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A UV Zeeman-effect polarizer Grant, Robert Wallace 1985

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A UV Zeeman-Effect P o l a r i z e r  by Robert Wallace Grant B . S c , The U n i v e r s i t y of Manitoba, 1979  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULITY OF GRADUATE STUDIES (Department of Physics)  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1985  © Robert Wallace Grant, 1985  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the  requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by t h e head o f my department o r by h i s o r her r e p r e s e n t a t i v e s .  It is  understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l n o t be allowed without my w r i t t e n  permission.  Department o f  \hy$\c<,  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3  Columbia  ii  ABSTRACT  I t i s shown that l i g h t emitted by mercury vapour at 253.7 nm can be p o l a r i z e d by passing the l i g h t through mercury absorption gas embedded i n a magnetic f i e l d which i s transverse to the d i r e c t i o n of propagation of the light.  The absorption l i n e s of the mercury are s p l i t by the Zeeman e f f e c t ,  so that the absorber has an absorption c o e f f i c i e n t which depends on both the p o l a r i z a t i o n and wavelength of the transmitted l i g h t . A complete theory f o r the H g  2 0 2  isotope i s presented and the r e s u l t s  are compared to measurements made with a natural mercury emitter and absorber.  The observations are i n q u a l i t a t i v e agreement with the theory  once isotope and hyperfine s t r u c t u r e of the isotopes i n n a t u r a l mercury are included i n the theory.  Quantitative a n a l y s i s was not possible because the  emission l i n e p r o f i l e s could not be measured with the a v a i l a b l e equipment.  iii  TABLE OF CONTENTS  Abstract  i i  Table of Contents L i s t of Figures Acknowledgements  i i i vi viii  Chapter One Introduction  Chapter Two I I . 1 The Zeeman E f f e c t 11.2 Line Broadening  1  3 4 14  Natural Broadening  15  Resonance Broadening  17  Doppling Broadening  21  The Combined Line P r o f i l e  22  11.3 Theory  24  11.4 The Wave i n a Transverse and P a r a l l e l F i e l d  50  The Transverse F i e l d  51  Parallel Field  57  iv  II.5 The P o l a r i z a t i o n of the Transmitted Wave  58  Completely P o l a r i z e d L i g h t  65  Unpolarized Light  67  P a r t i a l l y Polarized Light  68  Unpolarized Incident Wave  70  Simple Examples  72  L i n e a r l y P o l a r i z e d Incident Wave  75  Chapter Three 111.1 The Experimental Design  89  The Light Source  92  The Gas C e l l  93  The Magnet  95  The Aperture Stops and Lenses  96  P o l a r i z e r and Quarter-Wave Plate  97  Monochromator  98  Detector  98  E f f e c t of the Magnetic F i e l d on the Experimental Apparatus 111.2 The Experimental Procedure The Unstable L i g h t Source  99 100 104  111.3 Experimental E r r o r  106  111.4 Experimental Results  110  V  Chapter Four Comparing Theory and Experiment  120  IV. 1 The E f f e c t of Using Natural Mercury i n the Absorption C e l l E f f e c t of Nuclear Spin  121 121  IV.2 The Shape of the Experimental Incident Line  124  IV.3 The Experimental Results  125  Chapter F i v e Concluding Discussion  129  References  132  Appendix  135  vi  LIST OF FIGURES  Figure Numbers  Page  rel 1.  Relative Intensities I  of the Componet Lines f o r  observations along the y-axis  12  2.  O r i e n t a t i o n of the Magnetic F i e l d  27  3.  O r i e n t a t i o n of |u>' Co-ordinate System  32  4.  a) P o l a r i z a t i o n Measuring System b) P o l a r i z a t i o n O r i e n t a t i o n  62  5.  Emission, Absorption and Transmission Line P r o f i l e  73  6.  Geometry of P o l a r i z a t i o n E l l i p s e  78  CD  7.  J  as a f u n c t i o n of Magnetic F i e l d Strength f o r  Three Gas Temperatures, 300K, 350K and 375K D = 25 ( 1 0 ) cm J = J at zero f i e l d and i s independent of the xx yy strength of the f i e l d - 3  -1  86  CO  8.  J  as a f u n c t i o n of Magnetic F i e l d Strength f o r  Three Gas Temperature, 300K, 350K and 375K D = 40 ( 1 0 c m ) J = J at zero f i e l d and i s Independent of the xx yy -3  -1  strength of the f i e l d  87  vii  GO  9.  J  as a function of Magnetic F i e l d Strength f o r  Three Gas Temperatures, 300K, 350K and 375K D = 60(10- cm ) 3  J  xx  _1  = J at zero f i e l d and i s independent of the yy r  strength of the f i e l d  88  10.  Experimental  90  11.  Temperature Regulator f o r the Water Bath  12.  Im ( J ) as a f u n c t i o n of Magnetic F i e l d Strength xy J ' and J ' as functions of magnetic f i e l d strength xx yy for a n a t u r a l mercury absorption gas at 22.6°C  13.  Apparatus  94 113  b  14.  117  J ' and J ' as f u n c t i o n of magnetic f i e l d strength f o r xx yy e  a n a t u r a l mercury absorption gas at 30°C 15.  P o l a r i z a t i o n P as a f u n c t i o n of magnetic f i e l d for a n a t u r a l mercury gas at 22.6° and 30.0°C  118 strength 119  viii  ACKNOWLEDGEMENTS  Physics demands extended periods of i s o l a t e d s o l i t a r y work.  The reward  for t h i s e f f o r t comes from the time spent with others sharing ideas and learning.  T would l i k e to thank my friends and colleagues i n the Physics  Department f o r making my time there rewarding.  I would p a r t i c u l a r l y l i k e to  thank Dr. F.L. Curzon who suggested and supervised the work i n t h i s  thesis.  Without h i s unflagging support t h i s thesis would never have been completed. I would also l i k e to thank Mr. A. Cheuck f o r h i s assistance i n designing and constructing the experimental apparatus.  The assistance of  Dr. Bloom i n providing laboratory space and the electromagnet used i n the experiment i s also acknowledged. F i n a l l y , I would l i k e to thank Dr. F.L. Curzon, Susan Mah, Connie Zator, Young Yuen and Don Furseth f o r t h e i r help i n preparing t h i s  thesis.  - 1 -  CHAPTER I Introduction  A number of spectroscopic techniques for analysing l i n e shapes and t r a n s i t i o n p r o b a b i l i t i e s require high q u a l i t y l i n e a r p o l a r i z e r s .  These  p o l a r i z e r s c u r r e n t l y e x i s t f o r the v i s i b l e and near u l t r a - v i o l e t region, but because of the properties of the d i s p e r s i v e media used, they have low t r a n s m i s s i v i t y i n the f a r u l t r a - v i o l e t region. The object of t h i s study i s to examine the p o s s i b i l i t y of e x p l o i t i n g the Zeeman E f f e c t to produce p o l a r i z e r s which have a high t r a n s m i s s i v i t y i n the u l t r a - v i o l e t region.  The proposed technique involves the charater-  i s t i c s of the absorption spectra of a gas immersed In a magnetic f i e l d .  The  absorption spectra, when viewed i n a d i r e c t i o n transverse to the magnetic f i e l d , s p l i t i n t o a s e r i e s of Zeeman Components which are l i n e a r l y p o l a r i z e d e i t h e r transverse or p a r a l l e l to the magnetic f i e l d .  By adjusting the  degree of s p l i t t i n g and the shape of the absorption l i n e s the gas can act as a l i n e a r p o l a r i z e r f o r i n c i d e n t l i g h t at or near an absorption l i n e of the gas. The technique i s examined both t h e o r e t i c a l l y and experimentally.  In  Chapter I I a t h e o r e t i c a l model i s developed f o r the propagation of l i g h t In a gas Immersed i n a magnetic f i e l d .  The derived equations are more general  than i s required f o r t h i s study, but i t i s hoped that they may f i n d application i n further studies.  - 2 -  The t h e o r e t i c a l model i s used to c a l c u l a t e the e f f e c t of a gas composed of the H g  2 0 2  isotope, on the state of p o l a r i z a t i o n of an incident 253.7 nm  l i n e of mercury.  The 253.7 nm l i n e was selected because i t s properties are  reasonably w e l l known and therefore the c a l c u l a t i o n could be performed with a high degree of p r e c i s i o n . The t h e o r e t i c a l c a l c u l a t i o n s i n d i c a t e that the method can be used to produce h i g h l y l i n e a r l y p o l a r i z e d l i g h t with small transmission l o s s e s . The technique i s examined experimentally i n Chapter I I I . The state of p o l a r i z a t i o n of the 253.7 nm l i n e of mercury i s measured a f t e r the i n i t i a l l y unpolarized l i g h t traverses a gas c e l l f i l l e d with n a t u r a l mercury and immersed i n a magnetic f i e l d traverse to the d i r e c t i o n of propagation of the light.  While the use of n a t u r a l mercury rather than an even isotope  introduces a number of undesirable e f f e c t s , the experiment supports the conclusion that the technique can be used to produce high q u a l i t y polarizers.  -  3  -  CHAPTER I I  In t h i s chapter a t h e o r e t i c a l model i s developed to describe the production of polarized  l i g h t through the Z eeman E f f e c t .  The purpose of  t h i s model Is to account f o r the observed p o l a r i z a t i o n and i n so doing provide a framework f o r i t s study and assessment.  The f i n a l r e s u l t i s an  equation f o r the transmission of l i g h t through a gas Immersed i n a magnetic f i e l d , where the frequency of the l i g h t i s at or near a reasonance  frequency  of the atoms of the gas. The equation i n d i c a t e s how to optimize the p o l a r i z a t i o n and i n a d d i t i o n reveals the inherent l i m i t a t i o n s of t h i s method of p o l a r i z i n g l i g h t . The c e n t r a l r o l e of the Zeeman E f f e c t i n t h i s study n e c e s s i t a t e s a review of t h i s phenomenon.  However, since many comprehensive  accounts of  t h i s e f f e c t already e x i s t , ( see Condon and Shortley 1964, Sobel'man 1972, and Kuhn 1969) only a cursory review i s presented.  B r e v i t y i s achieved i n  part by omitting the mathematical d e r i v a t i o n s of most of the equations, and by presenting only those equations which are necessary f o r a c l e a r standing of the e f f e c t , or which are required i n the subsequent development.  under-  theoretical  - 4 -  I I . 1 THE ZEEMAN EFFECT  When atoms are placed i n an external magnetic f i e l d i t Is found that the f i e l d causes some of the s p e c t r a l l i n e s of these atoms to s p l i t i n t o a series of component l i n e s which are displaced about the frequency of the original line.  I f a s p e c t r a l l i n e i s observed along an axis which i s  transverse to the d i r e c t i o n of the magnetic f i e l d , i t i s found that a given component l i n e i s l i n e a r l y p o l a r i z e d with i t s e l e c t r i c f i e l d vector e i t h e r p a r a l l e l (n-component), or perpendicular (a-component), to the e x t e r n a l magnetic f i e l d vector.  I t i s these two r e s u l t s , the s p l i t t i n g of the  s p e c t r a l l i n e s and the p o l a r i z a t i o n of the component l i n e s , which together comprise the Zeeman E f f e c t . The Zeeman E f f e c t can only be s a t i s f a c t o r i l y accounted f o r by using a quantum mechanical representation of the atom i n an external magnetic f i e l d as given by: H = H + H o m  v  (2-1)  '  where H  i s the Hamiltonian of the Isolated atom, and H describes the o m i n t e r a c t i o n between the external magnetic f i e l d \ and the magnetic moment of the atomu.  H has the form m H  M  =  -J  .  \  (2-2)  The magnetic moment of the atom i s r e l a t e d to the t o t a l o r b i t a l angular momentum X, and the t o t a l spin angular momentum ^ of the electrons i n the  - 5-  t  atom through the equation ;  - - j p (t + ^  where  2  "I)  (2-3)  = |Le  (2-4)  i s the Bohr Magneton; m , i s the e l e c t r o n mass; and e i s the elementary g  charge. The Hamiltonian H should a l s o contain a term H' which i s quadratic i n m ^ B.  This term would account f o r the induced e f f e c t of atomic diamagnetism.  However, i t can be shown (see Messiah 1962, page 541) that f o r the lower energy states of the atom: H' 1CT  5  ZB  (Tesla)  (2-5)  m where Z i s the atomic number of the atom.  Thus, f o r the modest f i e l d s of  less than 1 Tesla considered i n t h i s study, the term H^ i s n e g l i g i b l y small and can be excluded from the Hamiltonian H. The Hamiltonian equation i s solved using perturbation techniques.  The  eigenstates of H are given by | ajm > where j i s the quantum number f o r the q  t o t a l angular momentum J of the s t a t e , mh i s the eigenvalue of  (the  component of the t o t a l angular momentum operator along the z a x i s ) , and a l a b e l s a l l the remaining quantum numbers of the s t a t e .  The eigenstates are  tAt t h i s point only atoms with a nuclear spin 1 = 0 are considered.  - 6 -  ( 2 j + 1) - f o l d  degenerate i n the quantum number m,  i . e . s t a t e s from the  same l e v e l have the same energy.^ The degeneracy of the s t a t e s d i c t a t e s the use of a degenerate p e r t u r b a t i o n t e c h n i q u e , which reduces the problem of f i n d i n g the f i r s t change i n the energy A  of each s t a t e to the d i a g o n a l i z a t i o n of a  p e r t u r b a t i o n m a t r i x , (see Merzbacher, In all  order  the r e p r e s e n t a t i o n  1970,  page 425).  | ajm > the p e r t u r b a t i o n m a t r i x , which  the s t a t e s of a g i v e n l e v e l ,  includes  i s a l r e a d y d i a g o n a l and the e q u a t i o n f o r  A E . reduces t o :  AE . = <a jml jm  J  r  1  H m  la jm>  1  J  1  = ~ -  <ajm|(L + 2%)  • B" | ajm>  u = <ajm|g j 5 • % | ajm> = a g . Bm o aj a  (2-6)  r  where, i n the l a s t  s t e p , the z a x i s was  chosen a l o n g 1$.  Lande g f a c t o r , i s a c o n s t a n t f o r each l e v e l ,  y  § j » known as the a  i . e . i t i s independent of m.  tThere Is some a m b i g u i t y In the meanings of the terms ' l e v e l ' and 'energy l e v e l ' , (see Condon and S h o r t l e y pages 97 and 385). To a v o i d c o n f u s i o n the term 'energy l e v e l ' w i l l r e f e r e x c l u s i v e l y t o the energy a s s o c i a t e d w i t h a p a r t i c u l a r s t a t e , or when the s t a t e i s degenerate to the energy a s s o c i a t e d w i t h the group of degenerate s t a t e s . The term ' l e v e l ' w i l l r e f e r to the group of s t a t e s which d i f f e r o n l y i n t h e i r m quantum number, even when these s t a t e s are no l o n g e r degenerate.  -  7  -  I m p l i c i t i n t h i s development i s the assumption that the change i n energy due to  i s much less than the energy separation between the l e v e l s ;  or e q u i v a l e n t l y , the perturbation due to H i s much smaller than the m perturbation due to the s p i n - o r b i t  interaction.  Where t h i s assumption i s no  longer v a l i d the neighboring l e v e l s must also be included i n the perturbation matrix.  For most of the l e v e l s of heavier atoms t h i s i s  necessary only f o r magnetic f i e l d s larger than 1 Tesla, which are not considered i n t h i s study.  The assumption i s e a s i l y s a t i s f i e d f o r the 253.7  nm l i n e of mercury (see the appendix). The e f f e c t of the magnetic f i e l d on the states can be deduced from equation (2-6).  The degenerate l e v e l s p l i t s i n t o (2j+l) d i f f e r e n t energy  l e v e l s , each of which corresponds to a unique value of m.  These new energy  l e v e l s are symmetrically displaced about the o r i g i n a l energy l e v e l since m = ±j, ±j -1, ...  The magnitude of the s p l i t t i n g i s l i n e a r i n B.  There i s no  f i r s t order change i n the energy l e v e l of a state with j = 0 or m = 0. The e f f e c t of the magnetic f i e l d on the l i n e spectra of an atom can now be examined.  The absorption or emission of a photon by an atom i s  associated with the t r a n s i t i o n from one energy l e v e l to another.  The  angular frequency of a s p e c t r a l l i n e r e s u l t i n g from such a t r a n s i t i o n i n the absence of a magnetic f i e l d i s given by OJ , where:  OJ  o  =  -J-  h  1  | E°  aj  -  E°,  ,  a ' j'  1  |  (2-7)  - 8 -  and E°^ and ja'j'm^  are the energy l e v e l s of the unperturbed states  |ajm> and  respectively.  In the presence of a magnetic f i e l d the energy l e v e l s E , become: a jm E . = E° . + AE . = E° . + g . a Bm (2-8) ajm aj jm aj a j *o ^ 6  and  U J  the l i n e s p l i t s i n t o a s e r i e s of component l i n e s ^ with angular  frequencies u) ., where: mm l  OJ  mm  or mm'  U  =  W  0 -^ +  B  (8 fa  g 'j. ') m  a  (2-9)  This equation describes the observed s p l i t t i n g of the s p e c t r a l l i n e s by the magnetic f i e l d .  However, the a d d i t i o n a l property of the Zeeman e f f e c t ,  namely the p o l a r i z a t i o n of the component l i n e s , i s only i n d i r e c t l y r e l a t e d to the presence of the magnetic f i e l d .  