  J  these  discharge and  discharge,  i n Table  VI.  i s t h a t measured  T=80°-K  157  102  A  2  116  A  2  2  0  167  Table  §4.3  2  VI - Collision  gauge.  131  0  A  Cross-sections  Polarization The  zero  A  the pressure  a  2  3  I t i s also  by t h e McLeod  T=300°K  A  and assuming  cross-sections that  a  1  magnetic  polaroid output, Si.  i n microns  10" mm H g ) . The  in  i n gauss  ^  gauss s e c  absolute field  was m e a s u r e d  i nthe light S , with Q  polarization  that  The p o l a r i z a t i o n  of the light  at  by p l a c i n g a s t a t i o n a r y  beam a n d c o m p a r i n g obtained  observed  without  P of the.light  the lock-in  amplifier  t h e second p o l a r o i d ,  i s then  obtained  from:  - 69 P =  , where  ..Under t h e b e s t at  the  roughly  half  discharge  Upper  the  Nj,  observed  may  intensities  be  from  state.  with  the p o l a r i z i t i s  Impurities  i n the  considerably.  wave  R,  (A r e p r e s e n t s  The r e l a t i v e  of a rotation,  of the  intensity  of  population  t o t h e sum o f t h e  the intensity  arising  from the  of the t r a n s i t i o n  P, o r Q) b y I ^ j ^ ,  " ^RCJ'-l)  a constant  increasing levels  P, a n d Q t r a n s i t i o n s  A(J)  function  the r e l a t i v e  J ' i s then proportional  Denoting  Assuming  a t 50u.  computed  o f t h e R,  J'  were  a t lOOu  of the r o t a t i o n a l  same u p p e r s t a t e .  N  pressure,  too drastically with  to the lower  of the state  widely  Qualitallvely,  the polarization  populations  state  found  Populations  does n o t v a r y  transitions  P varies  the electronic-vibrational  relative  excited  increasing  decrease  State  If molecule  with  thus  a n d 5% f o r t h e R ( 0 ) ,  respectively.  of that  also  10%,  of the discharge.  decreases  transmittance.  the polarizations  10%,  of order  and R(2) l i n e s  conditions  ation  §4.4  conditions,  20u p r e s s u r e  R(l),  T i s the polaroid  +  I  Q(J')  spectral  + I  P(J'+1)  response  of the  photomultiplier o  and  constant  monochromator  the  relative  intensities  measured Q(J)  t o -.10%  accuracy  i s completely  reduces t o :  absent  efficiency  over  of the relevant  t h e ~100A  transitions  and t h e p o p u l a t i o n s from  the  bands  scanned, were  N j , computed. hence  t h e sum  70  -  N  The  numbers  their in  J '  w  R(J'-1)  N ,  sum e q u a l s  temperature  unity  headed  while  immersed  t o those  suggests  that  fairly  I  P(J'+1)  by  those  are l i s t e d i n Table 300°K p e r t a i n headed  i n liquid  ulations  is  +  experimentally obtained normalized  T  the column  charge  I  VII.  o f the ground rule  Comparing  state  so  The  that numbers  t o t h e d i s c h a r g e a t room  b y 80°K p e r t a i n  nitrogen.  the selection  well  -  to the disthese  given i n Table  A J = 0,±2  popI  (§2.4)  derived i n chapter I I  obeyed.  J  N  J .  8 0 °K  0  .017*  .007*  1  .241  .281  2  .128  .193  3  .358  .406  .072*  .031*  .127  .052  . 4  5  Table  J ' 32.5°K N  V I I - Experimental  Upper  State Populations  P ( l ) a n d R ( 3 ) can. n o t be r e s o l v e d by o u r a p p a r a t u s . T h e v a l u e s g i v e n a s s u m e t h a t J'=4 h a s a t l e a s t half the population of J ' = 5 Probably the p o p u l a t i o n of J ' = 0 i s less than that s t a t e d , w h i l e t h e p o p u l a t i o n o f J'=4 i s g r e a t e r .  -  5  Experimental Discharge The  Stability  sensitivity applied.  not  ed  serious experimental  of the  An at  up  discharge  to  10%  fields  of about  the  the  of  i t i n the  effect  was  worst  d i s c h a r g e was  magnetic  I  0  ( 1 - C  2  H  2  curve H  2  2  gauss  )  1  has  a halfwidth only without  introduced component  the  -2  then  f o r -10  any  The  As  be  effect  gain i n  lowest  was  taking  might  be  preasures,  expectwhere  even without  +  ^  (  H  <H<_  T  any  of the of the  that the than  2  H  2  has )  resultant the  of  the  "  d  I  °  +  varies  apparent  i s of the  the  form  of  Lorentzian fitted  a percent  optical ~3%  into  the  smaller  than  s u r f a c e s , however, the u n p o l a r i z e d  Then, assuming  intensity,  amplifier ( 1 - C  gauss  The  light.  emitted  ), the  fraction  factor.  light  ,  )  +10 a  of  1  produced  a polarization  dependence  broader  gauss.  intensity 0(10 gauss~  C~  and  so  could  somewhat u n s t a b l e  with  )  ( I - C  lock-in  in intensity  field.  polarization  that  the  field  fittings.  at the  Assuming t h a t the as  ±15  is  magnetic  to permit  curve  shortcoming  to the  decrease  r e p o r d u c i b l e enough  account  -  Errors  most  observed  71  signal  the  same  produced  field by  the  form Po i + ( 2 y H T ) *  signal  has  a  true Lorentzian.  fitted The  curve  effect  is  somewhat estimated  -  to.produce  Magnetic  ed  magnetic  into  magnetic  general the  This  field,  causes  i.e.  proportional  small  the  field.  made  another only  accuracy which  across  are not quite  of the  effects In  should addition,  to the  More  applied  serious  o f t h e x-y  are  recorder,  proportional  introduce  to' n o n - l i n e a r i t i e s  i n the reading 1% r a n d o m  of data  error.  of measuring  to  a maximum  not effect  from  graphs  the magnetic  errors  error  available  field  lifetimes  lifetimes  of the magnetic  -  were  t o an error  -  of the states,  arises.~>  field  such  represent  T h u s a 2% s y s t e m a t i c  the r e l a t i v e  the absolute  Inhomogeneities  the linear  The g a u s s m e t e r s  o f .1 t o .2 g a u s s .  affect  will i n  .  addition  does  does  perpendicular  The n o n - l i n e a r i t i e s  capable  devid-  i n nearby  experiment.  