I t can only be properly accounted  for by examining the r a d i a t i v e process which governs the t r a n s i t i o n between two states.  Furthermore, while equation (2-9) describes a l l possible  t r a n s i t i o n s between two l e v e l s , these t r a n s i t i o n s many occur with markedly different probabilities.  I t i s the r a d i a t i v e process that determines the  p r o b a b i l i t y of a p a r t i c u l a r t r a n s i t i o n or equivalently  the strength of the  associated component l i n e .  tThe component l i n e s are produced by the t r a n s i t i o n s between s p e c i f i c states of two given l e v e l s .  - 9 -  These observations i n d i c a t e that a b r i e f digression  to the theory of  r a d i a t i v e processes i s necessary for a complete understanding of the Zeeman Effect.  For the purposes of t h i s study the only r a d i a t i v e processes of  i n t e r e s t are those described by e l e c t r i c dipole t r a n s i t i o n s , since these produce the most intense l i n e s . The  probability  of an e l e c t r i c dipole t r a n s i t i o n between the  states  |ajm> and  l a ' j ' m ^ r e s u l t i n g i n the absorption or emission of a photon  polarized  i n the d i r e c t i o n of the u n i t vector ^ and  d i r e c t i o n £ i s of the form: (Sobel'man page  W£ «  |"R.<ajm  where P i s the e l e c t r i c dipole  t r a v e l l i n g i n the  305)  |"p| a ' j ' m ^  |  2  (2-10)  operator  Z "P = -e E "rr  i  and  the sum  (2-11)  1  i s taken over the p o s i t i o n ^.of  each electron with respect to  the nucleus. Most of the i n t e r e s t i n g properties of the t r a n s i t i o n can be deduced from the matrix element <ajm | 1? j a ' j ' m ^ . Wigner-Eckart Theorem (see Messiah page 573), element and hence W*  In p a r t i c u l a r , by e x p l o i t i n g  the  the dependence of the matrix  on m and m' can be found as  follows:  - 10 -  <ajm | $ | a'j«m'> =  C(j'lm'q|jm)  / ( 2 j +1) x <aj | | P | | a'j'> e*  (2-12)  where <aj || P || a'j'>^, the reduced matrix element, i s a s c a l a r independent of m and m*, C ( j * l m'q|jm) i s a Clebsch-Gordon c o e f f i c i e n t (these are tabulated i n Condon and Shortley page 76), q = m-m',  e^ i s the complex  conjugate of the u n i t vector e defined as f o l l o w s ^ : q e = - /2 1  (i + ij);  e = k; o  e =^ -1 /  (i - ij)  (2-13)  T  From the properties of the Clebsch-Gordon c o e f f i c i e n t s I t follows immediately that <<xjm | "P ^'j'm'> = 0, and by equation (2-10) j - j ' = 0, ± 1;  j + j ' > 1 and; q = 0, ± 1  = 0 unless: (2-14)  Thus, only a r e s t r i c t e d c l a s s of states i s connected through e l e c t r i c d i p o l e t r a n s i t i o n s and the s e l e c t i o n r u l e s contained i n equations equation  (2-14) govern  (2-9).  t<aj || P||a'j'> Is r e l a t e d to the q u a n t i t i e s <aj:P:a'j'> introduced i n Condon and Shortley. The r e l a t i o n s h i p s are tabulated i n Sobel'man (page 85). I f these r e l a t i o n s are s u b s t i t u t e d i n t o equation 2-12 i t Is e a s i l y shown that equation 2-12 i s equivalent to the equations given by Condon and Shortly at page 63. ttThe properties of these s p h e r i c a l basis vectors are presented 1957 page 103.  i n Rose,  - 11 -  Equation (2-12) a l s o accounts f o r the p o l a r i z a t i o n and r e l a t i v e i n t e n s i t i e s of the Zeeman Components.  This can be seen most r e a d i l y by  studying the s p e c i a l case of the emission or absorption of a photon t r a v e l l i n g along the y - a x i s , i . e . \ = j . Any d i r e c t i o n of p o l a r i z a t i o n 1t can now be resolved completely i n t o components along the x and z a x i s , and no further r e s t r i c t i o n s are imposed by independently examining  along each  of these axes. I n s e r t i n g equation (2-12) into equation (2-10), and l e t t i n g  t=kwe  find:  "2  k  W  jV  and W, = 0  1  when  k  C ( j ' l m ' q |jm)  2  | <aj  |1  P  I! a'j'>  |  when q = 0  2  (2-15)  q = ± 1  Now l e t t i n g I = i we f i n d :  = 0  when  q = 0  and W  i  "  I  2l+T  C(J  '  lm,q  Thus photons p o l a r i z e d  '  jm)2  I  <«J  II I U ' J ' > I p  2  when q = ± 1  i n the k d i r e c t i o n r e s u l t e x c l u s i v e l y  from  t r a n s i t i o n s where q = 0 and those polarized along the x-axis r e s u l t exclusively  from t r a n s i t i o n s where q = ± 1.  (2-16)  -  F i g . 1:  Transition  -  12  Relative Intensities l of the Component Line, tor Observations along the y-axis r e l  T  I  R E L  7  r  .  ,(m' = m)  T  \  R  E  L  /  .  , (m' = m ±  1)  -  For  2J+1''  transitions  M II  between two l e v e l s , a j •* a ' j ' the term  P!|a'j*> |  remains f i x e d .  2  p r o b a b i l i t i e s of t r a n s i t i o n s equivalently,  13 -  T h e r e f o r e the r e l a t i v e  between the s t a t e s  o f the two l e v e l s , o r  the r e l a t i v e i n t e n s i t i e s o f the component l i n e s I  o n l y on the square of the Clebsch-Gordon c o e f f i c i e n t s . intensities Fig.  case o f \ a l o n g the y - a x i s are t a b u l a t e d i n  o r i g i n of the observed p o l a r i z a t i o n  now be u n d e r s t o o d .  a magnetic f i e l d , populated. and  These r e l a t i v e  1. The  can  f o r the s p e c i a l  produced by the Zeeman E f f e c t  In the absence of an e x t e r n a l p e r t u r b a t i o n , such as  the s t a t e s  ]a'j'm'> a r e degenerate i n m' and e q u a l l y  Thus a l l the t r a n s i t i o n s  a l l the s p e c t r a l  spectal  between two l e v e l s have the same energy  l i n e components have the same f r e q u e n c y .  l i n e components has a c h a r a c t e r i s t i c  polarization,  Each of the  but when these  components are added t o g e t h e r t o produce the observed s p e c t r a l characteristic polarizations light.  a l o n g the x a x i s  f o r every s p e c t r a l  polarized  relative intensities I  rel  A.  unpolarized  l i n e the t o t a l I n t e n s i t y  a l o n g each a x i s  studied polarized  a l o n g the z - a x i s .  i s found by summing the  rel and I , g i v e n i n F i g . 1, over a l l allowed Li  T h i s i s the expected r e s u l t  there i s no p r e f e r r e d The  to g i v e i s o t r o p i c  i s e q u a l t o the t o t a l i n t e n s i t y p o l a r i z e d  total intensity  o f m.)  cancel exactly  l i n e the  T h i s r e s u l t i s e a s i l y demonstrated f o r the s p e c i a l case  above by n o t i n g that  (The  depend  direction  values  s i n c e I t accords w i t h the p r i n c i p l e  i n space f o r an i s o l a t e d  s i t u a t i o n changes when a magnetic f i e l d  that  atom.  i s applied.  l i n e s no l o n g e r have the same frequency, making i t p o s s i b l e  The component to observe each  - 14 -  component independently.  Observations transverse to the magnetic f i e l d are  equivalent to the s p e c i a l case examined e a r l i e r of photons t r a v e l l i n g along the y-axis, since the z-axis was chosen along % i n equation (2-6).  The  n-components can be immediately i d e n t i f i e d as the q = 0 t r a n s i t i o n s and the a-components as the q = ± 1 t r a n s i t i o n s . The  r e l a t i v e i n t e n s i t i e s of the components w i l l s t i l l be given by  F i g . 1 provided the weak magnetic f i e l d approximation remains v a l i d , implying that the states do not deviate s u b s t a n t i a l l y from the unperturbed states  | a jm>.  The frequency of each component l i n e i s given by equation  (2-9). These r e s u l t s provided a complete d e s c r i p t i on of the main properties of the Zeeman E f f e c t .  They are used i n a l a t e r s e c t i o n , a f t e r the problem of  l i n e broadening has been reviewed, to examine the f e a s i b i l i t y of p o l a r i z i n g s p e c t r a l l i n e s through the s e l e c t i v e absorption of the component l i n e s .  II.2 LINE BROADENING  The  r e s u l t s presented i n the previous section p e r t a i n to the i d e a l i z e d  case of a p e r f e c t l y i s o l a t e d atom at r e s t .  In t h i s i d e a l i z a t i o n the atom  can only absorb or emit photons with the discrete energy of a t r a n s i t i o n between two s t a t e s .  I n p r a c t i c e an atom i s moving randomly through a gas  and i s surrounded by perturbing p a r t i c l e s which a l t e r i t s s t a t e . deviations  These  from the i d e a l have the e f f e c t of allowing an atom to absorb or  - 15 -  emit photons over a range of frequencies f o r each allowed t r a n s i t i o n .  This  e f f e c t i s known as L i n e Broadening and the function which expresses the dependence of the t r a n s i t i o n p r o b a b i l i t y on the angular frequency, w, of the photon i s the Line P r o f i l e P(w)* In the case of emission l i n e s the broadening processes are i n t i m a t e l y related to the s p e c i f i c mechanism used to excite the atoms and therefore cannot be described without reference to the p a r t i c u l a r l i g h t source.  Even  when the e x c i t a t i o n mechanism i s w e l l understood the c a l c u l a t i o n of the r e s u l t i n g l i n e p r o f i l e i s generally  very complex.  In t h i s study our chief  i n t e r e s t i s i n the absorption of l i g h t by a single element gas through resonance t r a n s i t i o n , i . e . t r a n s i t i o n s connected to the ground s t a t e .  In  t h i s case the problem assumes an unusually tractable form i n which the l i n e broadening Is dominated by only three w e l l understood mechanisms.  In t h i s  section these three mechanisms, N a t u r a l , Resonance, and Doppler Broadening, are treated  separately and then with some q u a l i f i c a t i o n s the r e s u l t s are  combined to obtain the complete l i n e p r o f i l e .  Natural Broadening  An atom can never be i s o l a t e d from the ubiquitous photon field.  The f i e l d i n t e r a c t s with the atom allowing  between the atom and the f i e l d .  radiation  energy to be  transferred  This transfer of energy i s , i n turn,  t The Line P r o f i l e corresponds to the experimentally observed s p e c t r a l l i n e shape only In the case of l i n e s i n an o p t i c a l l y t h i n m a t e r i a l .  - 16 -  accompanied by a t r a n s i t i o n  from one atomic s t a t e t o a n o t h e r .  of these allowed t r a n s i t i o n s  between s t a t e s i m p l i e s t h a t the s t a t e s a j have  a f i n i t e average or n a t u r a l l i f e t i m e x uncertainty  .T  p r i n c i p l e AE  Applying  aj.  the time-energy  , ~ h we f i n d that a f i n i t e  indeterminancy i n each energy l e v e l of the order AE ,. field  leads  to a f i n i t e  l i f e t i m e and a f i n i t e  t r a n s i t i o n must occur a t a range o f e n e r g i e s d i s c r e t e energy.  The e x i s t e n c e  i m p l i e s an  Thus the photon  l i f e t i m e means that a rather  A complete a n a l y s i s (see H e i t l e r  1962, page 994) shows t h a t a f i n i t e  lifetime  than a t a s i n g l e 1954 page 181, or M e s s i a h  l i f e t i m e manifests i t s e l f  i n the form o f  a L o r e n t z L i n e P r o f i l e L(u)).  L(oj)du) =  2rt  (co- u  0  )2  where y i s the f u l l width a t h a l f maximum.  This  l i n e p r o f i l e occurs  whenever a p e r t u r b a t i o n has the e f f e c t  of a l t e r i n g  and  Broadening.  Is t h e r e f o r e not unique t o N a t u r a l For N a t u r a l  (2-17)  + (L)2  the l i f e t i m e o f a s t a t e  Broadening y = y^ and i s c a l l e d the n a t u r a l l i n e w i d t h  where  -1 aj  +x  -1  a'j  When an atom i s Immersed the  i n radiation  which i t i s c a p a b l e of  l i f e t i m e of the ground s t a t e i s not s t r i c t l y  excepting  the case of v e r y  (2-18)  , f o r the t r a n s i t i o n a j •*• a ' j '  intense  radiation,  infinite.  absorbing,  However,  the l i f e t i m e of the ground  - 17 -  state Is much greater than that of any excited s t a t e . absorption y.~ N  Thus f o r resonance  % \ where a j i s the excited s t a t e . a3  Natural Broadening u s u a l l y makes the l e a s t s i g n i f i c a n t c o n t r i b u t i o n to the f i n a l l i n e p r o f i l e .  Resonance Broadening  C o l l i s i o n s among the atoms of a gas also lead to l i n e broadening.  In a  gas composed p r i m a r i l y of atoms of one element, the broadening of the resonance l i n e s of t h i s element Is dominated by c o l l i s i o n s among the atoms of t h i s element.  Consider two atoms d i f f e r i n g only i n that one i s i n  the ground state ( a j ) , while the other i s i n an excited state ( a j ) which 0  Q  has an allowed t r a n s i t i o n to the ground s t a t e .  A c o l l i s i o n between these  two atoms permits a resonance t r a n s f e r of energy, i . e . the excited atom decays to the ground state while the ground state atom i s driven i n t o the state a j .  This resonance t r a n s f e r e f f e c t i v e l y creates an a d d i t i o n a l mode of  decay f o r the state a j and hence leads to a shortening that s t a t e .  This c o l l i s i o n induced shortening  of the l i f e t i m e of  of the l i f e t i m e w i l l  independently produce a Lorentz l i n e P r o f i l e characterized by a f u l l width at half-max. Yr, y  R  has been t h e o r e t i c a l l y c a l c u l a t e d using a v a r i e t y of techniques.  The most complete treatments appear to be by A l l and Griem, 1965 and D'Yakonov and P e r e l 1965. A l i and Griem derive t h e i r r e s u l t using the  - 18 -  Impact and c l a s s i c a l path approximations f o r a state a j which has an e l e c t r i c dipole t r a n s i t i o n to the ground state a j • Q  Q  They f i n d :  e f 2  H  =  3  '  8  4  *  2JQ + 1 2jTT~  N  „  m ^ T *-  <" > 2  e o  19  i n cgs u n i t s , or e y  R  = 3.84*  2j + 1  N  f  2  j  4n;em w 0 e o  v  (2-20)  n  '  i n MKS u n i t s : where: N i s the density of atoms i n the ground s t a t e , me i s the e l e c t r o n mass, e  Is the electron charge,  f  i s the absorption o s c i l l a t o r strength, and  E Q i s the P e r m i t t i v i t y constant  D'Yakonov and Perel determine y  using i r r e d u c i b l e tensors to c a l c u l a t e  the r e l a x a t i o n of the nondiagonal density matrix of an excited atom i n c o l l i s i o n with an atom I n the ground state.  They also use the c l a s s i c a l  path and impact approximations and assume the existence of an e l e c t r i c dipole t r a n s i t i o n to the ground state.  Their r e s u l t does not e x h i b i t the  simple dependence on j and j of equation (2-20) and a numerical r e s u l t i s provided only f o r the case where j = 1 and j = 0 .  However, i f f o r t h i s  - 19 -  s p e c i f i c case, t h e i r r e s u l t i s put i n the same form as equation (2-20) they f i n d that the numerical factor corresponding to 3.84 has a value of  2.55.  The impact approximation, upon which these r e s u l t s are based, remains v a l i d provided the time between e f f e c t i v e c o l l i s i o n s i s much larger than the duration of an e f f e c t i v e c o l l i s i o n Q  , and only frequencies w i t h i n a range  _1  Aw from the l i n e centre are considered.  These two v a l i d i t y c r i t e r i a are  contained i n the equation from A l l and Griem:  R  R  K T (Jj-)3/1  0-  and, K  Max { y , AOJ} « Q  where,  (2-21)  i s the Boltzmann constant, T i s the temperature i n K°, and M i s the  mass of the atom. Y  R  has been experimentally determined by G. Stanzel 1974, for the 253.7  nm resonance l i n e of Mercury; he found:  Y  for N ~ 5 x 1 0  18  cm  -3  R  = (5.3 ± 0.5) 10  and T ~ 553 K°.  -9  N sec  cm  -1  3  (2-22)  Using the c h a r a c t e r i s t i c s of the  253.7 nm l i n e given i n the appendix, the t h e o r e t i c a l values given by A l l and Grien and by D'Yakonov and Perel are  respectively:  Y- = 6.03 x 10  -9  N sec  -1  cm  (2-23)  Y_ = 4.01 x 10"  N sec  -1  cm  (2-24)  K  3  and R  9  3  - 20 -  Thus both r e s u l t s are In f a i r agreement with experiment given that the v a l i d i t y c r i t e r i a are only weakly s a t i s f i e d at these temperatures and d e n s i t i e s , i . e . from equation (2-21).  Y ~ R  3 x 10  10  The t h e o r e t i c a l equations f o r y  R  were not immersed i n a magnetic f i e l d .  «  8X10  10  ~ 0  (2-25)  were derived assuming that the atoms In the absence of a magnetic f i e l d a  c o l l i s i o n between two l i k e atoms permits not only a resonance t r a n s f e r of energy, but also an a l t e r a t i o n i n the exited s t a t e .  The r e s u l t i n g excited  atom can be i n a state with an m quantum number d i f f e r e n t from that of the excited atom before the c o l l i s i o n .  When a magnetic f i e l d i s applied the  states no longer have the same energy, therefore a change i n the m quantum number of the excited s t a t e must be accompanied by a change i n the k i n e t i c energy of the c o l l i d i n g atoms.  