i n the x channel  a b o u t '1% i n In  i n this  region  of these  a slight non-linearity.  pen displacements  be  to the voltage  effects  Both  field  may  t o ohmic h e a t i n g  to hysteresis  earth's  non-linearities  but  not exceed-  miscalibration.  i n the discharge  materials.  however be q u i t e residual  and  i s due p a r t l y  and p a r t l y  ferro-magnetic  the  calibration errors  field  n o t be- q u i t e  coils.  coils,  field  non-linearities  The  as  halfwidth  Field  The  of  i n the apparent  1% f o r o u r c u r v e s .  ing  b)  changes  72 -  i nthe  - 73 -  discharge and  r e g i o n were n o t measurable w i t h  probably  broadening  The about in  c)  .1 g a u s s  e r r o r s i n the magnetic  the r e l a t i v e  i n the  should  be  discharge  field  will  lifetimes  from FIG.10  gauge w i l l We  differ  s t i l l  t o those  that  however,  expect  measured,  the p a r t i c l e  measurements  inside  difficult  and a c c u r a t e  Pressures  i n the discharge  proximity  o f t h e McLeod  less  those  than  probably  contribute  o r 1.5%  the pressure  from  error  the pressures s o t h a t we  greatest  cell  equally  f o r the liquid  would  be  difficult.  should, judging  The p r e s s u r e  no  a r e needed.  g a u g e , be no more t h a n  measured-.  t o be  introduce  the discharge  cell,  i n the  For the cross-  densities  calculations  measured  the pressure  i n the extrapolated lifetimes.  sections,  thus  Discharge  cell.  proportional  Pressure  negligible  lifetimes.  clear  . by t h e McLeod  errors  leading to  of the lineshapes.  3.5% ' e r r o r i n t h e a b s o l u t e  Pressure It  do n o t e x c e e d  our gausmeter  from the  20 o r 3 0 %  difference i s  nitrogen cooled  dis-  charge . d)  Temperature The affects free  i n the  Discharge  temperature only  path  temperature  o f t h e gas i n t h e d i s c h a r g e  the calculated  o f ground  state H  i s of the order  cross-section. 2  a t 50u p r e s s u r e of the size  The and  again mean 100°K  of the discharge  -  cell,  and t h e time  s e c o n d s , h e n c e we rapidly  expect  a power  inside  will  no  less gas  2 0 0 - 300  Since  rise  the ambient  computed  cm  - 2  cm  - 1  and a  cell.  thermal  /°K, the  about  2°K  the outside walls  i ti s inferred  have  higher exhibit  temperature  that the  a temperature  temperature  cross-section  t h e assumed  t o be  of the discharge o f 10W,  5  when t h e p r e s s u r e i n s i d e i s  microns,  i n the discharge w i l l  the  equilibrium  be a t a t e m p e r a t u r e  temperature  than  ceeding  in  o f .01 W a t t s  the outside wall.  great  i s of order 10~  translational  input to the c e l l  of pyrex  than  collisions  with the walls  conductivity walls  -  between  established  Assuming  n  by more t h a n  varies will  not ex10°K.  a s / T , a 10°K  affect  As  error  the results  only  slightly.  e)  Cascading Although  t h e r e i s no d i r e c t  levels  a r e n o t p o p u l a t e d by r a d i a t i v e  higher  energy  observed.  states,  from  t h e ground  Coherence  will  much  the  transitions  transitions  therefore that probability  have these  than  3d ! 1  from  ever  been  transitions  excitation  state.  e m i t t e d by one m o l e c u l e  leaving  i s absorbed  the d i s c h a r g e , the Hanle  be " n a r r o w e r "  because  lower  that  Narrowing  When l i g h t before  no s u c h  I t seems l i k e l y  occur with very  f)  evidence  than  the composit  the lifetime  system  effect  would  by  another  signal  indicate  has a l o n g e r l i f e t i m e  than  - 75  the  individual  in  these.levels  transitions stable  R.F.  molecules. because  to the  state  -  .This phenomenon does  there are  ground  to which  state,  i t can  no  electric  nor  not  occur  dipole  i s there a  meta-  decay.  Broadening The  the  presence  polarization  netic  field,  term,  providing  frequency. the  o f R.F.  fields  curves.  The  i n general  In the presence  i t i s possible the  will  R.F.  to take  frequency  polarization  o f a weak  account  of the  then  magStark  the  V>>CO^-OJ^,,  curve  broaden  Larmor  obtained i s of  form : 7  P  P  (compare  with  + c E" + Cw -oo . ) 2  r2  =  y  eq.(9)) where  on  the p o l a r i z a b i l i t y  v,  and  curves In  r=^- . i s r  a  The p  p  this  =  that 9.  by  experiment  a  factor  s h o u l d be Since  f i e l d , we »  T a  p  p  state,  lifetime  does  an  E  used.  but  depending  independent  yielded  from  of  these  not that  accuracy.  o f a p p r o x i m a t e l y 100-150V/cm Thus  of at  applied  conclude  experimental  constant  / p ^ T f ?  and' 3 0 0 - 4 0 0 V / c m w e r e changed  C is a  of the  apparent  2  y'  the  least  to  T a  pp  appear C E <<r 2  4  80, I  s  to  factor  2  i . e . the  changed  by  decrease  and  CE  T=T  app  k  i s  correction a  factor  of  at the  larger  within  the  - 76 -  §4.6  Helium Other  4*D  Lifetime  has  already  curve  convincing  been  a state  From  Helium  4 D a  the lifetimes  by magnetic  of a state  with  i n this  c o n v e n i e n t example  the atomic  proof that  d e t e r m i n e d by o t h e r w o r k e r s  those measured  A  Obtained  field In-  and o t h e r broadening  the lifetime  In particular  than  affected  R.F. f i e l d s  be t o remeasure  methods.  by  t h e most  are not seriously  homogeneitiesj would  That  Experiments  Perhaps measured  Compared W i t h  whose  a state  various  Hanle  thesis:should  whose  lifetime  using  a narrower  o f such  state  mechanisms  be  effect  chosen.  i s offerd  transitions  4*D •> 2*P  o  occurs to  a t 4922A.  those  high  observed  Hanle  narrowest  state  halfwidth thus  state  1  8  sec.  curve  s h o u l d b e some  observed  for H  typical  zero f i e l d  i s shown  i n F I G . 