This has the e f f e c t of decreasing the  p r o b a b i l i t y of a change i n the r e s u l t i n g e x i t e d s t a t e . equations f o r y  The t h e o r e t i c a l  should remain v a l i d , however, provided the magnetic f i e l d  induced l i n e s p l i t t i n g (u ), from equation (2-9) i s small compared to mm o the inverse of the duration of an e f f e c t i v e c o l l i s i o n , i . e .  u  ,«n«  mm  ~ wo «  Q  (2-26)  (see D'Yakonov and Perel and Sobel'man page 464). For many l i n e s t h i s i n e q u a l i t y i s not s a t i s f i e d at even modest f i e l d strengths of less than 1 T e s l a .  At these f i e l d strengths the Zeeman  - 21 -  s p l i t t i n g of the l i n e becomes s u f f i c i e n t l y great to cause a s i g n i f i c a n t decrease i n the p r o b a b i l i t y of a c o l l i s i o n induced t r a n s i t i o n to a s t a t e with a d i f f e r e n t m quantum number.  This i n turn decreases the o v e r a l l  p r o b a b i l i t y of a resonance t r a n s f e r of e x c i t a t i o n energy since In e f f e c t the number of states able to p a r t i c i p a t e i n the transfer has decreased. the  resonance l i n e width y  Thus  w i l l be smaller than predicted by equation  R  (2-20) when the i n e q u a l i t y (2-26) i s not s a t i s f i e d .  Doppler Broadening  Doppler Broadening of s p e c t r a l l i n e s i s the cumulative r e s u l t of the random motion of the atoms.  Ignoring a l l other broadening mechanisms, an  atom w i l l only absorb or emit photons of frequency I O as measured i n the q  atom's frame of reference. The frequency of these photons i n the reference frame of an external observer i s u , where according to the n o n - r e l a t i v i s t i c V Doppler P r i n c i p l e w = O J + ~ q  U Q  »  an(  ^  v  I  s  t n e  component of the atom's  v e l o c i t y along the d i r e c t i o n of observation. I f the d i s t r i b u t i o n f u n c t i o n for V i s f ( V ) dV then the Doppler l i n e p r o f i l e , D(u)), produced by a u n i t volume of atoms i s : D(u>) du = f ( — (w-w o  For  )) —  o  dw  (2-27)  a gas In e q u i l i b r i u m f ( V ) dV i s just the Maxwellian v e l o c i t y  d i s t r i b u t i o n and D(w) dto becomes:  - 22 -  D(w)dw = u  _ 1 / 2  D  0\  e x p [ - ( - g — ) ] dco  -1  (2-28)  2  o where D = U>q — i s known as the Doppler Width, and V =  B (—yj~)  V  2 K  T  (2-29)  l / 2  The Combined Line P r o f i l e  The combined l i n e p r o f i l e P(io) dw i s a convolution of the characteri s t i c l i n e p r o f i l e s of each broadening process. are  I f two broadening processes  s t a t i s t i c a l l y independent and have p r o f i l e s Pj^ (w-to ) and P (o)-w ), the 0  2  0  combined l i n e p r o f i l e P'(U)-U)Q) i s given by:  p  where P', P^ and P  2  ' ^ > ° J°L i<*> 2 ^ - ^ ) ^ p  p  (2-30)  are expressed as functions of frequency d i f f e r e n c e s ,  W-U)Q , rather than absolute frequencies u.  Natural and Resonance Broadening are s t r i c t l y independent processes and the combined l i n e p r o f i l e L'(w) i s again a Lorentzian with y now given by:  Y =YN  +  YR  (2-31)  - 23 -  Doppler and Resonance  Broadening are not s t a t i s t i c a l l y Independent,  because a s i n g l e c o l l i s i o n can produce both a resonance t r a n s f e r of energy and a change  in velocity.  A complete a n a l y s i s i n v o l v i n g  the  c a l c u l a t i o n of a c o r r e l a t i o n f u n c t i o n (Sobel'man, page 401) shows t h a t the c o r r e l a t i o n between these two p r o c e s s e s can l e a d to a c o l l i s i o n a l narrowing of  the Doppler width of the combined  line profile.  L «  and L =  the P(w)  However, where:  —  (2-32)  — - — i s the mean f r e e path and a i s the g a s - k i n e t i c c r o s s /2a N ° o  two p r o c e s s e s can be t r e a t e d i s therefore given  i n d e p e n d e n t l y . The complete l i n e  section,  profile  by:  P(u) =  L'(0)  D ( u , - 0 ) d0  (2-33)  or •(f) P ( U  )  -  T h i s type of p r o f i l e In  I_ -l/2 2% n  D-1 J " ~  Is known as a V o i g t  2  Sl__ dx (a)-u) -x)2 + (L)2  expect a u n i t  2  _  3  4  Profile. development  volume of atoms i n the gas t o absorb photons  over a range of f r e q u e n c i e s w i t h a p r o b a b i l i t y g i v e n by the V o i g t P r o f i l e equation (2-34).  )  0  c o n c l u s i o n , where a l l the assumptions made i n t h i s  remain v a l i d , we  (  In the presence of a magnetic f i e l d  the l i n e w i l l s p l i t  of  - 24  -  and each Zeeman Component w i l l have t h i s same Voigt P r o f i l e , apart from a s l i g h t narrowing due to the e f f e c t of the magnetic f i e l d on the resonance broadening of each m-component.  II.3 THEORY  In t h i s s e c t i o n a t h e o r e t i c a l model i s developed f o r the transmission of l i g h t through a gas immersed In a magnetic f i e l d where the frequency of the l i g h t i s at or near a resonant frequency of the atoms of the gas.  The  theory i s based on a s e m i - c l a s s i c a l approach s i m i l a r to one used i n a r e l a t e d context by Corney et a l . , 1965 and Dodd and Series 1961.  While more  s o p h i s t i c a t e d and rigorous methods i n v o l v i n g quantized f i e l d s are p o s s i b l e , l i t t l e a d d i t i o n a l information of importance to t h i s study i s gained at the cost of much greater complexity. Therefore, while acknowledging the merits of r i g o r , I have opted f o r a simpler and I hope more r e a d i l y comprehensible e x p o s i t i o n of the problem. Conversely most of the r e s u l t s obtained through t h i s development  can be  derived more simply by making perspicacious s u b s t i t u t i o n s i n t o the more familar equations d e s c r i b i n g the transmission of l i g h t through a gas.  It  was f e l t however that only through the step by step development of a theory, i n which a l l the assumptions and approximations are given e x p l i c i t l y , can the a p p l i c a b i l i t y of the r e s u l t be properly determined. systematic development  i s presented.  Thus a complete and  - 25 -  In t h i s development the l i g h t Is treated c l a s s i c a l l y as an electromagnetic (em) wave governed by Maxwell's equations, while the atoms are treated quantum mechanically.  The e x t e r n a l ( s t a t i c ) magnetic f i e l d i s  allowed to assume any o r i e n t a t i o n with respect to the d i r e c t i o n of propagation of the l i g h t .  While t h i s provides a more general equation  than i s necessary f o r t h i s study, and involves some a d d i t i o n a l complexity, the more general equation may be u s e f u l i n future research. For s i m p l i c i t y the theory i s developed using c.g.s. Gaussian u n i t s ; however, the important r e s u l t i n g equations are a l s o expressed i n MKSA The e l e c t r i c f i e l d , 1j(r,t), of a c l a s s i c a l em wave t r a v e l l i n g  units.t  through a non-magnetic, c u r r e n t - f r e e medium obeys the equation:  V ^(r,t) -V(v"l(r,t)) = — — c at 2 2  CE(r,t) + 4n$ )  where  (2-35)  c  2  i s the d i p o l e moment per unit volume.  In the equation a l l higher e l e c t r i c moments have been neglected.  This  corresponds to the e a r l i e r approximation of considering only e l e c t r i c dipole transitions. To solve equation (2-35) f o r "E("r,t), "P a magnetic f i e l d .  must be determined  for a gas i n  This i s accomplished by f i r s t f i n d i n g the average d i p o l e  a moment P  for a s i n g l e atom immersed i n a magnetic f i e l d and i n t e r a c t i n g  t J.D. Jackson's " C l a s s i c a l Electrodynamics" provides a t a b l e f o r converting any equation from one system to the other.  - 26 -  with an em wave.  The average dipole moment  corresponds to the  expectation value of the quantum mechanical e l e c t r i c dipole operator  (see  equation (2-11)). Thus,  V-<t|"*|t>  where  (2-36)  |t> Is the state of the atom at time t . The problem has therefore been reduced to s o l v i n g the time dependent  Schrodinger equation:  l h  |_  | t>  |> - H t  (2-37)  The Hamiltonian, H f o r t h i s atom i s given by:  uH  =  H  o  +  H  D  +  TT  8  aj ^ ' *  +  H  I  (  t  )  ( 2 _ 3 8 )  where: H  q  i s the Hamiltonian f o r the unperturbed atom, and H | ajm> = q  E .|ajm>. "o + + ^— g^.. J • B was introduced i n equation (2-6) and accounts f o r the e f f e c t s of a weak magnetic f i e l d .  With complete g e n e r a l i t y the magnetic  f i e l d can be confined to the x-z plane with the angle between % and the z axis denoted by p. (see F i g . 2).  R e c a l l from equation (2-6) i f p=0,  F i g . 2: O r i e n t a t i o n of the Magnetic F i e l d  - 28 -  The operator  i s introduced phenomenologically  to provide for the  exponential decay of the e x c i t e d states e ' , where 1/y i s the l i f e t i m e of a given s t a t e .  The f i n i t e l i f e t i m e can account f o r a l l s p e c t r a l l i n e  broadening mechanisms which produce a Lorentz p r o f i l e .  H  D  | ajm> = -(1/2) i h  T  (  (  .  Note that:  | ajm>  (2-39)  H^.(t) describes the i n t e r a c t i o n between the atom and the electromagnetic wave.  I f the wavelength of the em-wave i s much greater than  the s i z e of the atom (which i s true f o r o p t i c a l frequencies) then H ( t ) can be given by:  lyt)  = 4  • "E(r,t)  (2-40)  which i s analogous to the c l a s s i c a l expression f o r the energy of a d i p o l e i n an e l e c t r i c f i e l d . The i n t e r a c t i o n between the atom and the em-wave's magnetic f i e l d i s much weaker and has been neglected. To solve the Schrodinger equation the techniques of time-dependent perturbation theory are employed.  The state |t> Is expressed as a  superposition of states which are independent of time.  - 29 -  |t> = C ( t ) g  I g> + I C ( t ) |d> d  (2-41)  d  where: |g> represents |g> =  the ground state which has been made non-degenerate: i . e .  \t00>. While t h i s r e s t r i c t i o n i s not e s s e n t i a l i t greatly s i m p l i f i e s  the d e r i v a t i o n of  |t>.  The solutions for degenerate ground states are  Introduced l a t e r In an obvious, a l b e i t ad hoc, way for the s p e c i a l case of l i g h t t r a v e l l i n g transverse to the magnetic f i e l d . jd> i s simply a shorter notation for the excited states "d" represents  j ajm> where  a l l the quantum numbers of the s t a t e .  C ( t ) and C ( t ) are the time dependent p r o b a b i l i t y amplitudes for the g  states  d  |g> and  |d> r e s p e c t i v e l y .  C ( t ) = <g |t>  Since a l l the states are orthonormal:  and  C ( t ) = <d |t>  (2-42)  The p r o b a b i l i t y of an atom being i n a state other than the ground state i s very small, implying |C | ~ 1.  This indicates that the state of the atom  can be w e l l described by just the f i r s t order terms i n the  perturbation  expansion. Defining the time-Independent operator  H = H - Rj.U)  i t can be shown (see Messiah page 722) that to f i r s t  (2-43)  order:  - 30 -  "|H(t-x) |t> = e  "^H(t-t )  t  lg> +  J L _  "il(t -x)  0  /  ih~  d t o  e  0  H (t ) e  T  |g>  ( 2 - 4 4 ) ,  °  where the o s c i l l a t i n g e l e c t r i c f i e l d producing H ( t ) was switched on at time t = t at which time the atom was i n the ground s t a t e .  The terms of the form e - i - H | g> can be evaluated immediately since H  q  ]g> = E |g> where E q  q  i s the energy of the ground s t a t e .  ]g> = 0, since  H  there i s no broadening of the ground s t a t e , and |g> i s non-degenerate.  Thus:  -(E e  and equation  ( 2 - 4 4 )  lt> = lg>e  |g>  =  |g>  f l  (t-r) ( 2 - 4 5 )  e  becomes  - ^ E (t-x) o  g^^ 5 • $ ]g> = 0 since  1 ih  / J  t  dt e x o  lict-to)  h  E  o  (  t  0 - ^ ( 2 - 4 6 )  This equation f o r |t> although not d i r e c t l y solvable can be used with  - 31 -  equation (2-41) to obtain C. ( t ) and C ( t ) . a  To f i r s t order  g  -iE (t-0 Q  C ( t ) = <d |t> ~ <d |g> e d  1 t  ih ^  +  d t  4« "V ( t  o  I  < d  H t )lg>e  e  l (  "T  E  (t 0  "  T)  (2-47)  o  The f i r s t term i s zero since the states are orthonormal. The second term can be solved i f R |d> can be found. states the  Toward t h i s end we consider the  |u>'^ defined i n a new co-ordinate.system R'. R' i s chosen such that  z'-axis i s p a r a l l e l to the magnetic f i e l d 1i. Thus R' i s generated by a  r o t a t i o n about the y-axis through an angle {3 with respect to the o l d co-ordinate system; see F i g . 3.  In the new co-ordinate system the operator  H becomes: 5  =  H  o  +  D  H  +  ^ S  a  j  B J  z  ,  (2-48)  Therefore H |u>' - (E .- 1/2 i h y , + p, g .Bu | u>' = \ |u>' ^ aj ' a j o a3 u r  1  1  v  (2-49) '  <d| i s r e l a t e d to '<u| through the equation <d| - < ajml - Y ^u 1  J  1  D (0,p,0)'<aju I = Y D mu ' u mu j  K >  J  1  L  3 1  < ul'  v  (2-50)  where D^ (0,B,0) are elements of the r o t a t i o n matrix, (see Rose, 1957). u  fTo s i m p l i f y the notation the state |aju>' i s designated |u>* with the remaining quantum numbers implied.  F i g . 3:  O r i e n t a t i o n of |u>' Co-ordinate System  - 33 -  Inserting  C  d  ih  ( t ) =  these r e s u l t s into equation (2-47) gives:  d t  0 I mu u D  '  6  < u  I I< 0> b> e H  (2-51)  t  Now tranforming back to the o r i g i n a l co-ordinate system using the equation:  '<u|-I n  ^*(0,p,0) <ajn | - Jnu D^* nu n 1  u  <n|  (2-52)  we have . C (t, d  - ^  _  -~\ ( t - t ) dt I o  un  0^  . "  «  4  E (t-x)  • <n ! ( , , fe> . " B l  0  •  (2-53,  To evaluate the remaining matrix element <n | H^(t ) |g>, the time dependence of E ( r , t ) must be given e x p l i c i t y . sinusoidal  Let ^ ( r . t ) be given by the  wave;  Ifr.t) - V r )  +  ^  e  (2-54)  i w t  Where ^ ( r ) may be complex, but as defined E ( r , t ) i s r e a l as required f o r H ( t ) to be Hermitian.  With t h i s d e f i n i t i o n  <n | H ( t ) |g> =  (2-55)  T  <n| - "p • V r ) |g> e'^ + <n | -1.1*(t)|e  lwt  - - <n 1 J? lg> - 1l(r) e '  l w t  - <n|"P|g> . l * ( r ) e  i a ) t  - 34 -  The l a s t step i s permitted because ^ ( r ) a f f e c t s the states only through the The matrix element <n| 1»|g>  operator  i s just the familar e l e c t r i c dipole  t r a n s i t i o n matrix, and equation (2-12) i s applied at a l a t e r stage to give the e x p l i c i t dependence on n.  Invoking the e l e c t r i c dipole s e l e c t i o n rules  the element <n | "I |g> = 0 except f o r states  |n> with j = l since  |g> i s non-  Since <n 1> |g> • t l ( r ) i s independent of time the equation f o r  degenerate. C,(t) becomes a  C,(t)-£E d ih u  un  D mu^ V nu" *  e *  1  0 + E (r) e  J (.E(r) e T  e ^  V  < n | n >  V  1  (X - E ) t  B  n  ^>J e  dt  (2-56)  Q  The i n t e g r a l can be e a s i l y solved, with the r e s u l t  ¥  (X -E -h )t U  o  W  h t W ^ o  o  B(r) [ f  ]*  +  5- ( x u - E -  *Yr)  i  to)  Q  F<V o K  + h M  ]'  (2-57)  >  \ - F.  R e c a l l from equation ( 2 - 4 9 )  U  , -° = T- (E .- E + a g _,Bu - ( 1 / 2 ) i hv h  n  aj  o  o aj  The cases of i n t e r e s t occur when the frequency  aj  of the l i g h t i s at or near a  - E , . In t h i s case the second h term i n equation (2-57) i s n e g l i g i b l e compared to the f i r s t term and the resonant frequency of the atom, i . e . OJ ~  E  .).  a  Q  - 35 -  expression f o r C ( t ) i s given f i n a l l y by: d  d  Ih  u n  mu  nu  ^  J  6  <n p |g> .  e  Equations (2-41) and (2-44) can also be used to d e t u n e C ( t ) . f i r s t order  To  8  c g  ( t ) = <g fe> e  % h  E  °  (t-r )  +  (2-59)  - 36 -  Using arguments s i m i l a r to those used f o r C^Ct), i t i s apparent that the second term i s zero since <g|^|g> = 0. Thus,  4  E  n<  t - t  >  C (t) = e  (2-60)  n  The expressions f o r C ( t ) and C,(t) can now be i n s e r t e d i n t o equation g d (2-41) y i e l d i n g  |t> to f i r s t order. With  |t> known, <t f |t> can be found*.  Expanding <t $ |t> and r e t a i n i n g only f i r s t order terms: <t|p |t>=< |p ]g> c * ( t ) c ( t ) + £ < g rp |d> C * ( t ) C ( t ) g  g  g  g  + ^<d|1?|g>C (t) C * ( t ) g  d  (2-61)  d  The f i r s t term i s zero since <gfp|g> = 0 The t h i r d term i s also just the complex conjugate of the second term; hence <t|p|t> can be w r i t t e n as:  <t|P>> = 2 Re  < g|1?|d> C* ( t ) C ( t ) ) d  (2-62)  where Re(Z) stands f o r the r e a l part of Z.  tNote that <t|t> * 1, since the states |g> and |d> were made orthonormal; i . e . <d|d>=l. Thus, the proper expression i s ^ j ^ ^ . However, since Z C,(t) C, « 1 t h i s small numerical c o r r e c t i o n can be ignored. , d d  - 37 -  I n s e r t i n g the derived expressions f o r C ( t ) (equation 2-60) and C,(t) 8 d (equation 2-58) the expression i n brackets i n equation (2-62) becomes:  Z <g|$|d> C*(t) C. ( t ) = Z <g|P|d> D D* <n|?|g>.E(r'5 . d , mu nu d dun (1)  1  -icot *("  £ V 6  1 1  V  (  (1)  1  ^  ( X -E -ho) u o  -icox '  )  <~ > 2  63  The steady s t a t e s o l u t i o n , which i s the one of i n t e r e s t , occurs when the e l e c t r i c f i e l d has been I n t e r a c t i n g with the atom long enough f o r a l l transient e f f e c t s to have vanished.  In the above equation t h i s i s  equivalent to taking the l i m i t of x •*• - ».  