25.  2  o f 1 so that  30% n a r r o w e r  effect  by v a r i o u s  The  other workers  e x p e r i m e n t a l work  R.E. B a r d s l e y o f t h i s  are listed  laboratory..  The  I n Table V I I I ,  on t h i s  curve f o r  extrapolating the  t o z e r o p r e s s u r e i s s h o w n i n F I G . 26.' sec.  than the  thesis.  crossing  The c u r v e  8  relatively  has a l i f e t i m e o f  i n this  level  compared  and has a  a n d a Lande g f a c t o r  o b t a i n e d i s (3.97±•4)x10"  obtained  by  T h e He 4 D  of these A  this  4xio  effect  i s very bright  i n molecular hydrogen  polarization.  approximately its  This transition  line  lifetime lifetimes  f o r comparison.  was  performed  T(X10 3.8  8  sec)  ± .3  Author I.  Martlson  et a l .  3  Beam  h  3.91± .2  Descomps  4.1  ± .5  J.P.  4.7  ± •5  Pendleton  and  Hughes  3.0  ± .5  Klndleman  and  Bennett  3.5  ± .4  Fowler  al.  et  al.  Descoubes  et  3.9 ± .5  Brldgett  3.8  Allen  eta l .  3.66  Wlese  eta l .  3.97± .4  ours  ± .5  Table  and  3  3  -  foil ,  1969  Level  3 6  3 8  1  1964  of  decay  1967  observation of  decay  1969  u  Theoretical  4*D  1963 decay  Direct  of the  1965  observation of  1  Level  decay  coincidence  Directoobservation  0  Lifetime  1967  observation of  Delayed Direct  9  I960  Resonance  Crossing  Direct  3 7  •'  VIII  Date  Magnetic  5  King * k  Technique  1965  Crossing  State  of  Helium  -  Figure  25  78  -  Experimental Level-Crossing H e l i u m 4 D -»- 2 P. T r a n s i t i o n 1  1  Curve  for  the  HALFWIDTH (GAUSS)  Figure  26  -  4D J  Curve  Halfwidth  as  a Function  of  Pressure  - 80' -  CHAPTER . -  V  D I S C U S S I O N OF R E S U L T S AND  §5.1  Introduction Under t h e Born-Oppenheimer  the  lifetime of a state  ronic very For  CONCLUSION  t o depend p r i m a r i l y  and v i b r a t i o n a l p a r t s weakly  o f t h e wave  on t h e r o t a t i o n a l p a r t  the states  discrepancy  measured, however,  between  a p p r o x i m a t i o n we  on t h e e l e c t -  function  o f t h e wave there  expect  and  only  function.  appears  t o be a  large  the lifetimes of the J=l state  [(2.66+.1 ) x l 0 s e c ] a n d t h e l i f e t i m e s o f t h e J=2, a n d 3 _ 8  [(3•85±•15) 10~ sec] . x  states whether we  this  account  §5.2  discrepancy  We  Effects  will  could  experimental  first  errors  theory.  Primary  that we  t h e Lande  consider  whether  enough  we w i l l  are that  this  u n l i k e l y that  t o account therefore  Throughout  this  As i s w e l l  known, i n t h e absence  field  I couples  nuclear  t o J t o form  I+J-l,...., | l - j | .  i s applied  dis-  t h e Zeeman e f f e c t be l i n e a r a n d  Zeeman e f f e c t .  a n g u l a r momentum F = I + J ,  are  re-examine the  of non-zero  spin  there  f o r the a p p l i c a t i o n of the theory  g f a c t o r be. k n o w n .  the nuclear  dis-  f o r t h e 50%  the effect  magnetic  might  lifetime  ignored  field,  arises,  a n d , i f i t i s , how  I t seems v e r y  requirements  experiment  have  naturally  •  large  i n lifetimes,  this  i s real,  be r e a l .  crepancy  to  The q u e s t i o n  for i t .  Hyperfine  crepancy  8  they  become  spin  on t h e  o f an  a  external  total  When a  decoupled  discussion  large  and precess  -  seperately fields  about  g  For  =  using  which (J=2  F(F +D  the  where  u  g  the lifetime  too large  n  functions  so t h a t  -- u J ' H 0  (5.32±.2)xl0  8  sec.  lifetime  p  t h e two l i m i t i n g  where  x =  g =gj).  o f the energy  levels  cases  we may  t h e Zeeman e f f e c t  of the magnetic  consider  with  field  the inter-  i s non-linear.  i n a magnetic  field  VJe c o n s i d e r  H,  with  a coupling parameter  a.  + u l«H + a l - J  —  n  —  —  i s t h e n u c l e a r magneton we  J  c o m p a r e d t o t h e J=2  momenta I a n d J c o u p l e d it  _  yields  H a m i l t o n i a n "R o f a s y s t e m  angular  g j by  g  Besides  case  ~ K I +D  low  1 2 J  F  i s now m u c h  mediate  gp, a t very  g factor  2FCF+1)  gp t o c o m p u t e  dependence  field  + J(J + D  f  h a s 1=0  The g f a c t o r ,  to the high  the J = l state this  S  and  the f i e l d .  i s related  cr  8 l -  take  those  at high  and as z e r o  order  eigen-  field i.e.  |J,I,m ,m > J  The  non-zero matrix  I  elements  of this  < J , I ,rrij jm-j. |$| J , I ,rrij »nij> = y  m 0  j  H  Hamiltonian are + y  m n  j  H  + a^j^j  and <J,I,m ,m |-^| J , I , m ± l , m J  I  J  I  + l>  = / J ( J + l ) - i r i j ( n i j i 1) • •/I(I+l)-m (m +l) I  I  - 82 -  The systems and  secular  equation  may  be r e s o l v e d  o f d i m e n s i o n _<2I+1, w h o s e e i g e n v a l u e s  whose e i g e n v e c t o r s  Neglecting  the small  are the states  into  .are  T  (i.e.  these  = 0  t e r m y m-j-H t h e s e c u l a r n  f o r the states  are already  equations  J m= m  =  ±  I  =±l  m j  m= I  ±y  |-J,I,±1,±1>  eigenstates  j H  m  1  o f the system)  0  J =  0  m -=±l  -E  a  ]  0  = 0 -E  a  m =±l ][  and  J m- =0 m  m-j-= 1  m  =-l  j  m=  -yjJ-a-E  1  J=°  a  m  m =0  =  r  0  J m-j-=-l m  =  1  a  -E  a  I  J m-j-=-l  m  =  1  a  energies  o f the system.  for J=l E±y H-a  the  separate  yjH-a-E  are:  83  -  The of  magnetic  produce ed  field  level  of these  and  energies  i n FIG.27-  crossing  a r e shown as a  States.that  function  can i n t e r f e r e t o  e f f e c t s are connected  Am^=0  that  the time  excitation (2) xx  we  without shape,  (this  we  find  Lorentzian  substitute the f i e l d into  rule  merely  involved again  says  by d o u b l e  end-  can indeed  d i s t o r t i o n s have been  distortion  hyperfine  For  we  well  •95  g j j and assuming  moment  by an a o f .5 t o 10 The  H=0  f o r such  that  that  observed  with  an a.  out  since  entirely  the natural  ^j-gj •  of the  elements;  linewidth.  g„ f o r F = 2 , 3 , I f the 3 F  g f a c t o r of these contributes  would  No  the magnitude  can not r u l e  and  each  dispersive  experimentally.  the g factors  the average  MHz  cj> ( s e e ( § 2 . 5 ) i s c h a n g e d t o  comparable  ^j-gj?  are  the width  quadrupole  on t h e e x c i t a t i o n m a t r i x  the J=3 state  resolved,  r  observed  qualitative,  are respectively  I f f o r the  m. >>T).  the lineshape.  be e m p h a s i z e d  splittings  i s s o weak  J  d i s t o r t i o n s near  depends  are only  t h e I_«J c o u p l i n g  by b r o a d e n e d  such  equally  &  when t h e a n g l e  should  functions  lengthy c a l c u l a t i o n s that the  seriously distorting  It  wave  and use t h e s e l e c t i o n  use the e l e c t r i c  shows m a j o r  ours  that  i n changing  a f t e r some  obtained  dependent  the B r e i t formula,  45°,  4  equations  arrows.  We  Q  roots  -  and  states  states i s  to the signal  be e x p e c t e d  t o be much t h e  same. The  intermediate  coupling  case  requires  the  handling  Figure  27 -  Zeeman E f f e c t i n t h e Hyperfine S p l i t t i n g  Presence  of -  Small  85  -  21  of the  s t a t e s and  present  §5.3  Electronic  must  we  H  attribute  suffer  2  from  s t a t e s are  case  b  the  the  case • T  we  Z  b  "  T  to  zero  negative.  8  n  of  T  J=2  J=3  3d  I I , the  states  1  of II.  of  the  the  Hund's  i.e.  C (J)|A> z  and  | a r e  A and  splitting, if'. at  one  under  we  use  compared yield  J=l  Using  the  solutions to  e x p l a i n the  using the  x^.  least  that  s o l u b l e and  and  Born-Oppenheimer  using  T^,  cannot  now,  real,  E  splitting  But  the  is  C (J) •  i t i s clear we  discrepancy  coefficients  Now  only  c r o s s e c t i o n of  the  by  J  state vectors  +  z  f o r x^,,  exist  are  x ~4. 5xl0~ sec.  that  + B (j)|n>  n  hyperfine  The  with  states i n Appendix  for the.lifetimes  equations  collision  warranted  l i n e a r 'combinations  Z  solve  large hyperfine  the  A  of  i n Chapter  B (M) T  considered  If a  as  states.  z  Hence  failure  expansion  Z  equations  scheme  the  A (J)  attempt  these  II a n d  coupled  ( J )  assuming  Z,  are  lifetime  L-decoupling.  z  .  1  the  = A (J)|Z>  A^  Variation  the  mentioned  derived  coupled  |E'(J)>  that  i t to  As  3d*£  where  i n v o l v e d i s not  Wave F u n c t i o n assume  approximation. of  labor  data.  If we  the  -  to  the  the  x's  is  discrepancy.  obtained natural  J=l only  rather  assuming width,  and  8  f o r the  of  coupling  x^,~8xl0~ sec.  seems  obtained pair  simple  s t a t e becomes  states which  any  lifetime  those  g =.450 p  the  of  x  state, half  of  unlikely.  the  -  In in  Appendix  fraction have but  not  §5.4  I I , there  of  t o be  addition to  a  This molecular  of  J=l  the  of  of  this  a  few was  with  may  crossing,  the  and  considered  states  J=3  state  i n the  and  a  The  very  small  mixing  Vr'ould  which  is  rare  are  apparent of  thesis.  r e m o v e d by  searching  exceeds  20  method any  MHz  the be or  that  J=2  3*D  between  J=3  J=3;  no  splitt-  evidence  J=2  state  only  one  that  that  some o f  f o r the  levels  J=l  the and  field  of  the  only  i f the  lifetimes  should  i n v o l v e d has  Refinements  to  considerably.  the  a gx  can  be  ambig-  level  J=l  state.  hyperfine  so.  or  product  equipment  lacks  J=3  atomic  lifetime  be  applic-  providing — 7  state  the  states  the  f o r a non-zero  observable  the  the  a hyperfine  Because  results  of  and  and  i s the  of  collision  discrepancy  J=l  hyperfine  of measuring  molecular  their to  the  I t i s suggested the  measurement  assumes  structure, i t s lifetime certainty.  and  similar  f o r i f one  Work  first  lifetimes  those  in this  c r o s s i n g would  almost  the  excited states  preferably with  This  for Further  An  i n t e r p r e t a t i o n of be  level  splitting  expected.  obtained  i n the  to  scheme  lifetime.  f o r J=2  reported  lifetimes  accounted  MHz  hyperfine  accepted  The  of  ing  able  than  state  1  as  p o s s i b l y be  This  short  Suggestions  3d !,  Helium  may  states  coupling  "mixed"- i n t o  a very  t h e s i s has  hydrogen  lifetime  uity  be  for J=l  and  cross-sections.  any  may  simple  impossible.  Conclusion  state  -  the  state with  greater  86  i n the  could  range  probably  10  the —3 0  sec  extend  to  this  10  sec.  range  - 87 -  APPENDIX I THE TRANSITION MATRIX ELEMENTS  To compute t h e f a c t o r s f o r m u l a , we s h a l l n e e d  A and R  c  i n the B r e i t  f o r the e x c i t a t i o n the m a t r i x elements  o f t h e q u a d r u p o l e moment, Q  (2) , a n d f o r t h e d e c a y t h o s e o f  X X  the  dipole  moment, g * r . The  element  may be r e w r i t t e n  Q  (2) o f t h e q u a d r u p o l e moment t e n s o r  X X  i n terms o f s p h e r i c a l h a r m o n i c s , Y  whose m a t r i x e l e m e n t s a r e w e l l k n o w n . 3 1  .  Q(2)  Sex  v  Inparticular  _v +/ — » Y "2,2 2,-2 V 3 2,0  abbreviating J ,m  <J,m|Y |J',m-> £m  Y  J,m  2,0 J,m  x  2[3m -J(J+l)] 2  J ,m  v  2, ±2 J,m+2 Y  J 2  Y, J  = [6(J±m-l)(Jim)(J+m+1)(J+m+2)]^  J ,m  I 2,±2J J+2,m+2  [(J±m+1)(J±m+2)(J±m+3)(J±m+4)] J  +  2  J  V2  Y,  J J +2  are reduced m a t r i x elements independJ o f t h e m a g n e t i c q u a n t u m mumbers. • B e c a u s e t h e y e n t e r i n t o  where t h e f a c t o r s ent  J  [6(J+m+2)(J+m+1)(J-m+2)(J-m+1)]*/  2,0 J J+2,m J ,m  J« ,m»  - 88 A and R forth  0  only  a s common  and s e t e q u a l  The into be  then and  The  matrix  operator  i s similarly  and l o w e r i n g  R  =  operator.  , and  y  2  of R  21  are:  +  J-l,m+l  [(J±m)(J±m-l)] / R^_ 1  2  1  =  (R+J  J-l,m±l we  J,m  will  neglect  v  so  A, y > y ' ; s i n c e  = y ± l = y ' + l we  that  t h e common  fr ) ' = ^ x v y  J  both  find  states  that  (R ) ' --'u'+l y  n  -^y'+2  <• x y J  Thus v  Svy  f a c t o r Rj_-|_ i n a l l s u b s e q u e n t  Referring t o eq.(7)  calculations. terms  6  . R,. - R +  r  T  =  Again  Letting  r^tir^  R^ + R + ~  =  x  +  J,m ±  expanded  r^sincj)  x  elements  (R )  hence-  the x-axis:  r costj) +  setting  r  be i g n o r e d  = gi«coscj) + gj«sincf>  g*r  or  raising  g makes w i t h  g  will  to unity.  d i p o l e moment  the familiar  the angle  f a c t o r s they  =  (O  v icj>  (§2.3)  decay  we  see that  to a state v  v = y ' + l and y = y ' + 2  (r l ' = y v y^y  (iR ) ' --'y'+l y  y  ;  A  <•  - y'+2 ;  i nthe  with  -  Specific  Applications The  j  line  R(o) arises  from  the t r a n s i t i o n  sequence  J I I  j t  ^ -> ^  89 -  -»- Q  i . e . v = 0,  F o r m=-l  y = l, y ' = - l ,  a n d m=±l  • y m m y ' " ~^ 2,2^1^-1 / 3 ^ 2 , o ) o ' - l Q  r  Q  •  Vv vy ff  g  thus  For  Y  =  f R Jl  1  1  ' ~  - 0,0  1  Y  f RR  i -Jl l  A ( l , y ,y',v) =  3  l6e  2 1  6  e  2  1  * - -2 ?e e  "  *  m=+l  g , g y'v vy &  = 2e  to  2±4> 2i<J>  and-  A(+l,y,y',v) = l 6 e '  thus  H I A(m,y,y',v) = 32e mvy>y '  R  2 i t t >  2icJ)  o - iQ^.ilMg.^ol^l^ilMs^ol' + l Q ^ n g . ! ^ +  I « l , l l  2  l 6  1  >  2  +  0  = 101.3 Hence P  D  = .632 The  i.e.  line  R(2) a r i s e s  y'=-2 } => v = - l y = 0J m=0  °> -> 2  1  from  t h e sequence  J 0  J ' 2  J " 1  - 90 -  I I • A(m,y,y',-1) = my>y  96e  2 1 < | ,  1  similarly I I A(m,y,y',+1) = 9 6 e my>y '  U I . A(m,y,y',v) = mvy>y ' and  R  2 1 ( J )  1926 * 21  = 640  0  192 hence P  0  =  =  .6  The  t r a n s i t i o n R(2) arises  The  non-zero  combinations  o f m a t r i x elements  ->  -V  V  -1  -y  -*-  -2  -1  -»-  0  1  -V  2  1  ->-  0  -»-  0  ->  m  and m a k i n g t h e u s u a l  I y>y  from t h e sequence  ci>  -»-  0  ->•  1  1  ^  3  ^  here are  -1  substitutions,  A(±l,y,y',±2) =  lS^O/ITe  2 1  *  1  *  1  I A(±l,y,y',0) = 5 7 6 e y>y J A(0,y,y',±l) = l 4 4 0 e y>y'  2 1 4 ,  1  H I A(m,y,y',v) = l S ^ e e mvy>y'  or and  2 i < ! )  P  G  =  .275  2  R  0  = 112,992  - 91 -  APPENDIX I I THE  §A2.1  Energy  Levels  The written  3d  = E  2  Hamiltonian f o r the rotating  I.I  n,L,IA|,v T  I.I  T  m o l e c u l e may b e  frame + B ( J -L ) y y  2  y  j i AI n,Lj|A| v n , L I A | ,v  fixed  + B ( J -L ) x x  2  + B [ J + J + L + L + 2J L x x x y x x 2  3  = E  H  and E i g e n s t a t e s  I n the molecule E  S T A T E S OF  1  + B[J  2  L  2  - J  3  2  z  2  + L  2  - L  2  2  + 2J L ] y y  + J L~ + J~L ] +  z  +  2  where  j-  ..B = p  ,  r i s the internuclear u i s t h e r e d u c e d mass  L  i s the electronic  J  i s the t o t a l f i x e d frame  angular  J  1  IT Using the  Hund's  non-zero  case b b a s i s matrix  1  =  are  3 2  "pure  precession",  : 1 / 2  -[(J + A)(J±A + l ) ]  B[J(J+I)-A +L(L+I)-A ] 2  system  |J,±2>  =  |A >  |J,±I>  =  |n >  |J,O>  =  |E>  states  of  and assuming  = B[L(L+l)-A(A±l)]  J  f o rthe 3d  functions  elements  <L,A,J|;K|L,A,J>  these  axis  = L ±iL x y  3  For  i n the molecule  = J ±iJ x y  <L A,j|-Jt|L,A±l J>  Hence  momentum  a n g u l a r momentum  z i s the internuclear  distance  the states  2  |J,A>  ±  ±  the matrix  elements a r e :  of interest are  l / 2  - 92 -  a=<£|#|» -  = B [ J ( J + 1 ) ] •+ E  3 = <n |#|n >  = B [J(J.+ l ) - 2 ]  + E  n  6 = <A |«|A >  = B [J(J+l)-6] + E  A  ±  ±  ±  n  ±  A  "e = <Z|Jf|n >  = B  ±  n = <n |#|A > ±  We t h e n  = B  ±  [J(J+l)L(L+l)]  E n  n A  [ ( J + 2 ) ( J - l ) ( L + 2 ) ( L - l ) ] 1/2  obtain the eigenvalue  Transforming  1 / 2  equation:  a  e  e  o  0  e  6  0  n  6  e  0  3  o  n  0  n  0  6  o  0  0  n  0  t o a system.of  'A E ' X  B  = x>  VV  A  c  n +  x n  +  6  symmetric  and anti-symmetric  wave f u n c t i o n s : 1,  IT  v  A ±  The  l  (  above e q u a t i o n  A  +  ±  A  -  becomes, a f t e r  > some r e a r r a n g e m e n t  o ft h e  terms:  whose  a  /2e  0  0  0  /2e  3  n  0  0  0  n  6  0  0  0  0  0  3  Tl  0  0  0  Tl  6  solutions yield  the energies  = B-H-  A-  B"n-  and s t a t e vectors o f t h e  system. The  energies  o f t h e £ a n d II s t a t e i n t h e a b s e n c e  93 . -  -  of  any J°L c o u p l i n g a r e u n a m b i g u o u s l y  and  £(J=0)  known s i n c e  II ( J = l ) a r e u n p e r t u r b e d . Thus E and  The  E  = 111,804.63  v L  cm  = 112,064.91  n  observed  spectrum  _1  _  cm  1  can then  be u s e d  to find  B  , B , B , Lt  B  E n  ,  B  n  A  A  best  values  o b t a i n e d by t r i a l - a n d - e r r o r  A  =  B  £  =  B  n  = 27.76  B  A  =  B  Z I K  rtA  =  =  27.2  cm  26.39  cm  The  energies  g i v e n by D i e k e  The  f i t o f these  term  data  found  energies  listed  can probably  -D J (J+1) . 2  be somewhat.improved,  of .02 f o r D  t o t h e energy  e i g e n s t a t e s o f t h e system states  +  +  eigenvalues  may n o w b e f o u n d  Q  fits the  , and A  .  Denoting  state  they  =  A |E>  + B |n > +  C  (n )'=  A |E>  + B |n >  +  c |A >  ( A  A |E>  + B |n >  +  C  +  ) ' =  E  N  A  i n terms  theeigenstates  B^+0: E '  found  ±  )', ( A )' a c c o r d i n g t o which  +  f o r comparison.  of the centrifugal  A value  2  v  i n Table IX.  well.  "pure"  and  are listed  are also  1 8  J , by i n c l u s i o n  ±  (II  1  f o r the states  Corresponding the  1  27.54 cm"  thus  distortion  i cm"  28.11  energies  at high  i  cm  The  particularly  were:  I  112,488 cm  E  B  state  Zj  , and E .  