Examining the f u n c t i o n which i s  dependent on x, i t i s c l e a r that i t contains an o s c i l l a t o r y component and 1  the component e  /  2  W J  l  /  z  . Since l i m e  \ ? J  = 0 i t follows that the  expression i n brackets i n equation (2-63) becomes: -iwt ((X  u  ! E -ho)) o  <- > 2  64  when x •*• - ». The summation over d i n equation (2-63) includes a l l e x c i t e d s t a t e s . However, i f the frequency of the l i g h t i s near a p a r t i c u l a r resonance l i n e , as we have assumed, the c o n t r i b u t i o n from the states of neighbouring l e v e l s w i l l be minimal and the sum can be taken only over the states of the  - 38 -  relevant l e v e l .  (Where the l e v e l s are c l o s e l y spaced the sum should Include  the states of the neighbouring l e v e l s . )  With these modifications equation  (2-63) becomes.  E  «l* ^>  mun  ilKi ' 1  <«l*  (or^W) u  <2-65,  o  The matrix elements <g]~P ^n>, and <n|~P |g> can now be reduced using equation (2-12):  <g]P \a> = C(llmq|00) <aj||P|| a'j'> e*  (2-66)  Since |g> i s non-degenerate, q = -m; i . e . each unit vector corresponds to a s i n g l e allowed t r a n s i t i o n .  From tables (see Condon and Shortley, page 76)  the Clebsch-Gordon c o e f f i c i e n t s are found, to be:  C(llmq|00) = -  (2-67)  •3  Thus <g fy>  = -  <«j||p||  /3  tt  'j'>  e*  "™  *  (2-68)  u  From the property of the u n i t basis v e c t o r s , e = ( - l ) e t h i s becomes: u -u. p  <|-p g  ^> = l L _ < a j | | p | | a'j'> e  /3  ra  m  (2-69)  -  39 -  similarly <n $ |g> =  ^L- <aj||p|| a \ 1 ' > * e* /3  (2-70)  n  Using  these  equations,  (2-65)  becomes:  mun  u  0  As an a i d to f u r t h e r development, <t |~P |t> i s expressed complex p o l a r i z a b i l i t y  t e n s o r per atom, o f which i s d e f i n e d by the e q u a t i o n :  <t $ |t> - 2 R e ( a  It i s evident conveniently  from e q u a t i o n s  defined with  a  a mn  i n terms of the  • "E(r) e "  l u 3 t  )  (2-72)  (2-62) and (2-71) that a  respect  u  a  3  Is most  t o the u n i t v e c t o r b a s i s e . q  \<ai\ H 1 « ' J ' > 1 2 3(\ - E -hu>) u o  D  (D mu  p  Thus  (D* nu  >  K  and, •+  <t $ |t> • e = P • e = 2Re(E a E (r) e" ' m c m n m n n a  3  ;  i u ) t ;  )  v  (2-74) '  The d i p o l e moment per u n i t volume "P , which Is r e q u i r e d i n e q u a t i on (2-35),  can be found by summing ^  a  c  over the atoms i n a u n i t volume.  It: is " ~ s  - 40  convenient  -  to d e f i n e a complex p o l a r i z a b i l l t y  t e n s o r per u n i t volume a  by  the e q u a t i o n = 2Re(a • l ! ( ^ ) e "  \  I t f o l l o w s from the d e f i n i t i o n of "p  l u t  )  (2-75)  t h a t a_ i s g i v e n by the sum  of a  3  over a  u n i t volume of atoms, o r : a  At t h i s p o i n t we  =  mn  X a * 1=1 mn  v  n e g l e c t the s m a l l number of atoms i n the e x c i t e d  which i s e q u i v a l e n t t o i g n o r i n g induced e m i s s i o n s , and atoms i n the ground s t a t e . averaging,  (see J a c k s o n 1975,  page 226)  state,  take the sum  By p e r f o r m i n g the u s u a l macroscopic the d i s c r e t e sum  (2-76) '  over  spatial  can be r e p l a c e d by  an i n t e g r a l over the v e l o c i t y d i s t r i b u t i o n of a u n i t volume of atoms. Assuming a Maxwellian  v e l o c i t y d i s t r i b u t i o n and a c c o u n t i n g f o r the  floppier  s h i f t of each atom's n a t u r a l frequency as seen by an e x t e r n a l o b s e r v e r , ( i . e . a c c o u n t i n g f o r Doppler Broadening)  1  mn  3h  I 1  the I n t e g r a l i s g i v e n by:  ^4lli»ll-.4.xl2 S <• D 'D <aj||P||a'j'>| mu nu „ 2  J 1  1  11  J  N  Vi  1  wo  u  V%  £ -*  <o  Q  + -co  0  d  + ^g^Bu  "co  - (l/2)i  l,z  n  y  a j  v  (2-77)  - 41 -  , V ,1/2 = ( - ± - ) 2  recall V  Q  N = d e n s i t y f o the atoms i n the ground s t a t e M = mass of the atom V = component w  of the v e l o c i t y along  the d i r e c t i o n of  observation  o h < „r E ) E  m  n  u  aj  o  T h i s e q u a t i o n may  be s i m p l i f i e d  by i n t r o d u c i n g the complex e r r o r  f u n c t i o n W(a + i b ) (see Abramowitz and Stegun 1972, page  W(a  whence a  mn  a  +  ib) = I  (  C f. ih  a  295)  (2-78)  t  becomes  lNcu 1/2 mn  3h V  cjg  Q  |<aj||p||a'j'>!  1  J  2  2  1  D^Va nu  mu  u  + ib)  (2-79)  ^0 W  h  e  r  e  a  u  =  ( W  " "0 ' —  g  a j  B  U  )  (  2  "  8  0  )  - 42 -  V o) 0  It Is conventional  to introduce  the diraensionless q u a n t i t y f  3  o s c i l l a t o r s t r e n g t h f o r the t r a n s i t i o n a j relation  (2-81)  'aj  0  the a b s o r p t i o n  a ' j ' , which i s d e f i n e d by the  (Sobel'man page 302):  <jjQ  2m  = _ . 3he  1  |<aj1|Pl|a'j'>l  e  f  With t h i s d e f i n i t i o n a  iNcf =  a  mn  mn  e n 2  2 V  0  1 / 2  £ 0  m  w e  o  (2-82)  becomes, ( r e c a l l i=0):  § „„  2  2j+l  2  2  D mu  ( 1 )  D nu  ( 1 )  W(a u  + ib)  (2-83)  u  To f a c i l i t a t e f u t u r e development some elements of a a r e g i v e n explicitly.  From Rose, 1957 the m a t r i x :  4 (1 + cosp) Z  t> (0,p,0) = mu (1)  - sinp /2  2 (l-cosp)  — sinp /2  -^1-cosp)  cosp  sinp •2  — /2  sinB  j (1 + cosp)  (2-84)  - 43 -  Therefore:  a  ll  =  i K  0  i j ( l + c o s p ) W ( + i b ) + l / 2 s i n p W(a +ib)+ 2  2  a i  Q  + j (l-cosp^WCa^+ib)]  = IKQ [I  oc,^  (l-cos6) W( +ib) 2  a i  + | (1+cosp)  a  -ll  =  a  l-l =  i K  s i n 2  0  P  2  +  i  (2-85)  s i n 8 W(a +ib) 2  Q  W(a_ +ib)]  (2-86)  1  --i-W(a +ib)  [jW^+lb)  0  + | WU^+ib)] Ncf where:  KQ =  e u 2  2  v  0  m  (2-87)  1 / 2  (2-88) e  w  0  2  or Ncf  e * ' 2  1  Kg =  i n MKSA u n i t s 8 l t e  For most l i n e s  2  0  V  0  m  e  w  0  i n a gas of modest d e n s i t y l  the i n f o r m a t i o n i n the appendix: |a| ~ 10" mercury at the vapour p r e s s u r e E q u a t i o n (2-75) f o r P  c  (2-89)  2  5  a m n  l«l«  F o r example,  using  f o r the 253.7 nm. l i n e o f  o f mercury a t 20°C.  and e q u a t i o n (2-83) f o r a t o g e t h e r  c o n t a i n the  - 44 -  r e l e v a n t quantum m e c h a n i c a l r e s u l t s I n a form s u i t a b l e f o r f u r t h e r  analysis  u s i n g the c l a s s i c a l e l e c t r o m a g n e t i c e q u a t i o n (2-35) g i v e n e a r l i e r .  Insert-  ing  these q u a n t i t i e s i n t o e q u a t i o n (2-35) g i v e s :  V E(r,t)-v(v."E(r,t))  = — —  2  b  c  (ttf.t) + 4*(a • 1 : ( - r ) e c  2  lwt  far,t)+4Ttt>  )  at2  + a* • £ ' & ) e  i w t  ))  (2-90)  2  where E, , = ~E(r)e local  iu3t  +"E * ( r ) e  l w t  i s the t o t a l e l e c t r i c f i e l d  a t each  atom. Equation  (2-90) has the same form as the e q u a t i o n which d e s c r i b e s the  p r o p a g a t i o n of an em wave through an a n i s o t r o p i c c r y s t a l 1975, page 665).  I n an a n i s o t r o p i c c r y s t a l the e l e c t r i c f i e l d need not be  p e r p e n d i c u l a r t o the phase v e l o c i t y ; the e l e c t r i c  field  (see Born and Wolf  i . e . there may be a component o f  i n the d i r e c t i o n o f the phase v e l o c i t y .  If this  occurs  the energy o f the wave does not propagate i n the d i r e c t i o n of the phase velocity. The component  of the e l e c t r i c  field  along the d i r e c t i o n o f the phase  v e l o c i t y a r i s e s from the term v(V«^(r,t)) which i s zero f o r i s o t r o p i c materials.  From Maxwell's e q u a t i o n s :  V*"! = -4itV • "p c hence  (2-91)  - 45 -  V(V  • 1)  = -4nV  [V .(a • " E ' ( r ) e "  l a ) t  + a * , t'*(r)e )]  (2-92)  ia)t  Using t h i s e q u a t i o n i t can be shown t h a t where U ^ l ^ l and  the componet  component  the t e r m V ( V « ^ )  of E along the phase v e l o c i t y i s s m a l l compared  p e r p e n d i c u l a r to the phase v e l o c i t y .  to the  Thus, i n o r d e r to s i m p l i f y  g r e a t l y the s o l u t i o n t o e q u a t i o n (2-90), the term V(V»£) may  be o m i t t e d , and  the s m a l l d i f f e r e n c e between the d i r e c t i o n of the phase v e l o c i t y and energy flow may be i g n o r e d . In  the e q u a t i o n f o r ^  must be the t o t a l f i e l d the f i e l d  field  i s extremely  the e l e c t r i c f i e l d  a t the atom,  of the i n c i d e n t  n e i g h b o u r i n g atoms.  c  E l  o  c  a  l  '  i n t e r a c t i n g w i t h the atoms  This f i e l d  i s produced by both  em wave " E ( r , t ) , and the induced  The r e l a t i o n between the i n c i d e n t  field  field  and the induced  complex f o r h i g h f r e q u e n c i e s i n a n i s o t r o p i c m a t e r i a l s .  However where let I « 1 mn  the induced  field  i s much s m a l l e r than the i n c i d e n t  and the a p p r o x i m a t i o n 1i(r,t) = ^ , . ( r , t ) can be made. local approximations e q u a t i o n (2-90) becomes: field  V2(E  Or)  of the  e-  i w t  + iV)  e ^) 1  = -1 [ (e . l(r) c 5t2 ~  e"  With these  lut  2  + e*. lYr)  e ^)] 1  (2-93)  - 46 -  where  or  £ _ = ] _ + 4na_  7^ E ( r ) e  — c  + V  2  SYr) e  2  -  i w t  (2-94)  [e • E ( r ) e  at2  ( \ l Y r )e e  c  at  2  l*t(r)  ) = 0  (2-95)  and the d i f f e r e n t i a l  t h i s e q u a t i o n can be t r u e o n l y i f :  e"  - L_a_L_  l w t  c  and  i w t  ~  2  Since "E(r) and " l * ( r ) a r e l i n e a r l y independent operators are l i n e a r ,  J  ~  at  2  . V r ) S **) = 0 1  (  l i k e w i s e f o r the complex conjugate  term.  Without l o s s of g e n e r a l i t y e q u a t i o n  (2-96) can be s o l v e d f o r the case  of a plane wave p r o p a g a t i n g a l o n g the z - a x i s .  t -E e  (2-96)  2  The s o l u t i o n i s of the form  e **™^ 1  1  (  K  -  z  1  w  t  )  e,  where E E  _  and E  x  -+  + i  and E a r e the amplitudes y  respectively.  (  X  + IE _ ) /2 y  o f the wave a l o n g the x and y - a x i s  (2-98)  - 47 -  Substituting  t h i s s o l u t i o n i n t o e q u a t i o n (2-96) y i e l d s the two  homogeneous e q u a t i o n s :  e — u  o K E  2  - ( e  2  1  u  Ej + e _ 1  1  E_ ) = 0  (2-99)  x  2  and c — u  Solving  o K E  2  n  u  ^  E_ ) = 0  ( e _  2  K  a)  2  2  —  n  + e  u  ) ± [(en  " e . ^ )  2  2  gives:  + 4 2- - ]W2 e  l  1  -  (2-101) 2  where the f a c t t h a t  i  i  _  Substituting  (2-100)  L  these two simultaneous e q u a t i o n s f o r K  c —  E  - (e_  2  2  e-^ - i - i e  has been used.  t h i s e x p r e s s i o n back i n t o e q u a t i o n s (2-99) g i v e s  T< n - -i-i> ± [7; < n - -i-i> e  e  e  e  2 +  E  i-il  1 / 2  ;  where the s i g n s + and - a r e a s s o c i a t e d  ( " 2  w i t h the s o l u t i o n s K ^ and K 2  2  1 0 2  )  (see eq.  2-101) r e s p e c t i v e l y . The form o f t h e s o l u t i o n i s g r e a t l y s i m p l i f i e d by d e f i n i n g function  suggested by Corney, K i b b l e  and S e r i e s , 1965:  the complex  - 48 -  2e  n i  tan 9 =  (2-103) e  E whence  -  l l  - l  e  -1  cos 6/2  1  - — = "ln" 9"fl  f  o  r  t  h  e  s  o  l  u  t  i  o  n  s  K  +  E and  —  =  ^^Q^  f o r the solutloTl K  2  (2-104)  l  (2-105)  The g e n e r a l s o l u t i o n f o r the d i f f e r e n t i a l e q u a t i o n ( 2 - 9 6 ) i s thus  E = a+ (cos 9/2  + a_ ( - s i n 9/2 ^  + s i n 9/2 e ^ )  + cos 9/2 e _ j ) e  i ( K e  +  z _ a ) t )  )  1  (2-106)  The f a c t o r s a+ and a_ a r e determined by the boundary c o n d i t i o n s . and K_ a r e the r o o t s of e q u a t i o n decaying s o l u t i o n .  K+  ( 2 - 1 0 1 ) which g i v e an e x p o n e n t i a l l y  These e x p r e s s i o n s a r e examined i n d e t a i l i n a l a t e r  section. F o r the purpose o f t h i s study we can s e l e c t an i n c i d e n t e-m wave p r o p a g a t i n g along the z a x i s and g i v e n by the e q u a t i o n :  . i ( K z-wt) I = (E (to) i + E (to) j ) e x y 1  (2-107)  where, E ( w ) = A (to) e x  x  X  , and  (2-108)  - 49 -  A (u)) i s the amplitude,  (a r e a l v a l u e d f u n c t i o n o f the frequency u) and <b x  x  is  the phase of the x component.  E^(w) i s d e f i n e d a n a l o g o u s l y .  We choose the boundary of the gas a t z = 0 normal t o the z = a x i s , t h e r e f o r e the e l e c t r i c no r e f r a c t i o n .  field  Equation  i s continuous  (2-107) can now be equated t o e q u a t i o n (2-106) a t z  = 0 and s o l v e d f o r a+ and a_.  —  •T  (E  x  Thus:  + IE ) e_, + —  = ( a cosG/2 - a_ s i n 9/2) e +  a c r o s s the boundary and there i s  y  L  1  V2  (-E + IE )e, X  y  1  + ( a s i n 9/2 + a_ cos 9/2) e_ +  x  (2-109)  w i t h the s o l u t i o n s  E  a+ =  IE ( s i n 0/2 - cos 0/2) + — £ (cos 0/2 + s i n 0/2) /2 /2"  E  a_ =  IE ( c o s 0/2 + s i n 0/2) + — ^ (cos 0/2 - s i n 9/2) •2 /2  (2-110)  The r e f l e c t e d wave has been omitted because we a r e p r i m a r i l y in  (2-111)  interested  the change I n the t r a n s m i t t e d wave w i t h the a p p l i c a t i o n o f the magnetic  field.  It i s clear  t h a t the change I n the r e f l e c t i v i t y of the gas w i t h the  a p p l i c a t i o n of the f i e l d t r a n s m i t t e d wave.  i s s m a l l , hence there i s a n e g l e g i b l e e f f e c t on the  -  The  final  50 -  s o l u t i o n f o r the em wave i n the gas can now be g i v e n :  ^ E IE E = { [ - ^ < s i n 9/2 - cos 9/2)cos 6/2 + — £ (cos 9/2 + s i n 9/2)cos 9/2]e_, /2 /2 1  E IE + [ — ( s i n 9/2 - cos 9/2) s i n 9/2 + —2L (cos 9/2 + s i n 0 / 2 ) s i n 9/2le_,} /2 /2 1  s  i(K+z-ut)  +  -E -iE { [ - ^ (cos 9/2 + s i n 9/2) s i n 9/2 + —-2. ( •2 ' /2  E IE +[-JS(cos 9/2 + s i n 9/2)cos 9/2 + - ^ ( c o s • 2 /2  c o s  9/2-sin9/2) s i n 9/2]e 1  9 /2-sin9 /2)cos 9/2]e } -1  i ( K e  - " Z  t i ) t )  (2-112)  II.4 THE WAVE IN A TRANSVERSE AND PARALLEL FIELD  The p h y s i c a l e f f e c t s o f t h e gas upon the i n c i d e n t l i g h t a s c e r t a i n e d from e q u a t i o n s reduces  (2-112).  While  the e q u a t i o n i s q u i t e complex, i t  c o n s i d e r a b l y i n the case where the magnetic  t r a n s v e r s e or p a r a l l e l cases a r e o f p a r t i c u l a r  field  t o the d i r e c t i o n o f p r o p a g a t i o n . interest  can now be  i s either S i n c e these two  they a r e examined i n d e t a i l .  - 51  -  The T r a n s v e r s e  When the magnetic f i e l d  i s p a r a l l e l to the x - a x i s , i t i s t r a n s v e r s e t o  the d i r e c t i o n of p r o p a g a t i o n . o r i e n t a t i o n of the f i e l d equations  (2-85),  Field  From our e a r l i e r d e f i n i t i o n of the  t h i s corresponds  to an angle 8 = n/2.  Returning to  (2-86) and (2-87) and s u b s t i t u t i n g t h i s v a l u e f o r 6 we  find  a  ll  a  l-l  =  a  =  -l-l  a  -ll  =  i K  "  0  i K  0  i j< l w  a  +  i j< l W  a  l b  +  >  i b  +  1  )  /  2  +  " \  +  i b  i b  T h i s i n c r e a s e d symmetry s i m p l i f i e s equation  )  )  J  +  ( - i + ib))  w  + J  a  w  (2-113)  ( - i + ib))  (2-114)  a  e q u a t i o n (2-112).  R e c a l l from  (2-94): e  =6  ran  mn  + 4-rua mn  v  (2-115) '  hence e  ll  Thus the complex f u n c t i o n  =  e  -l-l  a  d  0 = e  l 1  l-i  "  e  -1 -1  and we have the simple r e s u l t 6 = i t / 2 . e q u a t i o n (2-112) y i e l d s :  E  -ll  =  e  i-i  [ e q u a t i o n (2-103)]  2 e  tan  n  =o (REAL)  S u b s t i t u t i n g t h i s value  (2-116)  into  - 52 -  IE (  +e-)  ei  e  i  (  +-  K  w  t  )  /2 + — /2  but  (e  - e j  n  e  (2-117)  from the d e f i n i t i o n of e^ t h i s i s simply  ^ i ( K _ z - tot) _ i(K+z - tot) j E = E (to) e i + E (to) e x y  Thus the e l e c t r i c  field  of the i n c i d e n t wave d i v i d e s  components along the x and y a x i s . in  general  will  f a c t o r s K_ and  The p r o p e r t i e s  of these two  into  two  components  K+.  R e c a l l equation  i s c o n t a i n e d i n the e x p r e s s i o n s f o r  (2-101):  2  ,2 _ K  Invoking  naturally  be d i f f e r e n t as they are governed by the two d i f f e r e n t  The e f f e c t of the magnetic f i e l d K_ and K+.  (2-118)  ^  ±  the unique symmetry of e  J %  ' h i  m  f o r the t r a n s v e r s e f i e l d  n  ±  e  l - l  we have:  <  2  "  1 1 9  >  hence: c — to  ,  2  2  K  2  = 1 + 4it ( a  n  ± a  i  n  )  (2-120)  - 53 -  The p h y s i c a l consequences of t h i s e q u a t i o n are most  readily  demonstrated by i n t r o d u c i n g the c o n v e n t i o n a l parameters n , the index of +  refraction  and k , +  the a b s o r p t i o n c o e f f i c i e n t  JK  - (n  ±  ±  where^:  + i 1/2 £ k )  (2-121)  ±  It follows that:  -I^-k|  n2 ±  a)  2  c . ^-n k +  -  <£- Re ( K ) c  (2-122)  ±  1  2  a) ^-Im c 2  +  =  (K )  (2-123)  +  2  S o l v i n g these e q u a t i o n s  f o r n and k u s i n g the approximation — -t  la  I « mn  1 we  find  n  k  Thus from e q u a t i o n  ~ 1 + Zn Re ( a  +  +  ~  4* £ Im (a  u  u  ± a-) l  (2-124)  l  ± a )  (2-125)  in  (2-121):  tThe f a c t o r 1/2 J J m u l t i p l y i n g the a b s o r p t i o n c o e f f i c i e n t k i s i n t r o d u c e d to conform w i t h the c o n v e n t i o n a l d e f i n i t i o n of k as the c o e f f i c i e n t governing the e x p o n e n t i a l damping of the intensity,, I , which i s p r o p o r t i o n a l to the square of the e l e c t r i c f i e l d , i . e . I = I Q e ~ . +  z  -  £ K  Substituting (2-113) and  =1  ±  54  -  + 2% ( a  ± a  n  the complete e x p r e s s i o n s f o r (2-114), we  £-K+  = 1 + 2n i Y  ^•K_  = l + 2niK  Q  0  [1/2  w(a  W(aQ  + ib)  (2-126)  from  + i b ) + 1/2  x  W(a_  1  + lb))  from e q u a t i o n  through  assumes I t s f i n a l  form.  content of t h i s e q u a t i o n .  the term a  The  i n the f u n c t i o n s W(a  magnetic  + ib). u  (2-80) a  u  c  =  ^0 (u) - oj - T — g .Bu) 0 h aj n  6  v  The  (2-127)  (2-128)  u Recall  equations  i n t o e q u a t i o n (2-118) the e q u a t i o n f o r  examine the p h y s i c a l  f i e l d manifests i t s e l f  )  and a^-^  p r o p a g a t i o n t r a n s v e r s e to the magnetic f i e l d us now  n  find  With these e x p r e s s i o n s i n s e r t e d  Let  i  Oo u  consequences of t h i s are c l e a r l y i l l u s t r a t e d by c o n s i d e r i n g f i r s t  absorption c o e f f i c i e n t s  k+ and k_  from equations  (2-125),  the  (2-113) and  (2-114):  k =4it KQ ^ Re (1/2 W(a +  k  _  = to h  + i b ) + 1/2  x  ^  R  e  (  W ( a  0  +  W(a_  i b )  )  1  + ib))  (2-129)  (2-130)  -  55 -  Examining k_, which determines the amplitude of the x component of the wave, we f i n d t h a t the magnitude of the a b s o r p t i o n  v a r i e s over a frequency  range determined by Re (W(aQ + i b ) ) which i s independent of the f i e l d . (Note that Re ( w ( z ) ) i s p r o p o r t i o n a l t o a V o i g t  function.)  I n c o n t r a s t the  amplitude o f the y component of the wave depends on k+ which i s a f u n c t i o n of the magnetic f i e l d . f u n c t i o n s Re(w(a  x  ^0  As the f i e l d  + i b ) ) and Re(w(a_  i s increased 1  + ib)) s h i f t  the peaks of the two t o the f r e q u e n c i e s  u)  Q  +  ^0  j j - g j B and co - ^ — a  respectively.  0  absorption  w i l l decrease a t the frequency w  Q  frequencies  d i s p l a c e d symmetrically  about u  = k where k i s the f a m i l i a r a b s o r p t i o n a magnetic f i e l d , The  Hence the magnitude of t h e  0  w h i l e i n c r e a s i n g a t the two  coefficient  i n d i c e s o f r e f r a c t i o n n_ and n  +  = k_  f o r a gas not immersed i n  +  determine the phase v e l o c i t y of the In the presence of a magnetic  * n_ and a wave p r o p a g a t i n g through the gas w i l l e x p e r i e n c e a  r e l a t i v e phase s h i f t  between i t s x and y components.  Examining the e q u a t i o n s f o r k discussions extension  +  (See Sobel'man, 1972 p. 381)  x and y components of the wave r e s p e c t i v e l y . field n  Note that when 1$ = 0, k  .  of the Zeeman e f f e c t  o f the e q u a t i o n s f o r K  degenerate ground s t a t e .  +  and k_ i n the l i g h t of our e a r l i e r  ( s e c t i o n II.1) +  leads  and K_ t o the more g e n e r a l  instance  of a  I n t h a t d i s c u s s i o n we noted t h a t a components _  c o u l d only absorb photons p o l a r i z e d p e r p e n d i c u l a r ( i . e . along  t o an obvious  t o the magnetic  field  the y - a x i s ) , w h i l e u components c o u l d o n l y absorb photons  p a r a l l e l t o the f i e l d  ( i . e . along  s u r p r i s i n g t o observe t h a t  the x - a x i s ) .  the e x p r e s s i o n  for k  I t i s t h e r e f o r e not i s simply  proportional to  -  56 -  sum of the l i n e broadened a-components w h i l e k_ i s p r o p o r t i o n a l of  the l i n e broadened it-components f o r t h i s non-degenerate c a s e .  t o the sum Thus  there  emerges an i n t i m a t e r e l a t i o n s h i p between the a b s o r p t i o n c o e f f i c i e n t s and the Zeeman components. rigorously,  (see Sobel'man 1972).  Extrapolating and  K  T h i s r e l a t i o n s h i p can be demonstrated  this relationship  t o the more g e n e r a l degenerate case  K  +  become  -  K  where m - m  1  - 1 + 2wiKn £ , C , W(a ,+ i b ) w mm mm mm  (2-131) '  1  +  u)  = ± 1  £-K_ = 1 + 2 * 1 ^ E , C° co mm mm u  W(a  + ib)  mm  (2-132)  where m - m' = 0  mm'  a  The c o n s t a n t s C°  mm  c  In the next s e c t i o n  1  mm  m , )  and  1  m'-m  =0  (2  ~ ) 133  , a r e the r e l a t i v e i n t e n s i t i e s of the Tt and a  which have been n o r m a l i z e d such  ° . mm  where  examined.  W  , and C  components r e s p e c t i v e l y ,  L-t mm  ^0  " 0 - —B K f " V j '  = U  that  E , C , = 1 mm' mm 1  where  (2-134) v  m'~m=±1  p r o p a g a t i o n p a r a l l e l t o the magnetic f i e l d i s  - 57 -  The P a r a l l e l  The magnetic f i e l d 8=0.  i s p a r a l l e l t o the d i r e c t i o n  From e q u a t i o n s  (2-85),  a  l-l  =  -ll  a  "  = IKQ W(  n  (2-135)  0  + ib)  3l  = IKQ W(a_  a-!-!  1  (2-136)  + ib)  (2-137)  (2-94)  and  from e q u a t i o n (2-103)  and  therefore  ^-l  =0  (2-138)  tanB  = 0  (2-139)  0 = 0  Using t h i s r e s u l t ,  + "E E =  + 1  equation  E  e  7  /  (2-112) reduces t o  i(K+ - u t ) e  2  E  + IE  + 1  Thus the wave d i v i d e s n a t u r a l l y i n t o From e q u a t i o n  of p r o p a g a t i o n when  (2-86) and (2-87),  a  hence from e q u a t i o n  Field  y  /  e  i ( K - u>t) e  2  (2-140) "I  two c i r c u l a r l y p o l a r i z e d  components.  (2-101) c — C 0  2 and S- K_  2  K Z  (  2 +  = e  2  n  0  ^  = e . ^  (2-141)  - 58 -  Again making the assumption  ^ K  +  |a| «  1 we f i n d  = 1 + 2n i K g W(a  ^- K_ = 1 + 2n i %  x  + ib)  (2-142)  WCa-! + i b )  (2-143)  These e x p r e s s i o n s t o g e t h e r with e q u a t i o n (2-140) c o m p l e t e l y d e s c r i b e the p r o p a g a t i o n of l i g h t p a r a l l e l t o the magnetic f i e l d . c i r c u l a r l y polarized of  component i s a f f e c t e d  Note that  each  by a s p e c i f i c t r a n s i t i o n , u = ± 1 ,  the atom. When a magnetic f i e l d  indices  of r e f r a c t i o n  frequencies. effect  and w i l l  absorb  The d i f f e r i n g i n d i c e s  of faraday r o t a t i o n  results  i s a p p l i e d these two components have  can be extended  l i g h t over a d i f f e r e n t  of r e f r a c t i o n  of l i n e a r l y p o l a r i z e d  t o the s i t u a t i o n  lead  range of  t o the f a m i l i a r  incident  of a degenerate  analogy w i t h the case of a t r a n s v e r s e f i e l d .  different  light.  These  ground s t a t e by  When t h i s i s done the f i n a l  e q u a t i o n i s i n complete agreement with t h a t o b t a i n e d by Camm and Curzon, 1972,  using a d i f f e r e n t In t h e next  effect  approach.  section  these r e s u l t s  o f a gas on an i n c i d e n t  II.5  a r e used  em - wave.  THE POLARIZATION OF THE TRANSMITTED WAVE  To t h i s p o i n t the theory has been developed monochromatic p l a n e wave. t r a n s v e r s e magnetic  to examine the p o l a r i z i n g  field.  Recall  f o r the s p e c i a l  case of a  e q u a t i o n (2-118) f o r the case of a  - 59  i(K_z E = E  where K  and K_  +  x  -  tit)  (u)e  i(K z +  I + E (u>) e y  are g i v e n by e q u a t i o n s  E (u>) = A (w)e x x  j  (2-127) and  I* (2-108):  -ut)  (2-128) and  (2-118)  from  equation  i<j> X  E (w) y  where A  and A are the amplitudes x y i n c i d e n t em wave. r  = A (to)e y  and <b y  y  and <b x  are the phases of  T  the  r  A c h a r a c t e r i s t i c o f monochromatic plane waves i s t h a t t h r e e q u a n t i t i e s , examined at a g i v e n p o i n t z In space, remain are the two A e  - l / 2  k  - l / 2 +  k  and A e  These three independent  z  and  the phase d i f f e r e n c e g i v e n by;  q u a n t i t i e s are s u f f i c i e n t  the s t a t e of p o l a r i z a t i o n of the l i g h t 1970,  p. 485).  constant  They  amplitudes,  z  -  constant f o r a l l time.  to determine  completely  r e p r e s e n t e d by t h i s wave,  (Klein,  S i n c e f o r a monochromatic plane wave these q u a n t i t i e s  i n time, a monochromatic plane wave must have a f i x e d  are  s t a t e of  polarization. A strictly  monochromatic plane wave i s , however, a  i d e a l i z a t i o n which i s never  realized  (2-118) must be m o d i f i e d s l i g h t l y r e a d i l y accomplished  i n practice.  mathematical  Consequently  to a c c o r d w i t h r e a l i t y .  by t r a n s f o r m i n g e q u a t i o n (2-118) i n t o  quasi-monochromatic plane wave:  equation  T h i s i s most the  -  i(K_z  60 -  - tot) _  E = E (to,t)e x.  i(K z +  +E  1  y  - tot)_ (2-144)  j  (to,t)e  where E (to,t) and E (io,t) a r e s l o w l y v a r y i n g f u n c t i o n s o f time compared t o x y e^  and are  U t  d e f i n e d by the  expressions:  i<t> E (to,t) = A ( u , t ) e x  E  (t) (2-145)  X  x  (w,t)  = A (w,t)e  where the m o d i f i e d amplitudes and phases are  (2-146)  y  now s t o c h a s t i c  functions of  time. T h i s new r e p r e s e n t a t i o n o f the l i g h t wave r e q u i r e s d e f i n i t i o n of p o l a r i z a t i o n plane waves s i n c e  than was n e c e s s a r y f o r s t r i c t l y  i t admits the  A number o f e q u i v a l e n t of p o l a r i z a t i o n o f l i g h t .  sophisticated  nonchromatic  p o s s i b i l i t y of unpolarized  light.  schemes have been d e v i s e d t o r e p r e s e n t the s t a t e The scheme used i n t h i s study i s a f t e r a method  presented i n Born & Wolf ( 1 9 7 5 ) , the  a more  p. 544 where the  degree o f c o r r e l a t i o n between the  p o l a r i z a t i o n i s related to  irregular fluctuations  o f A , A , <b , x y' x T  and  <)>y.  An a t t r a c t i v e f e a t u r e o f t h i s scheme i s that  parameters are  closely related  t o the  e x p e r i m e n t a l l y measured  because the t h e o r e t i c a l development c l o s e l y models the t e c h n i q u e used t o a n a l y s e t h e l i g h t .  the t h e o r e t i c a l quantities  experimental  T h i s f e a t u r e proves t o be p a r t i c u l a r l y  advantageous when we compare the r e s u l t s of theory w i t h experiment.  -  The  theory  61 -  i s developed by c o n s i d e r i n g the f o l l o w i n g i d e a l i z e d  experiment, (see F i g . 4 ) . The I n c i d e n t l i g h t , having of gas i n a t r a n s v e r s e magnetic f i e l d ,  traversed a length, X ,  passes through a compensator  (such as  a q u a r t e r wave p l a t e ) which r e t a r d s the phase of the y-component by an amount £ with  respect  to the phase of the x component.  Using  equation  (2-144) the wave i s g i v e n by:  -»• iK X i O . E = [ E (oj,t)e " i + E (u,t)e e^j] x y  The wave next passes through a p o l a r i z e r w i t h an angle <\> w i t h r e s p e c t the p o l a r i z e r  t o the x a x i s .  u  (2-147)  C  i t s pass d i r e c t i o n  i n c l i n e d at  Whence the component p a s s i n g  through  i s g i v e n by:  iKA E(<KS,t) - [ E ( u , t ) e x  iK Jt +  cos cp + E ( o j , t ) e y  F i n a l l y , the i n t e n s i t y instantaneous  i ( -z - t ) .e  intensity^,  e ^ s i n 4,] . e  i ( — z - u)t) (2-148) C  of the wave Is measured by a d e t e c t o r .  The  I ' , o f the wave i s p r o p o r t i o n a l t o the square o f  the r e a l p a r t o f the complex e l e c t r i c f i e l d  I* - Re  2  f u n c t i o n , E(cj>,£,t), i . e .  (E(<|>,S,t))  (2-149)  t T e c h n i c a l l y , t h i s q u a n t i t y i s the f l u x d e n s i t y or i r r a d i a n c e and not the i n t e n s i t y , (see K l e i n , 1970 a t pp. 122 and 508). However, the two a r e c l o s e l y r e l a t e d and many authors i n c l u d i n g Born and Wolf use the term intensity. T h e r e f o r e , f o r convenience, the term i n t e n s i t y i s used h e r e .  a)  LIGHT SOURCE  /  | B  GAS CELL IN TRANSVERSE MAGNETIC FIELD  / COMPENSATOR  POLARIZER  DETECTOR N3  b)  F i g . 4:  a) P o l a r i z a t i o n Measuring System b) P o l a r i z a t i o n O r i e n t a t i o n  - 63 -  However, a r e a l d e t e c t o r , such as a p h o t o m u l t i p l i e r or a p h o t g r a p h i c p l a t e , does not measure the i n s t a n t a n e o u s i n t e n s i t y ; intensity I.  I t measures a time averaged  T h i s e x p e r i m e n t a l l y measured i n t e n s i t y ,  d e s c r i b e d by t a k i n g the time average of I  1  as d e s c r i b e d by the e x p r e s s i o n :  I = < I ' ( t ) > = ^/Jf where T i s the a v e r a g i n g time With the l i g h t  I, i s t h e o r e t i c a l l y  I ' ( t ) dt  (2-150)  period.  s o u r c e and d e t e c t o r used i n t h i s s t u d y , t h e f u n c t i o n s E  and Ey d e f i n e d by e q u a t i o n s (2-145) and (2-146) undergo  a l a r g e number of  random f l u c t u a t i o n s w i t h i n t h e response time T of the d e t e c t o r .  It i s  t h e r e f o r e r e a s o n a b l e , f o r t h e sake of mathematical s i m p l i c i t y , t o take the limit  T •*• °° r a t h e r than t h e a c t u a l response time of the d e t e c t o r i n  calculating  the time average of I ' . That i s , I i s g i v e n by: I = <I'(t)> = l i m jfj_l  which  I'(t)dt  i s p r o p o r t i o n a l to lim T-*»  j=rj I |Re ( E f o ; t ) |* d t  U s i n g t h i s e x p r e s s i o n f o r the time a v e r a g i n g , i t i s p o s s i b l e  2<Re  (see  Born and Wolf Thus,  2  (U<\>,l;t)>  = <m,Z;t)  . E*(cK£;t)>  (2-151)  to show t h a t :  (2-152)  1975 a t p. 498)  the e x p e r i m e n t a l l y measured i n t e n s i t y  I(<\>,l;u) =  I i s s i m p l y g i v e n by:  <E(4>,£,t) . E*((p,C;t)>  (2-153)  -  where the unimportant  64 -  c o n s t a n t o f p r o p o r t i o n a l i t y has been s e t e q u a l t o 1.  I n s e r t i n g e q u a t i o n (2-148) i n t o the above, we have:  * i ( K - K_*)A I(<K5;w) = < E E > e cos x  *  r  + [ <E  x  * E  y  2  x  i( _K  > e  K  iH  ^ i ( K - K*)A 4. + < E E > e sin y  * i(Ke * + <E E > e y x  y  K*H  x  2  4, (2-154)  e ^ ] cos 4, s i n 4, 1  E q u a t i o n (2-154) can be w r i t t e n more s u c c i n c t l y as  I(4>,£;u>) = J ^ c o s + (J xy  + J  2  + J e yx  1 5  y  y  sin  2  4,  ) cos 4. s i n 4,  (2-155)  where: J  = <E E*> XX  J  the  , J  yy  , J  e  l(K+-Kj)Jl  X X  = <E E*> y y - <E E*> x y  g  i(K_-Kj)Jl  xy  e  l(K+-K*H  yx  = <E E*> y x  j  xx  l(K_-K_)*  yy  j  The f o u r terms J  e  xy  and J form the elements of a m a t r i x c a l l e d yx  coherency m a t r i x J . J  =  J J ( j X X j ) X  yx It  i s e v i d e n t from the d e f i n i t i o n s of J  J  xy  y  ( -156a) 2  yy xy  and J that: yx  = J * yx  (2-156b)  - 65 -  F u r t h e r , i t f o l l o w s from e q u a t i o n (2-154) that the t o t a l i n t e n s i t y I ( i . e . T  the i n t e n s i t y measured w i t h the p o l a r i z e r removed) i s g i v e n by:  I  It i s c l e a r  = J  T  xx  + J  (2-157)  yy  from e q u a t i o n (2-155) that any two  light  beams w i t h  same coherency m a t r i x are i n d i s t i n g u i s h a b l e i n an experiment p o l a r i z e r and a compensator.*  Thus t h i s m a t r i x can be used  u n i q u e l y the s t a t e of p o l a r i z a t i o n of the l i g h t . illustrate  t h i s w i t h some examples.  m a t r i x of the i n c i d e n t  light  the  involving a to d e f i n e  L e t us b r i e f l y d i g r e s s t o  For s i m p l i c i t y we use the  coherency  which from equations (2-107) and  (2-144) i s  simply: <E J  = i  <E  x y  V,  x  E  *  >  *>  <E  x  E  two  polarized  light  amplitudes and two phases  amplitudes and be expressed  (2-158)  <E E *> y y  Completely P o l a r i z e d  Completely  >  y  Light  o c c u r s when the random f l u c t u a t i o n s of the  are c o r r e l a t e d such that the r a t i o of the  the d i f f e r e n c e i n the phases  are time independent.  This  may  as  t i n i n t e r f e r e n c e experiments the waves may be d i s t i n g u i s h e d by t h e i r c h a r a c t e r i s t i c coherence l e n g t h and s p a t i a l coherence. However, s i n c e these are i n t r i n s i c p r o p e r t i e s of the l i g h t source which are o n l y m i l d l y a f f e c t e d by the a b s o r b i n g gas, they a r e not c o n s i d e r e d h e r e .  - 66 -  A (t) -2 = q A (t) x where q and x  a  r  e  constants. J  P  and x  = 4> ( t ) - 0 ( t ) x y  In t h i s case  = <A  >  2  i  may be w r i t t e n ;  ( \ q e ^  X  (2-159)  * q  e ± X  )  (2-160)  2  The s u p e r s c r i p t P has been added t o denote completely p o l a r i z e d It  i s e a s i l y shown t h a t  monochromatic plane wave.  light.  t h i s m a t r i x i s i n d e n t i c a l to t h a t of a  S i n c e two l i g h t beams with the same  strictly  coherency  m a t r i x are i n the same s t a t e of p o l a r i z a t i o n i t f o l l o w s that the m a t r i x J' does indeed r e p r e s e n t c o m p l e t e l y p o l a r i z e d p r o p e r t y of p o l a r i z e d  light  light.  