The  E  Jl  +  E  +  n  +  A  E  | A  +  >  +  N  A  | A  +  >  tend  a s B^^-^-0  above, of the  -  J  Energy from above t h e o r y (cm ) - 1  94  -  Observed energy .(cm )  Energy from above t h e o r y (cm )  r  E  .  1  Observed energy (cm~ ) Y  E  0  111804.63  111804 .63  . 1  1796.64  1797 . 1 1  2  1819.54  1819 .78  .3  1888.90  1885 .07  4  2009.38  1997 .49  n  -  IT  n  n~  1  112127.50  112127 .23  112064.91  112064 .91  2  2279.27  2 2 7 4 .24  2140.99  2139 . 6 1  3  2472.01  2 4 6 3 .04  2265.06  2264 .09  4.  2709.58  2695 .70  2440.74  2441 . 1 2  5  2669.87  2671 .10  6  2953.38  2953 . 2 1  A  A  +  A~  A"  +  2  112533  112528 .75  3  2769  2766 .60  2734 .12  2735 .56  4  3075  3070 .18  3005.42  3010 .39  5  3335.01  3338 .57  6  3721.97  3716 .85  Table  The levels in  IX - Energies of the 3 d  coefficients  A^,  o f the symmetrized  Table  B^,  C^,  states  1  Complex  112517 .95  of  f o r the f i r s t E ' , (II ) ' , a n d  H  2  s i x rotational (A ) ' a r e  listed  X.  Using functions  112522.96  these  expansions, another  i s p r o v i d e d by t h e r a t i o s  check  on t h e s e  wave  of the i n t e n s i t i e s  of the  J  C  .901  1 2  '  .830  A  E  n  B  n  c  n  A  A  A  A  C  0  0  -.434  0  .434  .901  0  -.550  .090  .537  .746  -.394  .148  • 375  .915  .603  -.569  .260  .524  .811  3  .787  -.602  .137  .  4  .759  -.629  .169  .557  .492  -.669  .338  .602  5  .739  -.646  .192  .550  .413  -.726  .389  .642  6  .725  -.657  .210  .543  .357  -  .424  .664  Table  X  - Expansion  560  0  B  Coefficients  for  .760  the  3d  1 +  States  •  .724 •  .660 .616  -  R and P l i n e s values  arising  of the states  factors  will  transition  96 -  from  t h e same u p p e r  a r e almost  be a l m o s t  identical  identical  matrix elements,  A  and w i l l  state. the  Because  Franck-Condon  be s e t t o u n i t y .  „ f o r t h e 3 d ( E ) ->2p E 1  T t  T  theB  ,  1  The  will  t h e n be g i v e n by A  j*->j"  =  A (J)<E,J"|P|E,J'> E  where P i s t h e d i p o l e  The  + B ( J ) < E , J " |p|n,j'> E  m o m e n t o p e r a t o r summed o v e r  individual  m a t r i x elements  a l l directions  a r e g i v e n by  J.K.L,  M a c D o n a l d 3 3 as : < E , J [p E, J + l > =  [4CJ+D/2J+1] / 1  = 0  < E , J IP E ,J>  Q(J)  <E, J IP E ,J-1> =  [4J/(2J+1)]^  2  R(J-l)  <n,j IP E , J + 1> =  [3J/C2J+1)] /  2  P(J+D  <II,J p E ,J>  /3  =  1  Q(J)  <n,j [p E ,J-1> = - [ 3 ( J + D / 2 J + l ] The  theoretical  Table the  2p*E s t a t e s . i nTable The  sensitive  energy  because test  s i xrotational  thus  found  states'  arelisted i n  transitions to  experimentally are also  XI f o r comparison. levels  tests  of the large  latter  R(J-l)  1 / 2  intensities  Those measured  not stringent  because the  relative  XI f o r the first  listed  are  P(J+1)  2  and r e l a t i v e  intensities  of lines  o f t h e f o r e g o i n g theory; t h e former  number o f p a r a m e t e r s  used  t o f i tthe data,  of the experimental inaccuracy.  i soffered  b y t h e Zeeman  effect  A more  o f these  levels.  97  -  Transitions  I(Theor.)  -  I(Exptl.) 80°K  Theor.  300°K-  -"-RCJ'-D I  P(J'+1)  R(0) P(2)  2.74 1.07  42  46  13  17  R(D  3. 20 0.47  40  P(3)  5  R(2) P(4)  3.31 .26  80 3  R(3)  3.33 .16  20  P(5)  R(4) P(6)  3.32 .11  26  R(5) P(7)  3.31 .09  39  Table  Exptl. I  R(J'-1)  P(J'+1) . 80°K 300°X I  2.5  3.2  2.7  33 5  6.8  8  6.6  93 5  13  26  X I - R e l a t i v e I n t e n s i t i e s o f P and R i n t h e 3d*E -*• 2 p E ( 0 / 0 ) B a n d  18.6  Transitions  J  §A2.2  The Zeeman The  Effect  Zeeman  effect  Hamiltonian,it 5  m under  Hund's L  case  .  H  -  b coupling,  = (L'J)(J-H)  =  t h e term -2-LgJz + L  j 2  -  L , J , L-, and J z z '  frame.  The n o n - z e r o  assuming pure  5  J  °— —  +  where  , may b e w r i t t e n m  :  +  L ' H may b e r e w r i t t e n a s +  J " +• L " J  J  ^  elements  of $  m  are then  2  <A,J,m |R |A,J,m > J  m  J  =  g (A,A)y Hm J  H  -  t o t h emolecule  precession): Vo'H A = j^j+^rrij  >  -  area l l referred  matrix  +  0  J  fixed  (again  - 98 -  <A,J,m |$ |A±l,J,m > T  J  = — 2J(J+1>  T  m  J  To magnetic  alized  gj(A,A±l)y Hmj 0  s o l v e now f o r t h e e n e r g i e s  field,  may b e a d d e d  we  can f o l l o w e i t h e r  This  procedure  due t o t h e m a g n e t i c  zero-field  splittings.  set  there  this  the  procedure.  independent  test  courses,  this  place  at a  matrix  complete  (sum o f t h e i s then  and t h e g j ( A , A ) y H  products.  D  leaves  just  the trace invariant,  hence  t h e sum £ g j = £ g j ( A , A ) w h e r e t h e  over  whether  f o r our  c a n be drawn  The t r a c e  a l l coupled  states.  o f any c o u p l i n g p a r a m e t e r s  t o determine  look  of the non-diagonalized  of the matrix  to the  i s n o t t h e case  conclusion that  energies  diagon-  i ft h e energy  comparable  L e t us i n p a r t i c u l a r  any c o u p l i n g scheme,  .summation t a k e s  were  common J a n d mj = 1.  elements)  Diagonalization  is  Although  sum o f t h e z e r o - f i e l d  under  field  i s an i n t e r e s t i n g  of states with  diagonal  o f two  w o u l d be n e c e s s a r y  splittings  from  of the states i n a  t o Vi ( § A 2 . 1 ) a n d t h e p e r t u r b a t i o n m a t r i x  again.  states,  /L(L+l)-A(A±l)/(J+A)(J±A+1) . •  This  and serves  conclusion as a  the set of states considered  i s a  complete s e t .  The individual carried scheme  argument  above  g-factors unless  out.  the complete  A simpler procedure  considered  perturbation  does n o t y i e l d  i n f o r m a t i o n on t h e  diagonalization i s  f o rfinding  g j under  i n §A2.1 i s t o u s e n o n - d e g e n e r a t e  theory  with  the coupling  first  the perturbation Hamiltonian $  .  