From (2-160) note  that a  i s that  Det  J** = J J I xx yy  J J =0 xy yx  v  (2-161) '  P the r e l a t i o n s h i p s between and monochromatic p l a n e p waves we can denote the v a r i o u s s t a t e s of by the f a m i l i a r terms By e x p l o i t i n g  P a p p l i c a b l e to monochromatic plane waves. elliptically  r e p r e s e n t s an  I f the wave i s l i n e a r l y p o l a r i z e d x = uni where  p o l a r i z e d wave.  m = 0, ± 1 , ±2 ... and the coherency <A  Thus, i n g e n e r a l  > x  (  m a t r i x becomes 1  q  / i\q (-1) m  _2 2  )  The d i r e c t i o n of p o l a r i z a t i o n makes an angle a  <x  0  (2-162)  q  = arctan [ (-l) q] m  Q  w i t h the x - a x i s , where:  (2-163)  - 67 -  For c i r c u l a r l y  light x  polarized  = m n/2 and A = A and the m a t r i x i s x y  g i v e n by: < A  x  >  (+1  l )  where the upper and lower s i g n s r e p r e s e n t polarized light  (2-164)  1  right  and l e f t  respectively.  Unpolarized  We d e f i n e u n p o l a r i z e d  light  light  as l i g h t whose i n t e n s i t y i s independent o f  the angle <J> of the p o l a r i z e r , and o f the phase s h i f t  I(4>,£,(jj) = constant  It i s evident  circularly  from e q u a t i o n  i.e.  f o r a g i v e n co.  (2-165)  (2-155) and (2-156b) that t h i s  i s t r u e i f and  only i f J  An e q u i v a l e n t light  = J yy  y  propagation. given by:  and  J = J yx xy  =0  and perhaps more I n s i g h t f u l d e f i n i t i o n  i s a wave whose components  <E E *> = 0 x  xx  and E  y  a r e completely  f o r any c h o i c e of x, y axes p e r p e n d i c u l a r I t f o l l o w s t h a t the coherency m a t r i x  of u n p o l a r i z e d incoherent  so that  t o the d i r e c t i o n o f  f o r unpolarized  light i s  - 68 -  note:  Det  Partially  In  # 0  Polarized  g e n e r a l the s t a t e of a l i g h t  (2-168)  Light  beam l i e s between the two extremes o f  c o m p l e t e l y p o l a r i z e d and u n p o l a r i z e d l i g h t and t h i s i s known as p a r t i a l l y polarized  light.  I t can be shown that p a r t i a l l y  polarized  light  can be  u n i q u e l y expressed as the s u p e r p o s i t i o n o f a c o m p l e t e l y p o l a r i z e d and an u n p o l a r i z e d wave, (Born and Wolf, 1975, p.  550).  That i s  U P J = J + J  where In accordance  (2-169)  w i t h m a t r i c e s (2-160) and (2-167) we can d e f i n e :  and j P  = ( *?) D  U The m a t r i x elements the f o l l o w i n g  of J  ( -^D 2  P and J  a r e g i v e n i n terms of the elements  o f J by  equations.  A = 1/2 ( J + J ) - 1/2 / ( J +~J )2 - 4 Det J xx yy xx yy  (2-172)  - 69 -  B = J  x  x  " A  (2-173)  C = J  y  y  - A  (2-174)  D - J  U The elements of J  x  y  D* = J  y  (2-175)  (2-176)  x  P and J  thus p r o v i d e a unique r e p r e s e n t a t i o n of the s t a t e  of p o l a r i z a t i o n of the l i g h t . I t i s u s e f u l t o i n t r o d u c e one f i n a l parameter, the degree of p o l a r i z a t i o n P. portion I  P i s d e f i n e d as the r a t i o of the I n t e n s i t y of the p o l a r i z e d  j_ to the t o t a l i n t e n s i t y of the l i g h t . pol — I_, T T  p =  =  That i s :  B +C J  xx  + J  yy  note 0 < P < 1  L / / ;  (2-178)  When P = 1 the l i g h t unpolarized.  (2-17T) "  K A  i s c o m p l e t e l y p o l a r i z e d and when P = 0 i t I s  The degree of p o l a r i z a t i o n i s independent of the c h o i c e of x,  y axes. We can now use these d e f i n i t i o n s to examine the p o l a r i z i n g e f f e c t of a gas i n a t r a n s v e r s e magnetic m a t r i x i s g i v e n by  field.  From e q u a t i o n (2-155) the coherency  - 70 -  -k A  i(K_-K*)A  <E E > e x x J =  (2-179) I(K - K_*)A <E E * > e y x  where e q u a t i o n (2-121) f o r k± has been used. I f the s t a t e of p o l a r i z a t i o n o f the i n c i d e n t wave i s known the above matrix w i l l describe gas.  c o m p l e t e l y the s t a t e of the wave a f t e r t r a v e r s i n g the  Two cases a r e o f p a r t i c u l a r i n t e r e s t :  when the i n c i d e n t wave i s  l i n e a r l y p o l a r i z e d and when the wave i s u n p o l a r i z e d . cases i n order t o e s t a b l i s h the c o n d i t i o n s  We examine these two  under which the gas may a c t as a  polarizer.  Unpolarized Incident  When the i n c i d e n t wave i s u n p o l a r i z e d  <E E*> xx  =  <E E*> y y  and  where I (u)) i s here d e f i n e d 0  Thus, from m a t r i x  -  Wave  [from m a t r i x  (2-167)]:  1/2 I ( ) u n  u  *  <E E > = 0 x y  as the t o t a l I n t e n s i t y o f the i n c i d e n t wave.  (2-179): e J  = j  IQ  -kJl  0 (2-180)  (O) )  0  - 71 -  One r e s u l t i s immediately evident from t h i s matrix.  The gas a f f e c t s  the state of p o l a r i z a t i o n only through the absorption c o e f f i c i e n t s k k_.  +  and  Moreover, i t i s also c l e a r how the gas i s able to p o l a r i z e the l i g h t .  Consider a s i t u a t i o n where at a given frequency w;  k_Jl»l and k A « l  (2-181)  +  then: J ~ 1/2 ^(o,) ( °  J)  (2-182)  Thus the l i g h t at frequency OJ becomes l i n e a r l y p o l a r i z e d along the y - a x i s . In fact only the c o n d i t i o n k gas to act as a p o l a r i z e r .  »  k  and not K 9 «  1 i s necessary f o r the  However i t i s h i g h l y d e s i r a b l e that the  p o l a r i z e r have as high a transmlttance as possible and i n t h i s example t h i s i s achieved when  k Jl. •*• +  0.  I f the conditions of (2-181) can be arranged, two l i m i t a t i o n s to the technique are s t i l l apparent.  The l i g h t can only be l i n e a r l y p o l a r i z e d and -kjt  the l i g h t w i l l never be completely p o l a r i z e d since e zero.  i s never e x a c t l y  That i s , there w i l l be a strong l i n e a r l y p o l a r i z e d component along  the y-axis and a weak unpolarized component.  In the representation (2-169)  t h i s can be w r i t t e n as; 0  -kjl 0 J = I (u))/2 0  -kJl  +  IQ  (w)/2  0 (2-183)  0 (e  _ k + A  -e"  k _ J l  )  C l e a r l y then the l a r g e r k j l the higher the degree of p o l a r i z a t i o n .  - 72 -  Simple  To demonstrate r e a l i z e d we  Examples  t h a t the c o n d i t i o n s  g i v e n by (2-181) can i n f a c t  examine e q u a t i o n s (2-129) and  l i n e s w i t h a non-degenerate  k  ground  = 4 i Ko ^ Re  +  (2-130) which g i v e k_ and k  for  +  state:  (1/2 W(a  Re (w(ag  k_ = 4it Ko  be  x  + i b ) + l/2W(a_  1  + ib)  + ib)  L e t us assume f o r the moment that a l i n e can be found such that at f r e q u e n c y u) , k_A  »  Q  1 f o r a gas of r e a s o n a b l e l e n g t h and d e n s i t y .  demonstrated mercury.)  later  t h a t t h i s i s indeed the case f o r the 253.7 nm  on k  which  is sufficiently  +  unpolarized  i t i s c l e a r t h a t i n p r i n c i p l e we l a r g e t o cause k A +  can a p p l y a magnetic  the c e n t r a l f r e q u e n c y u> . 0  1 at the frequency O J .  «  Examining  we  the e q u a t i o n f o r k  the f i r s t  We  want  over i t s e n t i r e frquency range and not j u s t a t  require  l i g h t has a l i n e  shape  that the c o n d i t i o n s k i »  1, (see 2-181) both remain v a l i d over the frequency range  over Av,  Thus an  q  That i s i f the I n c i d e n t  w i t h a c h a r a c t e r i s t i c w i d t h Av  p r o f i l e W(aQ  field  Our o b j e c t i v e i s , however, much more a m b i t i o u s .  the gas to p o l a r i z e the l i g h t  that  magnetic  i n c i d e n t beam would become p a r t i a l l y p o l a r i z e d when passed  through the gas.  evident  l i n e of  In view of the e a r l i e r d i s c u s s i o n of the e f f e c t of the  field  k A «  (It i s  and assuming  c o n d i t i o n can be met  k Z »  over Av  1 and  Av.  1 at u> , i t i s 0  i f the a b s o r p t i o n  line  + i b ) i s s u f f i c i e n t l y broad so as to remain n e a r l y c o n s t a n t  (see F i g . 5a).  Of course i f k £ i s v e r y l a r g e at  the c o n d i t i o n  -  73  -  FREQUENCY  Fig.  5:  E m i s s i o n , A b s o r p t i o n and  •  Transmission Line Profine  - 74  k_A over  »  1 c o u l d be s a t i s f i e d  -  even w i t h a s u b s t a n t i a l change i n W(a  + ib)  Q  Av. While  satisfy  i t i s desirable  the f i r s t  condition; k £ equation f o r k  condition,  «  +  +  to have a broad l i n e p r o f i l e  i n attempting to  j u s t the o p p o s i t e i s t r u e of the  1 over Av.  F o r i f W(a  became broader  2  + i b ) and W(a_  ( r e c a l l that W(ag  x  a p p l i e d t o s h i f t the c e n t r e s of the two that o n l y t h e i r many purposes over Av  In  a l s o be important that k £ over Av  +  +  w i l l alter  + i b ) and  field  must  be  f a r enough from u)  so  Q  over the range Av.  remain  +  s i n c e changes i n k £  transmitted  line profiles  much s m a l l e r wings c o n t r i b u t e to k  I t may  + i b ) i n the  + i b ) , W(a^  -W(a_2 + i b ) a l l have the same b r e a d t h ) , a l a r g e r magnetic  second  For  reasonably constant  the l i n e  shape of the  light.  c o n c l u s i o n any attempt  gas must r e c o n c i l e  the two  to o p t i m i z e the p o l a r i z i n g p r o p e r t i e s of the  conflicting  c o n d i t i o n s k JL »  1 and k A  «  +  1 over  Av. The not  f o r e g o i n g comments would, of c o u r s e , be l e s s important i f we  alter  earlier  the l i n e p r o f i l e width a c c o r d i n g to our d e s i g n .  d i s c u s s i o n of l i n e broadening we  depended on the temperature  and d e n s i t y of the gas.  these two parameters "can l e a d magnitude, which polarizer's There  established  could  However i n our  t h a t the l i n e  width  Moderate adjustments i n  to changes i n the l i n e width of an o r d e r of  p r o v i d e s the l a t i t u d e n e c e s s a r y f o r o p t i m i z i n g the gas  performance. i s another s i t u a t i o n which  Consider the case where k A »  can g i v e r i s e to p o l a r i z e d  1 at frequency w , 0  light.  but the l i n e p r o f i l e  of  - 75 -  the a b s o r p t i o n l i n e Re(w(aQ + i b ) ) i s much narrower than the width Av of the Incident of  line,  the l i n e  (see F i g . 5b). F o r the x component, the l i g h t  i s almost  completely  line  i s mostly  transmitted.  used  to spread the l i n e p r o f i l e s  absorbed,  (Note  that t h i s  Re(W(a  effect  inhomogeneous, but h i g h l y i s o t r o p i c i s maintained  over  i n the wings of the  F o r the y component a weak magnetic f i e l d i s  frequency range Av r a t h e r than t o s p l i t previously.  but the l i g h t  a t the c e n t e r  1  + l b ) ) and Re(W(a_  1  + i b ) ) over the  them away from Av as was done  can be enhanced by u s i n g an magnetic f i e l d . )  Av and the t r a n s m i t t e d l i g h t  In t h i s way k+Jl »  1  i s p o l a r i z e d a l o n g the  x-axis. The  obvious  disadvantage  of t h i s  technique  w h i l e p o l a r i z i n g the source l i n e , a l s o d i s t o r t s c o m p l e t e l y a b s o r b i n g the c e n t r e of the l i n e . polarizing light source  line  i s t h a t the a b s o r b i n g gas, i t s shape by almost  Thus, t h i s  technique f o r  I s o n l y of use i n s t u d i e s where i t i s a c c e p t a b l e f o r t h e  t o be d i s t o r t e d ,  ( a l t h o u g h i n a p r e d i c t a b l e way) by the  polarizer. L i n e a r l y P o l a r i z e d I n c i d e n t Wave  C o n s i d e r now the e f f e c t light  beam.  polarized  T h i s s i t u a t i o n can be c o m p l e t e l y i l l u s t r a t e d  case where the i n c i d e n t to  of the gas on a l i n e a r l y  light  i s , from m a t r i x  by c o n s i d e r i n g the  i s p o l a r i z e d a t an angle o f 45° w i t h r e s p e c t  the x - a x i s . That  incident  (2-162) and e q u a t i o n  (2-163):  -  The  coherency m a t r i x  76  -  f o r the t r a n s m i t t e d  - k Jl e  (e  i - (n_c  light  n )£  i s then g i v e n  +  ~ 1 / 2 (k+  +  by  k_H  e  (2-184) ,  where e q u a t i o n  i ^  and k A  «  +  1 we  unpolarized  J = 0.  light  unpolarized optimize the two two  I f we  obtain e s s e n t i a l l y  k_)A  )  e  n . +  to remains  completely  a g a i n imagine a s i t u a t i o n where k I » the same r e s u l t  as i n the case of  1  an  i n c i d e n t wave, namely  will  0 0  0 1  i n c i d e n t beam.  In p a r t i c u l a r ,  the e a r l i e r  the p o l a r i z i n g p r o p e r t i e s of the gas cases are not  (2-184).  (2-185)  behave as a p o l a r i z e r much as i t d i d i n the case of  e x a c t l y the same.  Is the appearance of the  matrix  +  at frequency  J ~ (IQ(W)/2)  Thus the gas  +  been used to d e f i n e  transmitted  s i n c e Det  - 1/2(k  +  (2-121) has  Note that the polarized  ( n - n_)A  To  The  demonstrate the e f f e c t  s i t u a t i o n where k i » f r a c t i o n of  1 and  light  the  after  t r a v e r s i n g the gas  k^Jl «  the l i g h t  of these  +  i n the  to  However  c h i e f d i f f e r e n c e between  1 over Av .  along  comments on how  remain v a l i d h e r e .  i n d i c e s of r e f r a c t i o n n  an  the  coherency  terms c o n s i d e r  again  For u n p o l a r i z e d i n c i d e n t  the x a x i s which was  appeared as an u n p o l a r i z e d  not  component,  absorbed [recall  the  - 77  (2-183)]. not  I f the i n c i d e n t l i g h t  appear as an u n p o l a r i z e d To h e l p i l l u s t r a t e  case the t r a n s m i t t e d  i s p o l a r i z e d , however, t h i s f r a c t i o n  component s i n c e DET  this  light  an angle a  with  Q  does  0.  remains l i n e a r l y p o l a r i z e d but  +  = n_.  i t is  In  this  now  the x - a x i s , where from  (2-163)  <z  = arctan [ ( - l ) ( j ) ' ] m  0  1  Re(J (  a n d  (See Born and Wolf, 1975, Using  J =  s i t u a t i o n imagine f i r s t that n  p o l a r i z e d along a d i r e c t i o n forming equation  -  the equations  p.  -  1 ) m  \Re(7  =  1  If n  +  * n_,  elllptically  ))  xy  1  (2-174) f o r B and  m 0  )  27)  (2-173) and  ct  (2-186)  2  = a r c t a n [ (-1)  A(k  1 / 2  (e  C:  - k  )  '  )]  (2-187)  (which i s g e n e r a l l y the c a s e ) , the t r a n s m i t t e d  p o l a r i z e d and  light  the shape of the e l l i p s e i s governed by  is  the  factor: 6  The  ellipse  the c o - o r d i n a t e ellipse  =  (n_ - n H  (2-188)  +  i s i n s c r i b e d i n t o a r e c t a n g l e whose s i d e s are p a r a l l e l axes and  whose l e n g t h s are 2/C  touches the s i d e s at the  points  and  2/B,  (See F i g . 6 ) .  to The  -  F i g . 6:  78  -  Geometry of P o l a r i z a t i o n  Ellipse  - 79 -  (± /B, ± /C cos6) and (± /B cos6 , ± / F )  The  (2-189)  a n g l e , <J>, which the major a x i s of the e l l i p s e makes w i t h the x - a x i s i s  given by the e q u a t i o n tan 24> = ( t a n 2a ) Q  (2-190)  cos6  Thus, i n g e n e r a l , when the i n c i d e n t l i g h t i s l i n e a r l y p o l a r i z e d w i l l produce  elliptically  polarized  the gas  l i g h t , where the shape o f the e l l i p s e i s  a f u n c t i o n of the frequency o f the l i g h t . The  d i s c u s s i o n t o t h i s p o i n t has been f a c i l i t a t e d by u s i n g a  r e p r e s e n t a t i o n which d e s c r i b e s the s t a t e o f p o l a r i z a t i o n of the l i g h t specific  frequency w.  convenient one.  at a  T h i s r e p r e s e n t a t i o n i s n o t , however, always the most  F o r i n s t a n c e i f the s t a t e o f p o l a r i z a t i o n changes over the  width of the i n c i d e n t  line  p o l a r i z a t i o n o f the e n t i r e  (Av) i t i s d i f f i c u l t line.  Furthermore  t o a s s e s s the degree o f  the t h e o r e t i c a l r e s u l t s a r e  not compatible w i t h the e x p e r i m e n t a l r e s u l t s s i n c e an experiment  necessarily  measures the I n t e n s i t y over a range o f f r e q u e n c i e s and not a t a d e f i n i t e frequency w.  To overcome these d i f f i c u l t i e s we i n t r o d u c e a s l i g h t l y  modified r e p r e s e n t a t i o n .  I n s t e a d o f c o n s i d e r i n g the i n t e n s i t y at a s p e c i f i c  frequency OJ as i n the d e f i n i n g e q u a t i o n (2-154), we observe taken over the e n t i r e frequency range of the l i n e .  the i n t e n s i t y  That i s , we a r e  i n t e r e s t e d i n the i n t e g r a l of e q u a t i o n (2-154) over on.  T h e r e f o r e we d e f i n e  (2-191)  - 80 -  I f o n l y a p a r t of the l i n e  i s of i n t e r e s t  the l i m i t s of the i n t e g r a l  can be changed a c c o r d i n g l y . In complete analogy w i t h  the e a r l i e r r e p r e s e n t a t i o n we can d e f i n e a  CO  coherency m a t r i x  J  by the e q u a t i o n  I(4>» .,) - J° c o s i xx x  + (J e xy  + Jyx  ib + f sin ^ yy  2  e  ) ' cos  ib T  2  sin  6 ^ (2-192)  ib T  and  CO  *  / O <E XEX > e J  * (K_-K )A f <E E > e dio 'o x y +  do  =  (2-193) * / <E E >e • ^ o y x  from which a l l the remaining they were i n the e a r l i e r  H  , dio  f°° <E E*>e ° y y  -k A dco  parameters can be c a l c u l a t e d I n the same way as  representation.  