order  99  -  Thus;  f o r the g  J  Z'  state  HS^T^S •,J ,  =  = A (J) 2  g j  m  |*f  j  E  I S • J ,  m  + B (J)  (Z,Z)  2  + A (j)B (j)[ E  E  of the in  z  g factors Z'  Table  state  j  (z,n )  J  thus  g  found  +  M J  (n  A ) +  J  f o r the  u s i n g the  >  +  +  g j  +B (j)c (j)[g (n The  -  ••  +  n )  +  +  5  C (J) 2  g j  (A  +  A ) +  3  (n ,z)] +  g j  +  (A ,n )] +  g j  first  coefficients  +  six rotational from  Table  levels  X are  listed  XII. g-theory  J 1  •  771  .901  2  •  5^1  .571  3  .409  .455  4  .320  .387  5  .245  .331  6  .233  .287  Table  XII -  The  factors  qualitative  g  g-values  of the  not  f i t the  is  worthwhile  do  form  data  A  State  J  basis.  The  showing  a  experimentally obtained  n e a r l y as w e l l  t o examine whether  complete  3d Z  d e r i v e d above, although  agreement w i t h those  do  a  g-experimental  as  the  pure  might Z,IT,  case  be  and b  expected. A states  coupled  3 0  , It  considered  states  have  2  g-factors  j  (  g-factors  i s 0.5.  •  Hence Looking  f o r the  J=l states,  at Dieke's  the  experimental  sum  of  data,  the  -  gj(£)  =  that  ±.901  these  therefore  not  The help  as  which  shows  a  In Dieke the on  3 0  were  spectrum  with  of  R(2)  level,  not of  the  added  our  set  of  other  2  3m  to  good  of  basis  that  24,500  grating  be  sum;  functions  no  we is  way can  not the  set  of t h e . v i b r a t i o n a l be  quite ^  perturbation  with  both  small.  treatment  electrons  excited,  splitting.  r e s u l t of  in a  0.5  a  to  v i b r a t i o n a l l e v e l s to  state  ensure  plate  the  g-factors  arithmetic  or  gauss- f i e l d  spetrograph.  measuring  was  errors,  reproduction and  R(2)  ( a ) , and  parallel  (b)  to  obtained  from  R(0),  the  by  photographed  R(l)  showing  R(0),  A  reported  of lines  the  28.  gj  the  R(l),  are:  gi  =  .900  g  2  =  .597  g  3  =  .452  be  appears  to'give  integral  *E  Zeeman  the H  a  g-factors,  lines  should  There  i n c l u s i o n i n the  i s shown i n FIG.  splitting in  be  overlap  for  order  The  It  ±.500.  polaroid perpendicular  field  and  the  small  a Jarrel-Ash  a" p o r t i o n  that  -  B. ., , f o r v=v'±l s h o u l d A,v;A',v'  3*K  the  can  addition  candidates  include  (IT ) =  x  conclude  wavefunctions  Other  g  g-factors  complete. will  and  100  mentioned  measured  is  agreement w i t h  that  as  open t o those  of  R(4)  overlaps  question. Dieke.  R(l),  These  the  values  are  -  102  -  APPENDIX.Ill OTHER  In discussed the J=2  addition to.the  before,  on t h e 3 d !  states  1  a l s o done  on  3d n~(v=0)  and on t h e  1  level.  3d'n"(v=0)  The  the 3d  reason  l r  [  was  J=2  felt  a n d 3d*A  State  that  a rough  would  be  3,  lifetimes.  is  negative.  of  30u  1  D a t a was  and 20u.  The  to rather  obtained  gauss  average  they  The  o f 3.6±.3  halfwidth  the  J = l , and t h e J=2 easily  resolved  polarization for this  of this  line  should  b u t as o n l y suffice.  at  line line  pressures  each  a rough  The  and  These h a l f w i d t h s a r e  and t h e e r r o r s quoted  extrapolation yields gauss.  The  estimate  halfwidths  a t 30u p r e s s u r e ,  a t 20u p r e s s u r e .  of 5 runs  only.  most  lifetime  e x t r a p o l a t i o n of the h a l f w i d t h i s  a r e 4.805±-045 g a u s s  4.385±.026  tistical  The  large errors,  (±20%) i s r e q u i r e d obtained  brightest,  1  of the  useful i n determining 1  3 d I l > 2 p E (0-*0) .  t h e Q(2)  subject  The  knowledge  i n the 3 d !  f o r the discrepancy  is  the  done  3 levels  1  It  and  work  some p r e l i m i n a r y w o r k w a s  3 K ( v = 2 ) J = l , 2, a n d  §A3.1  of  STATES  slope,  a  are  sta-  zero-pressure , suggests  a  o  cross-section value widths  of  of -170A . 2  .412 I s t h e n  of the line  Table I I I .  The  lifetime  3.84xio~ sec.  obtained  8  on each  using  The  DIeke's  individual  run are l i s t e d  ghalf-  i n  -  Q(2)  -  o f 3d' n-^2p E 1  Pressure  1  30u  20y  4.75^58  4.37138  4.81734  4.37775  4.68231  4 .40187  4.90070  4.30700  4.87179  4.52766 4 . 3 8 5 ± .026  805±. 045  Average  3 0 y<  Pressure Transition  Average  R(2)  103  a t 20y 30y  Table  3  1  R(0)  R(D  R(2)  3.01  4.23  5.88  3.51  4.25  6.29  3.22  4.13  5.97  3.26  3-97  5.61  3.14  4.54  3.14  4.13  5.59  3.21± . 0 8  4.21+.08  5.77  has  .  5.31  Hi =5-36 2  H  1 2  =7.66  I I I- Level (3d !! 1  C r o s s i n g Curve and 3 K ) X  Halfwidths  -  §A3.2  T h e 3*K  State  Although state  has a l r e a d y  Table  I I I .  states v=2,  We  with  been published * , 1  attempted  little  long  were  lines  required  lineshapes.  could  n o t be  The  error  i s only  f o r these  f o r the state  resolved  gave  The s p l i t t i n g  i n .  a r i s e s because the  barely  which  are l i s t e d  the g-factor  The g - f a c t o r  the large  observed  the data  3  t o measure  success.  Zeeman p a t t e r n exposures  a v a i l a b l e o n t h e 3*K  the information  J=l i s .28±.05;  distorted  104 -  rise  and because  t o somewhat  on t h e h i g h e r  rotational  measured.  lines  observed  are the R(0),  R ( l )and R ( 2 )  3 K(v=2)->2p E(v=5) . 1  1  the R(2)  Only trapolated (1.6±.1'5)  t o zero 10  - 8  sec.  line  has had i t s h a l f w i d t h ex-  pressure; at zero  i t s gx p r o d u c t i s  pressure.  Assuming '  es  section which 8xio  - 8  we w o u l d sec.  obtain 2.46 state, is  o f 100A  2.7  , I t sg-factor compute  Assuming  t h e same  f o r the J = l state gauss.  Assuming  the halfwidth gauss.  assumptions  of  t h e same  cross.17  from  approximately  lifetime  a halfwidth  and g = . 2 8 , A  at zero  we  pressure  of  cross-section f o r the J = l  extrapolated  On t h e b a s i s  this  i s approximately  a lifetime  a  .  