Note that the t o t a l i n t e n s i t y o f the x and y components of the i n c i d e n t l i n e a r e g i v e n by: 1° - /  <E E*> do  and  o  and  1° = f <E E*> dio y ' y y  (2-194)  0  the t o t a l i n t e n s i t y o f the i n c i d e n t l i n e , I , i s g i v e n by x An  important  + 1° y  consequence of the f a c t  i n t e n s i t y over a range o f f r e q u e n c i e s  (2-194a)  that a r e a l d e t e c t o r measures the  and not a t a s i n g l e frequency  i s made  - 81  apparent i n t h i s r e p r e s e n t a t i o n . measured range of f r e q u e n c i e s , incident light partially  i s completely  p o l a r i z e d at any  be only p a r t i a l l y Consider  For i f K  then,  +  and K_  (from m a t r i x  £  0.  constant  2-193) even I f  g i v e n frequency  the s i t u a t i o n where the gas incident light.  over  u),  i s only  the o v e r a l l l i n e  Is  i s found to  experimentally.  a c t s as a p o l a r i z e r by  The  the  the  That i s , even though the l i g h t  p o l a r i z e d when i t i s measured  the x component of the  are not  p o l a r i z e d , the t r a n s m i t t e d l i g h t  p o l a r i z e d s i n c e Det  completely  -  absorbing  o v e r a l l e f f e c t i v e n e s s of the  gas  as a p o l a r i z e r f o r the e n t i r e l i n e can be determined from the degree of p o l a r i z a t i o n P^  and  the f r a c t i o n a l change i n the i n t e n s i t y of the p o l a r i z e d  CO  J component — — 1° y  CO  .  An  i d e a l p o l a r i z e r would have P  the e n t i r e l i n e would be c o m p l e t e l y component would be u n a t t e n u a t e d . J  p o l a r i z e d and  In g e n e r a l  =1  J and —22. = 1. 1° y  the t r a n s m i t t e d  That i s  y  i t can be assumed t h a t  the  yy  s m a l l e r — — , the g r e a t e r the d i s t o r t i o n of the l i n e shape of the i n c i d e n t i° y l i n e as a r e s u l t  of t r a v e r s i n g the gas.  the c o n d i t i o n of the gas  P  Therefore  i t i s d e s i r a b l e to choose  such t h a t  J •> 1 and - 2 2 - , . 1° y  1,  J which i m p l i e s - 2 2 1°  To i l l u s t r a t e the i d e a s presented  +  in this section J  c a l c u l a t e d as f u n c t i o n s of the magnetic f i e l d  1/2  yy  and  J  xx  are  s t r e n g t h f o r the 253.7 nm  line  - 82 -  of mercury  u s i n g the p r o p e r t i e s l i s t e d i n the appendix.  c o n s i s t s of a 3 cm l e n g t h o f H g  .  2 0 2  The a b s o r b i n g gas  The n u m e r i c a l c a l c u l a t i o n i s performed  f o r a number of gas temperatures, w i t h the gas d e n s i t y determined known vapour p r e s s u r e . has a Gaussian l i n e For  For s i m p l i c i t y  the i n c i d e n t l i g h t  from the  i s u n p o l a r i z e d and  profile.  convenience, the e q u a t i o n s used i n the n u m e r i c a l c a l c u l a t i o n a r e  collected  below.  From the m a t r i x J° (2-193)  J  xx  = J <E E > e ^ d o > , J = / <E E > ^ d e v here 1 = 3 cm. o x x ' yy o y x J  J  From e q u a t i o n s (2-177),  (2-173), (2-174) and (2-172);  ,f  OO  CD  4[ J P  - [ i -  But, s i n c e the I n c i d e n t  light  A  x  x  CO  J  - J  CO  J  1  yy—siJ2Ei]i/2  (r xx + ryy' )  ( 2  _  1 9 5 )  2  i s chosen to be u n p o l a r i z e d , from m a t r i x  (2-180), J° = J° = 0 and xy yx  CO  I  p  Furthermore,  =  f o r an u n p o l a r i z e d  I  v  CO  -  J  CO CO xx yy J + J xx yy  (2-196)  i n c i d e n t beam, from e q u a t i o n s (2-166) and  (2-157) <E E*> = <E E*> = I ( t o ) / 2 . xx y y o ' n  J  -  The Gaussian l i n e p r o f i l e  83 -  selected  f o r the i n c i d e n t source l i n e i s  g i v e n by:  - 2ff ( ¥ >  <V>  1  /  - r f " "  2  1  2  hpV]  where D i s the h a l f - w i d t h and the p r o f i l e has been n o r m a l i z e d such t h a t e q u a t i o n (2-194a) the t o t a l i n c i d e n t  1° = /"  from  i n t e n s i t y 1° i s e q u a l to 1, i . e .  2 <E E*> dw = 1  (2-198)  From e q u a t i o n s (2-129) and (2-130) k+ = AitKg ^ Re [1/2 W ^ + i b ) + 1/2 W(a_  + i b ) ] , k_ = 4nK  x  where, from e q u a t i o n s (2-88) and (2-89)  ^  =  Ncf  e n 2  1 / 2  1 i 8ne V m w2 0  0  Ncf  n  MKSA u n i t s o r  e  e it 2  Kg =  1 / 2  i n cgs u n i t s and 2V m a)2 0  e  Q  £ Re [ w ( a + i b ) ] 0  - 84  -  N = the d e n s i t y of the atoms i n the ground c = the speed of f  light  = the a b s o r p t i o n o s c i l l a t o r  e = the e l e c t r o n E m  space  = the e l e c t r o n mass  e  O)Q = the resonance 2 K  V  s t r e n g t h (see e q u a t i o n ( 2 - 8 2 ) ) .  charge  = the P e r m i t t i v i t y of f r e e  Q  state  R  (see e q u a t i o n  (2-7))  T  = [ ~ — )  0  f r e q u e n c y of the l i n e  i  /  (see e q u a t i o n (2-29))  2  and  = the Boltzmann c o n s t a n t T = the temperature  In degrees  Kelvin  M = the atomic mass. W(a^  + i b ) i s the complex e r r o r f u n c t i o n d e f i n e d by e q u a t i o n (2-78) and  e q u a t i o n s (2-80) and ^0  c b  =  ^ o ~  UQ gj  (  1  /  2  )  (  1  *5>  y  W  h  e  r  e  = the Bohr magneton (see e q u a t i o n = the Lande g f a c t o r  a  B = the magnetic y  aj  (2-4))  f o r the s t a t e a j , (see e q u a t i o n  . = the t o t a l L o r e n t z broadening width,  =  Y  N  +  Y  R  (2-6))  f i e l d strength  From e q u a t i o n (2-31) Y  (2-81)  (see e q u a t i o n  (2-39))  from  - 85 -  where y^ i s the n a t u r a l l i n e width d e f i n e d by e q u a t i o n (2-18) Y  R  In  i s the resonance broadening w i d t h . the c a l c u l a t i o n the A l l and Griem e x p r e s s i o n (2-19) was used. CO  In F i g . 7, J three d i f f e r e n t  y  y  i s p l o t t e d as a f u n c t i o n o f magnetic f i e l d  gas temperatures, 300K, 350K and 375K.  u n a f f e c t e d by the magnetric f i e l d zero f i e l d . and  strength for  Note t h a t J is xx  and has a c o n s t a n t value equal t o J  The h a l f width o f the source l i n e , D, i s 25 1 0 c m . - 3  - 1  In  - 3  cm  -1  and 60 1 0  - 3  cm  -1  line  respectively.  the next c h a p t e r the p o l a r i z i n g p r o p e r t i e s of a mercury gas a r e  examined e x p e r i m e n t a l l y .  at  Figs. 8  9 a r e the same as F i g . 7 except that the h a l f width of the source  has been i n c r e a s e d t o 40 1 0  yy  J^ as a f u n c t i o n o f Magnetic F i e l d S t r e n g t h f o r Three Gas Temperatures, 300K, 350K and 375K y  D = 25 ( I D " )  cm-1  3  J  1—  - 1 —  xx  =  J  yy f i e l d and i s independent o f t h s t r e n g t h of the f i e l d  —1—  a  t  z  e  r  o  1.5 3.0 2.5 2.0 MRGNET1C FIELD STRENGTH IKGRU55)  1—  3.5  1.0  4.5  F i g . 8:  j " as a f u n c t i o n of Magnetic F i e l d S t r e n g t h f o r Three Gas Temperatures, 300K, 350K and 375K  o o  4.5  MAGNETIC FIELD STRENGTH IKGRU55)  'yy  J^y as a f u n c t i o n of Magnetic F i e l d S t r e n g t h f o r Three Gas Temperatures, 300K, 350K and 375K  - cm ) 3  _1  0 0  J at zero f i e l d and i s independent o f the s t r e n g t h of the f i e l d  ~!  2.5  —|— 3.0  MAGNETIC FIELD STRENGTH (KGRUSS)  - 1 —  3.5  —T  4.0  -1 4.5  -  89  -  CHAPTER I I I I I I . I THE EXPERIMENTAL DESIGN  A schematic diagram of the e x p e r i m e n t a l arrangement Fig.  10.  The o b j e c t of the experiment was  p o l a r i z a t i o n , or more s p e c i f i c a l l y c o r r e s p o n d i n g to the 253.7 nm mercury  experiment  to determine the s t a t e of  the coherency m a t r i x J ° , of l i g h t  l i n e of mercury  gas c e l l immersed i n a magnetic  As i s e v i d e n t  i s given i n  a f t e r the l i g h t  traversed  field.  from F i g . 10, the experiment c l o s e l y p a r a l l e l e d  described  i n the l a s t  a  the  ideal  c h a p t e r , and the method f o r e x p e r i m e n t a l l y  d e t e r m i n i n g the coherency m a t r i x i s i n d i c a t e d by e q u a t i o n (2-192) which defines J .  Recall:  I(cb,£) = J  cos 4> + J 2  xx  sin cb + ( J e " ^ xy 2  yy  1  + J  e ^) 1  xy  sintb coscb  (where the s u p e r s c r i p t » has been omitted f o r c o n v e n i e n c e ) . Measuring the i n t e n s i t y I((b,£) f o r f o u r independent combinations o f the v a r i a b l e s cb and £ l e a d s to f o u r independent equations which can be for  J  xx  , J  yy  , J  yx  and J  xy  .  solved  While t h e r e are many a c c e p t a b l e c h o i c e s of the  f o u r combinations of <\> and £ , the f o l l o w i n g set of measurements was the  used In  experiment:  a. ) P o l a r i z e r a l o n g x - a x i s , thus I, (0,0) = J xx  1  b. ) P o l a r i z e r a l o n g y - a x i s , thus T^Cp/lyO)  =  c. ) P o l a r i z e r set at 45°, thus I ( u / 4 , 0 ) = 1/2 3  d. ) P o l a r i z e r s e t at 45° and quarter-wave along x - a x i s , thus ^ ( i t / 4 , n/2)  (3 1) (J + J ) + Re(j ) xx yy xy  plate inserted with fast  = l/2(J  x x  + J  y y  ) - Im ( j  )•  axis  F i g . 10:  Experimental  Apparatus  - 91  -  These f o u r e q u a t i o n s a r e l i n e a r l y independent and e a s i l y s o l v e d f o r J  xx  , J  yy  , Re(Jxy) and Ira(Jxy), y i e l d i n g the complete coherence m a t r i x ,  The e x p e r i m e n t a l s e t of x-y axes was  a r b i t r a r i l y s e l e c t e d , hence  the  e x p e r i m e n t a l x - a x i s does not n e c e s s a r i l y l i e a l o n g the d i r e c t i o n of the magnetic f i e l d  as i n the t h e o r e t i c a l d e s c r i p t i o n .  detail in a later Briefly,  section.  the e x p e r i m e n t a l method was  propagates through a c e l l magnetic f i e l d . propagation.  The  T h i s i s e x p l a i n e d i n more  filled  The magnetic light  as f o l l o w s .  A beam of  light  w i t h mercury vapour which i s immersed i n a  field  i s t r a n s v e r s e to the d i r e c t i o n of  next t r a v e r s e s the quarter-wave p l a t e and  polarizer,  which are arranged i n one of the f o u r p r e s e l e c t e d combinations d i s c u s s e d e a r l i e r i n (3-1).  The  light  s e l e c t s the l i n e of I n t e r e s t .  then passes through a monochromator which Finally  measured by a p h o t o m u l t i p l i e r and an  the i n t e n s i t y of the e n t i r e l i n e i s  oscilliscope.  - 92  The  I t was  e s s e n t i a l t h a t the  stable, intense, unpolarized  L i g h t Source  light  and  -  source used i n the experiment produce a  reasonably  narrow 253.7 nm  line.  The  source most commonly used i n experiments of t h i s k i n d i s a microwave excited e l e c t r o d l e s s discharge  lamp, (see G.  extreme s e n s i t i v i t y of n e i g h b o r i n g precluded Two  i t s c o n v e n i e n t use other  discharge  light  tube and  S t a n z e l , 1974).  experiments to microwave r a d i a t i o n  i n this  experiment.  sources were c o n s i d e r e d ;  a standard  w i t h n a t u r a l mercury and  low v o l t a g e  argon.  However, the  The  a high voltage  mercury  l a b o r a t o r y mercury lamp  lamp was  filled  selected after a  finally  s p e c t r o s c o p i c a n a l y s i s demonstrated t h a t the lamp produced a narrower 253.7 nm The  l i n e than the  s t a b i l i t y of the 253.7 nm  monitoring  l i n e produced by the  i t s i n t e n s i t y over a p e r i o d of 30 hours.  warm-up p e r i o d of two as much as 30% constant;  tube at the r e q u i r e d i n t e n s i t y .  30 hour p e r i o d .  there were l o n g p e r i o d s ,  of s t a b i l i t y were f o l l o w e d  by  to 20%  persisted despite e f f o r t s  examined  an  t y p i c a l l y greater  Initial  than f o u r hours,  observed.  by  not  These l o n g  during period  r e l a t i v e l y b r i e f p e r i o d s , d u r i n g which changes were observed.  These p e r i o d s  of  instability  to l o c a t e the source of the i n s t a b i l i t y  e l i m i n a t e i t . U l t i m a t e l y , the e x p e r i m e n t a l accommodate the I n s t a b i l i t y .  by  found to d r i f t  However, the d r i f t was  which no measurable change i n the i n t e n s i t y was  section.  Following  h o u r s , the i n t e n s i t y of the l i n e was  over the  i n the i n t e n s i t y of up  lamp was  procedure was  This adaptation  i s described  simply in a  and adapted  later  to  The Gas  The gas c e l l was  Cell  a c y l i n d e r , 3.7 cm long and 2.5 cm i n d i a m e t e r ,  c o n t a i n i n g a d r o p l e t of n a t u r a l mercury and i t s vapour. was  added, the c e l l was  B e f o r e the mercury  purged of i m p u r i t i e s by h e a t i n g i t as i t was  evacuated by a h i g h vacuum pump.  The c e l l was  equipped w i t h h i g h  quality  q u a r t z windows which, even under the s t r a i n of the p a r t i a l vacuum, had a negligible effect  on the s t a t e of p o l a r i z a t i o n of the l i g h t .  S i n c e the mercury vapour p r e s s u r e (and hence the gas d e n s i t y ) i s h i g h l y temperature s e n s i t i v e , controlled.  T h i s was  the temperature of the gas c e l l  had to be  a c c o m p l i s h e d by immersing the c e l l  r e g u l a t e d water b a t h s k e t c h e d i n F i g . 11.  Water was  carefully  i n the temperature  c i r c u l a t e d p a s t the gas  c e l l and i n t o a l a r g e dewer v e s s e l which c o n t a i n e d a h e a t i n g c o i l . e l e c t r o n i c m o n i t o r i n g d e v i c e was  c o n s t r u c t e d which c o n t i n u o u s l y measured  temperature of the water and a u t o m a t i c a l l y switched the h e a t i n g c o i l off  An the  on and  when the temperature reached c e r t a i n f i x e d s e t t i n g s above and below the  d e s i r e d temperature.  By t h i s means the mercury temperature c o u l d be s e t at  any v a l u e between 20°C and 30°C w i t h an a c c u r a c y of ±1/2°C.  Mercury  c o n d e n s a t i o n on the windows of the c e l l e x c l u d e d temperatures g r e a t e r than 30°C.  TEMPERATURE MONITOR TEMPERATURE READING HEAT SENSOR INPUT  HEATER COIL OUTPUT  HEAT SENSOR-  TEMPERATURE CONTROL  GAS CELL WATER BATH DEWAR VESSEL WATER HEATER COJL 4  PUMP  F i g . 11:  Temperature R e g u l a t o r f o r . t h e Water Bath  - 95 -  The Magnet  The electro-magnet Associates  and  i t s current  supply were manufactured  of the f i e l d ,  field.  The magnetic  field  was  .3% over the l e n g t h of the  and, an Incremental Gaussmeter H a l l Probe.  by f i r s t  showed that  the  cell.  c a l i b r a t e d u s i n g both a R o t a t i n g C o i l Gaussmeter  w i t h a r a t e d a c c u r a c y of .1%, ( a t the c a l i b r a t e d f i e l d  adjusted,  a  Measurements of the u n i f o r m i t y  u s i n g an I n c r e m e n t a l Gaussmeter H a l l Probe,  never v a r i e d by more than  f i e l d was  Varian  f o r use i n N u c l e a r Magnetic Resonance s t u d i e s and thus p r o v i d e d  v e r y s t a b l e and homogeneous magnetic  field  by  s t r e n g t h of 25  KG)  To ensure a c o n s i s t e n t r e s u l t  both d u r i n g the c a l i b r a t i o n and  the a c t u a l  s e t t i n g the c u r r e n t at a l a r g e p r e s e l e c t e d v a l u e and  the  experiment, then l o w e r i n g  the c u r r e n t to the l a r g e s t of the c u r r e n t s e t t i n g s to be used i n the calibration.  The  pre-determined the experiment. The  two  then lowered s e q u e n t i a l l y through  the  c u r r e n t s e t t i n g s which were used both f o r the c a l i b r a t i o n In t h i s way  v a r i a t i o n s due  range of e x p e r i m e n t a l f i e l d  the magnet a r e s i d u a l f i e l d  strengths.  of 90±1G was  and  to h y s t e r e s i s were reduced.  s e t s of measurements were found to be c o n s i s t e n t t o about  the complete in  c u r r e n t was  1% over  When no c u r r e n t  found to p e r s i s t .  flowed  The  The  first  Aperture  aperture stop,  (S  L  Stops and Lenses  i n F i g . 10)  encountered  by the l i g h t  served o n l y t o d e f i n e a s u f f i c i e n t l y narrow l i g h t beam t o prevent being r e f l e c t e d  The The  from  i n t o the d e t e c t o r from the w a l l s of the gas c e l l or the  poles of the magnet. t o t a l divergence  light  The second a p e r t u r e stop ( S ) l i m i t e d the beam t o a 2  of l e s s than  three l e n s e s L^,  .75°. and Lj (see F i g . 10)  purpose o f L± was to c o n c e n t r a t e  were a l l made o f q u a r t z .  the l i g h t e m i t t e d from the lamp i n t o a  narrow and i n t e n s e beam, hence a l e n s w i t h as s m a l l a f o c a l l e n g t h as p o s s i b l e was s e l e c t e d . through diameter  The l e n s  c o l l i m a t e d the beam b e f o r e i t passed  the p o l a r i z e r and the quarter-wave p l a t e and thus p e r m i t t e d the of the second stop to be made as l a r g e or l a r g e r than the f a c e o f  the p o l a r i z e r , which i n t u r n g r e a t l y i n c r e a s e d the i n t e n s i t y o f the l i g h t r e a c h i n g the monochromator. entrance  slit  The l e n s Lg was used to focus the beam on the  of the monochromator.  The f o c a l l e n g t h o f L  3  was such  that  the f-number of the l e n s was o n l y s l i g h t l y l a r g e r than the f-number of the monochromator. and  Thus e f f e c t i v e l y a l l o f the monochrometer's g r a t i n g was used  the d i s p e r s i o n of the instrument  approached i t s r a t e d v a l u e .  P o l a r i z e r and Quarter-Wave P l a t e  A. Glan-Thompson p o l a r i z e r , w i t h a t r a n s m i s s i v i t y of about 40% f o r a wavelength of 253.7 nm,  was  used i n the experiment.  