from  30u t o zero  o f o u r somewhat  i s i n surprisingly  good  pressure,  unwarrented  agreement.  -  105  -  R E F E R E N C E S AND  FOOTNOTES  A . C . G . M i t c h e l l a n d M.W. Zemansky; Resonance R a d i a t i o n and E x c i t e d Atoms ( C a m b r i d g e U n i v e r s i t y P r e s s , L o n d o n , 1961)  93-153.  pp.  G. M.  Lawrence  a n d D.B.  Savage; P h y s . Rev.  1 4 1 , 67  (1966)  • I . B r e w e r , C.G. J a m e s , R.G. B r e w e r , F . E . S t a f f o r d , R.M. a n d G.M. R o s e n b l a t h ; R e v . S c i . I n s t r . 2 3 , 145-0 (1962) R.G.  B e n n e t t a n d P.W.  Only  a few  P..  o f t h e numerous  Thaddeus  J.P.  D a l b y ; J . Chem. P h y s .  and  Barratt;  R.  papers are c i t e d  Novick;  l e Journal  40,  de  Physique  (1964).  1414  below:  13_6_ A,  P h y s . Rev.  Berg  87  (1964 20_,  e t l a Radium  633  (1959).  W.  Demtroder;  D.R.  Zeits.  Crossley  and  f. Physik  R.N.  Zare;  166,  (1962).  42  P h y s . Rev.  Letters  18,  942  (1967).  A.  Marshall,  22,  445  R.L.  de  Zafra,  K.R.  German' a n d  S.J.  Silvers,  Phys. .52,  R.N.  T.H.  5385  J.W.S. R a y l e i g h ; H. H.  Stroke;  R.W.  Wood a n d  W.  Hanle  G.  Breit;  Zare;  P h y s . Rev.  Letters  A.  Roy.  1207  (1969).  and J.C.  Soc. A  pp.  (1922). 55-60.  103,.396  (1923).  (.1924). 439  Zemansky;  Roy.  1900  Soc.  93  S o c . Amer. 1 0 ,  Proc.  102,  Roy.  30,  f. Physik  H. W.B.  Jean-Pierre  23_,  K l e m p e r e r ; J . Chem.  Oct. 1966,  Proc.  a n d M.W.  Skinner;  Soc.  Today,  Elet;  J . Opt.  (1965).  a n d W.  Letters  P h y s i c s o f t h e One-and T w o - E l e c t r o n P u b l i s h i n g Co., Amsterdam, 1969)  Proc.  Physics  Zeits;  Lombard!  P h y s . Rev.  Bergeman,  A.C.G. M i t c h e l l  1485  Metcalf;  (1970) .  J.C. P e b a y - P e y r o u l a ; Atoms ( N o r t h H o l l a n d pp. 348-361.  M.  H.  (1969)  (1925).  loc.'cit. 112,  Pebay-Peyroula;  642  pp.  258-318.  (.1926).  Compt. Rend.  26l,  -  Descoubes; C R .  Acad, Sc. P a r i s  259,  327  (1964).  -  106  -  17  P a t r i c k C a h i l l , R i c h a r d S c h w a r t z , a n d A. N o r m a n Phys. Rev. L e t t e r s 1 9 , 283 (196.7).  is  G.H.  Dieke;  494  (1958).  19  P.A.  P r a n k e n , Phys. Rev. 1 2 1 , 508  (1961).  20  G.H.  Breit;  21  F.D. C o l e g r o v e , P.A. F r a n k e n , R.R. P h y s . R e v . L e t t e r s 3, 4 2 0 ( 1 9 5 9 ) .  22  M.  23  G. H e r z b e r g ; S p e c t r a o f D i a t o m i c M o l e c u l e s ( D . V a n Company I n c . 1 9 6 6 ) s e e e s p e c i a l l y p p . 2 1 8 - 2 2 6 .  24  P.M.  J . Mol. Spetr.  2,  Revs. Modern Phys. 5,  B o r n a n d R.  Oppenheimer;  Davidson; Proc.  Von I . Kovacs  26  P.G.  L e w i s a n d R.H.  Roy. Soc. A 1 3 8 ,  S c h e y , K.  Smith;  8_4, 4 5 7  580  Sands; (1927). Nostrand  (1932).  and I t s S p e c t r u m .(Yale  a n d A. B u d o ; H u n g . A c t a  B u r k e , H.M.  (1933).  Ann. P h y s i k  W. R i c h a r d s o n ; M o l e c u l a r H y d r o g e n University Press, 1935). 25.  91  Jette;  Physica  1,  1  (1949).  Phys. Rev. 129., 1258  (1963)  A b r i e f account o f the s t a t e o f low energy electron-atom s c a t t e r i n g t h e o r y as o f 1 9 6 4 i s g i v e n by E. G e r j u o y i n • P h y s i c s T o d a y 1 8 . , 24 ( M a y 1 9 6 5 ) . 27  L.D. L a n d a u a n d E.M. L i f s h i t z ; Q u a n t u m M e c h a n i c 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. I n c . R e a d i n g M a s s . 1 9 5 8 , s e c o n d e d i t i o n ) §145. Although the authors give only i n e l a s t i c c o l l i s i o n cross-sections, following through t h e i r d e r i v a t i o n , t h e m a t r i x e l e m e n t s a r e e a s i l y f o u n d u n d e r t h e same c o n d i t ions.  28  R.R. B o c k e m u e h l , G e n e r a l Warren, Michigan.  29  W. H a p p e r a n d E . B . S a l o m a n ; P h y s . R e v . l 6 0 , . 2 9 ( 1 9 6 7 ) . The a u t h o r s o f t h i s p a p e r a t t r i b u t e t h e p h a s e s h i f t plates t o A. L u r i o , R. G a r w i n , a n d A. P a t l a c h o f I . B . M . W a t s o n L a b o r a t o r y , C o l u m b i a U n i v e r s i t y , New Y o r k .  30  G.H. 92,  Dieke, 81  S.P.  Motors Research  Cunningham,  Laboratories,  a n d F.T. B y r n e ; P h y s . R e v .  (1953).  31  L.D.  L a n d a u a n d E.M.  32  J-.H. V a n V l e c k ;  33  J.K.L. MacDonald;  Lifshitz;  loc.cit.  P h y s . R e v . 33., 4 6 7 Proc.  §10,7.  (1929).  Roy. Soc. A 1 3 8 , 193  (1932).  -  107  I . M a r t i s o n , W..S. Biclcel, L. L u n d i n , R. B u c h t a , a n d A m e r . 60_, 3 5 2 (1970).  -  J . B r o m a n d e r , H.G. Berry, I . B e r g s t r o m ; J.Opt. Soc.  B. D e s c o m p s , J . C . P e b a y - P e y r o u l a a n d Rend. Acad S c i . 2 5 1 , 9 4 l (1961). J.P. Descoubes; Physics l o c . c i t . p. 34l.  of the  W.R. P.T. 87  J. Brossel;  Compt.  One-and T w o - E l e c t r o n 138  P e n d l e t o n and  R.H.  Hughes;  Phys.  Rev.  Kindleman  W.R.  Bennet;  Bull.  Amer. P h y s .  and  A,  683  K.A.  Bridgett  A92,  75  59,  (1965) 8,  Soc.  (1963) .  R.G. F o w l e r , T.M. H o l z b e r l e i n , C.H. J a c o b s o n a n d S . J . B . C o r r i g a n ; P r o c . P h y s . S o c . ( L o n d o n ) A84, 539  L.  Atom  T.A.  King;  Proc. Phys.  Soc.  (London)  (.1967)  Allen, 842  and  (1964).  D.G.C. J o n e s ' , D.G.  Schofield';  J . Opt.  Soc.  Am.  (1969).  W.L. W i e s e , M.W. S m i t h a n d B.M. Glennon; Atomic T r a n s i t i o n P r o b a b i l i t i e s , V o l . 1 NSRDS-NBS4 ( U . S . G o v t . P r i n t i n g O f f i c e , Washington, D.C, 1966). F.W. D a l b y and North-Holland,  J . v a n d e r Linde;. C o l l o q u e Ampere Amsterdam, 1969.  XV,  

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