The p o l a r i z e r c o u l d  be  set a t any angle i n the p l a n e normal to the o p t i c a l a x i s w i t h an a c c u r a c y of b e t t e r than 1°. The quarter-wave p l a t e was 253.7 nm wavelength.  The p l a t e was  that the r e l a t i v e phase s h i f t + m 2% where m =  designed s p e c i f i c a l l y f o r use at the a t r u e quarter-wave p l a t e i n the sense  between the two components was n/2  1, 2, 3..... .  T h i s p r o p e r t y was  and not it/2  n e c e s s a r y to ensure  that  the r e l a t i v e phase s h i f t never exceeded the f i n i t e coherence l e n g t h of the l i g h t , which i n t u r n ensured t h a t the coherence l e n g t h d i d not m a n i f e s t itself  i n the e x p e r i m e n t a l measurements.  Most of the i n t e n s i t y measurements r e q u i r e d o n l y the p o l a r i z e r and not the quarter-wave p l a t e .  T h e r e f o r e the quarter-wave p l a t e was  fixed  to a  p i v o t i n g mount which enabled i t to be moved i n and out of the l i g h t beam without r e a l i g n m e n t .  Measurements taken w i t h and w i t h o u t the quarter-wave  p l a t e were made c o m p a t i b l e by d e t e r m i n i n g the a t t e n t u a t i o n of the l i g h t the quarter-wave p l a t e .  I t was  found that the i n t e n s i t y of the l i g h t  by  was  d i m i n i s h e d by a f a c t o r of 1/1.14, r e g a r d l e s s of the a n g l e of the q u a r t e r wave p l a t e ' s f a s t a x i s .  Thus a l l measurements taken w i t h the quarter-wave  p l a t e i n p l a c e were m u l t i p l i e d by the f a c t o r 1.14 the measurements taken w i t h o u t the quarter-wave  b e f o r e b e i n g combined  plate.  with  - 98  -  Monochromator  The monochromator was in  the f i r s t  the entrance  order. slit  the i n t e n s i t y . sufficient  The  a Spex 1800,  253.7 nm  l i n e was  s e t at 140 \im and  This resulted  to exclude  w i t h a r a t e d d i s p e r s i o n of observed  the e x i t  slit  10A°/mm.  i n the 3rd o r d e r  with  set a t 500 p,m to maximize  i n a bandpass of about 1.7A°  which  a l l o t h e r d e t e c t a b l e l i n e s emitted by the  was source.  Detector  The  i n t e n s i t y of the l i n e ,  i s o l a t e d by the monochromator, was  by o b s e r v i n g the o s c i l l i s c o p e t r a c e of the output p l a c e d at the monochromator's e x i t  slit.  by  of a p h o t o m u l t i p l i e r  To reduce n o i s e the l i g h t  chopped to g i v e a s e r i e s of 2 msec, p u l s e s . n e c e s s a r i l y excluded  measured  the l o n g observed  was  S h o r t e r p u l s e l e n g t h s were  r i s e time of the  oscilliscope  trace. The  l i n e a r i t y of the d e t e c t o r system was  neutral density f i l t e r s range found  t e s t e d u s i n g a s e r i e s of  to produce a range of i n t e n s i t i e s encompassing  i n the experiment.  A least  squares  f i t to a s t r a i g h t  line  performed on the data w i t h a r e s u l t i n g l i n e a r - c o r r e l a t i o n c o e f f i c i e n t r = .998  f o r the 14 d a t a p o i n t s , (see Bevington,  l i n e a r i t y allowed  the h e i g h t of the o s c i l l i s c o p e  1969).  the was of  T h i s h i g h degree of  t r a c e to be taken  as  -  99  -  d i r e c t l y p r o p o r t i o n a l to the i n t e n s i t y over the i n t e n s i t y  range o f  interest.  E f f e c t o f the Magnetic F i e l d on the E x p e r i m e n t a l Apparatus  To minimize the e f f e c t of the magnetic f i e l d the  p h o t o m u l t i p l i e r , each of these d e v i c e s was  shield.  The magetic f i e l d  strength inside  on the l i g h t source and  surrounded by a u-metal  the p.-metal s h i e l d of the l i g h t  source was measured u s i n g an I n c r e m e n t a l Gaussmeter found t h a t the f i e l d of  was  as always l e s s  than .7% of the f i e l d  the magnet p o l e s , which, even f o r the l a r g e s t  experiment, would  l e a d to a n e g l i g i b l e  H a l l Probe.  I t was  at the c e n t r e  f i e l d s used i n the  Zeeman S p l i t t i n g of the s o u r c e  lines. The e f f e c t of the f i e l d removing the a b s o r p t i o n c e l l field  was  increased.  on the remainder of the system was  by  and m o n i t o r i n g the i n t e n s i t y as the magnetic  Even w i t h f i e l d s up to 4 KGauss t h e r e was  d e t e c t a b l e change i n the  tested  intensity.  no  -  100  -  I I I . 2 THE EXPERIMENTAL PROCEDURE  The He-Ne alignment l a s e r was p o s i t i o n e d to d e f i n e an o p t i c a l p a r a l l e l t o the s u r f a c e s  of the poles of the magnet and through the c e n t r e  of the gap between the p o l e s . light  A l l of the o p t i c a l i n s t r u m e n t s ,  source and the e n t r a n c e s l i t  centred  on the o p t i c a l a x i s u s i n g  l i g h t source were p l a c e d moved across  to the monochrometer, were the alignment l a s e r .  Initially  The l e n s L  on moveable mounts a l l o w i n g the l i g h t  were a d j u s t e d  u n t i l the f o c a l p o i n t of  was  on a r e g i o n of e x c i t e d gas i n the lamp which produced the most 253.7  i n c l u d i n g the  the o p t i c a l a x i s and the l e n s L^ to be moved a l o n g  These two d e v i c e s  axis  x  and the  source to be the a x i s . positioned intense  nm. The l e n s Ig was p l a c e d  manoeuvered to the p o i n t the maximum i n t e n s i t y .  on a moveable mount which enabled i t to be  i n the plane normal to the o p t i c a l a x i s which gave I t was  found that when the angle of the p o l a r i z e r  was changed, the p o s i t i o n of Lj which gave the maximum i n t e n s i t y a l s o changed due t o a d e f l e c t i o n of the beam by the p o l a r i z e r . effect  Although  this  c o u l d be s i g n i f i c a n t l y reduced by c a r e f u l l y p o s i t i o n i n g the p o l a r i z e r  i n I t s mount, i t c o u l d not be completely  eliminated.  Thus whenever the  p o l a r i z e r was r o t a t e d the l e n s 1^ was r e p o s i t i o n e d to g i v e the maximum intensity. In general This property  a monochromator p o l a r i z e s the l i g h t which i t d i s p e r s e s .  of the monochromator was s t u d i e d because i n the course of the  experiment i t was n e c e s s a r y to send l i g h t p o l a r i z e d a t d i f f e r e n t angles  into  - 101 -  the monochromator and the measured I n t e n s i t y of t h i s l i g h t would be a f f e c t e d if  the monochromator a c t e d as a p o l a r i z e r . To examine the e f f e c t ,  unpolarized light  from  the a b s o r p t i o n c e l l was removed so t h a t o n l y the  the source was i n c i d e n t on the p o l a r i z e r .  p o l a r i z e r was s e q u e n t i a l l y r o t a t e d through the 253.7 nm l i n e was measured.  I t was  180° w h i l e the max.  found  u n a f f e c t e d by the angle of the p o l a r i z e r .  t h a t the max.  The  I n t e n s i t y of  intensity  was  T h i s r e s u l t was g r e e t e d w i t h some  s c e p t i c i s m and the measurements were repeated f o r the 763.5 nm l i n e of argon.  In t h i s case the I n t e n s i t y v a r i e d by a f a c t o r of more than f o u r over  the 180° r o t a t i o n . to  S i n c e the p o l a r i z i n g e f f e c t  be h i g h l y frequency  polarizing effect, the r e s u l t  of a monochrometer i s known  dependent, w i t h f r e q u e n c i e s where t h e r e i s no  (see K.  R a b i n o v i t c h et a l . , 1965 and G.W.  f o r the 253.7 nm l i n e was regarded  S t r o k e , 1963),  as one of those r a r e i n s t a n c e s  i n e x p e r i m e n t a l p h y s i c s where nature c o n s p i r e s to l e s s e n the work of the experimenter. The x-y axes of the p o l a r i z e r were s e l e c t e d a r b i t r a r i l y s e t t i n g the x - a x l s a l o n g the f i e l d  r a t h e r than  as i n the t h e o r e t i c a l a n a l y s i s .  T h i s was  done because the o n l y method a v a i l a b l e f o r a l i g n i n g the axes was to f i n d the p o l a r i z e r angles which produced p o l a r i z e d by the gas c e l l to  the maximum and minimum i n t e n s i t y f o r l i g h t  i n the magnetic f i e l d ,  an a c c u r a c y of a few degrees.  I t was f e l t  and t h i s c o u l d o n l y be done  t h a t r a t h e r than i n t r o d u c i n g  t h i s f u r t h e r source o r e r r o r i t was s i m p l e r and more p r e c i s e to c a l c u l a t e the maximum and minimum i n t e n s i t y from  the a r b i t r a r y s e t of axes.  from e q u a t i o n (2-192) u s i n g the J o b t a i n e d  - 102 -  The  experiment r e q u i r e d the quarter-wave p l a t e , when i n p l a c e , t o be  o r i e n t e d such t h a t i t s f a s t a x i s was p a r a l l e l t o the d e s i g n a t e d the p o l a r i z e r . as f o l l o w s .  The most s e n s i t i v e method found f o r a c c o m p l i s h i n g  t h i s was  Another Glan-Thompson p o l a r i z e r ( P ) was p l a c e d a l o n g the 2  o p t i c a l a x i s j u s t past for  x - a x i s of  the second a p e r t u r e  stop.  With the monochromator s e t  the i n t e n s e l i n e o f the He-Ne l a s e r , and the quarter-wave p l a t e  p o s i t i o n e d out of the l a s e r beam, the p o l a r i z e r Pj^ i n F i g . 10 was r o t a t e d until  the measured i n t e n s i t y was minimized, i . e . the p o l a r i z e r s were  crossed. x-axis.  T h i s p o s i t i o n of the p o l a r i z e r  slow axes were a l i g n e d w i t h  I t was found t h a t u s i n g p o s i t i o n e d along Of course  the x - a x i s .  T h i s c o u l d o n l y occur  i f the f a s t  the pass d i r e c t i o n s of the two p o l a r i z e r s .  t h i s method the quarter-wave p l a t e axes c o u l d be  the x-y axes o f the p o l a r i z e r Pj^ t o w i t h i n 1 ° .  t h i s method cannot d i s t i g u i s h between the s i t u a t i o n where the  fast axis i s along  the x - a x i s and the s i t u a t i o n where the slow a x i s i s along  Returning  t o the equations  (3-1) i t i s e v i d e n t  e f f e c t o f t h i s u n c e r t a i n t y i s t o l e a v e the s i g n of l m J the handedness of the e l l i p t i c a l wave i n doubt.  and  as the  normal to the o p t i c a l a x i s u n t i l the  measured I n t e n s i t y was a g a i n a minimum.  of l i t t l e  designated  The quarter-wave p l a t e was then swung i n t o p o s i t i o n between the two  p o l a r i z e r s , and r o t a t e d i n the plane  and  was h e n c e f o r t h  x y  t h a t the o n l y  , or e q u i v a l e n t l y  I t was f e l t  that t h i s was  importance and no attempt was made t o d i s t i n g u i s h getween the f a s t  slow axes of the quarter-wave p l a t e . As a check on the e x p e r i m e n t a l  technique  a mock experiment was  performed u s i n g the Glan-Thompson p o l a r i z e r P~ which was known to produce  - 103  highly linearly polarized l i g h t . the p o l a r i z e r P c e l l was  2  was  -  With no  c u r r e n t f l o w i n g i n the magnet,  p l a c e d between the p o l e s of the magnet w h i l e  p l a c e d between the second a p e r t u r e  c e r t a i n t h a t the r e s i d u a l f i e l d the c e l l was  stop S  complete s e t of measurements (3-1) = 1.00  ±  .03  and  of the magnet would not  surrounded by a p.-metal s h i e l d .  the data that P  2  was  the  the l e n s L .  To  affect  gas,  2  the Hg  With t h i s arrangement  taken.  I t was  gas  found a f t e r  be  the  analyzing  and  CO  J  xy  = -.02  (an e x p l a n a t i o n of the e r r o r e s t i m a t e s accuracy  of the experiment the  polarized. performing the gas  This r e s u l t  The  o p t i c s , was  initally  the gas  c e l l , which was  uniform  magnetic f i e l d  x - a x i s as i n ( 3 - l a ) and height  l i g h t was  found to be completely  entirely  light  of known s t r e n g t h . the  l i n e was  the instruments  from the source temperature and  trace corresponding  the i n t e n s i t y was  of measurements had the p o l a r i z e r and  passed  The  by  immersed i n a  to the t r a n s m i t t e d  c u r r e n t i n the magnet was  a g a i n measured.  been taken c o v e r i n g f i e l d s  had  as  through  the  l e n s Lj p o s i t i o n e d f o r maximum i n t e n s i t y ,  recorded.  was  positioned  With the p o l a r i z e r a l o n g  lowered to the next s e t t i n g at which the magnetic f i e l d measured and  system  negligible.  performed w i t h  unpolarized  the  linearly  d e p o l a r i z a t i o n of the l i g h t ,  h e l d at a constant  of the o s c i l l i s c o p e  of the 253.7 nm  Thus to w i t h i n  i n d i c a t e d not o n l y that the e x p e r i m e n t a l  a c t u a l experiment was  i n F i g . 10.  .13  i s given l a t e r ) .  as expected, but a l s o t h a t any  or I n t e r v e n i n g  The  ±  the  intensity then  p r e v i o u s l y been  A f t e r a r e p r e s e n t a t i v e set from about 250  the quarter-wave p l a t e were s e t at the next  G to 3500 G, combination  - 104  d e s c r i b e d by (3-1) and settings. complete  T h i s was  the p r o c e s s was  r e p e a t e d f o r the same magnetic  field  done f o r the f o u r combinations d e s c r i b e d by ( 3 - 1 ) .  The  s e t of measurements p r o v i d e d a l l of the i n f o r m a t i o n n e c e s s a r y f o r  d e t e r m i n i n g J as a f u n c t i o n of f i e l d problems  -  s t r e n g t h at a g i v e n temperature.  c r e a t e d by the u n s t a b l e l i g h t  source are c o n s i d e r e d below.  The U n s t a b l e L i g h t  The I n s t a b i l i t y of the l i g h t  Source  source a f f e c t e d  the e x p e r i m e n t a l procedure  in  a number of ways.  of  i n s t a b i l i t y o c c u r i n g d u r i n g the t a k i n g of a g i v e n set of measurements,  the  First,  The  number of measurements i n the set had to be l i m i t e d  complete  set to be taken i n about 3 h o u r s .  end and p e r i o d i c a l l y light  the  source had remained  measurable  at a f i x e d s e t t i n g was  Finally,  at the b e g i n n i n g , the  measured to ensure t h a t the i n t e n s i t y of  c o n s t a n t throughout the set of measurements.  change i n t h i s i n t e n s i t y was  measurements was  Second,  to permit the  d u r i n g the course of the experiment, the i n t e n s i t y of  the  not  i n o r d e r to reduce the p r o b a b i l i t y of a p e r i o d  observed the complete  When a  s e t of  rejected.  i t might  appear t h a t because  the magnetic  e x a c t l y r e p r o d u c i b l e , g r e a t e r p r e c i s i o n would  measurements (3-1) were taken b e f o r e the f i e l d  s e t t i n g s were  r e s u l t i f a l l four  s t r e n g t h was  However, because a s i g n i f i c a n t p e r i o d of time was p o l a r i z e r and the quarter-wave  field  changed.  r e q u i r e d to r e s e t  the  p l a t e and to r e a l i g n L j , the time n e c e s s a r y  -  105 -  to complete a s e t of measurements would be g r e a t l y  increased i f t h i s  approach were adopted and the i n s t a b i l i t y i n the i n t e n s i t y o f the source would b e g i n t o I n f l u e n c e the r e s u l t s ,  thereby n e g a t i n g any g a i n i n  precision. In a d d i t i o n  t o the experiment u s i n g an u n p o l a r i z e d l i g h t s o u r c e , an  attempt was made t o study incident gas  l i g h t beam produced by i n s e r t i n g  cell.  polarizer further  the e f f e c t o f the gas on a l i n e a r l y P  2  i n front  o f the  U n f o r t u n a t e l y , t h i s attempt was u n s u c c e s s f u l because the P  2  had a low t r a n s r a i s s i v i t y and w i t h the i n t e n s i t y of the l i g h t  d i m i n i s h e d by the p o l a r i z i n g  resulting  the p o l a r i z e r  polarized  p r o p e r t i e s of the gas c e l l the  i n t e n s i t y was too weak t o p r o v i d e a meaningful  measurement.  intensity  -  106 -  I I I . 3 EXPERIMENTAL ERROR  The with  experimental  i n t e n s i t y measurements c o u l d n o t , of course,  absolute p r e c i s i o n .  measurements p r i n c i p a l l y  Experimental  be made  e r r o r was i n t r o d u c e d i n t o the  from three s o u r c e s .  First,  there was e r r o r i n  r e a d i n g the h e i g h t of the o s c i l l i s c o p e t r a c e , i n p a r t because of the i n h e r e n t i m p r e c i s i o n of such a r e a d i n g , but p r i m a r i l y because the background n o i s e of the p h o t o m u l t i p l i e r caused the h e i g h t of the t r a c e t o f l u c t u a t e . Second, the r e p o s i t i o n i n g o f the equipment between each o f the f o u r of measurements d e s c r i b e d by equations alignment  of the equipment.  (3-1) u n a v o i d a b l y  series  p e r t u r b e d the  F i n a l l y , because of the e f f e c t s of h y s t e r e s i s ,  the same electrogmagnet c u r r e n t s e t t i n g s may not have produced p r e c i s e l y the same magnetic f i e l d It i s d i f f i c u l t  strengths  f o r each o f the f o u r measurements ( 3 - 1 ) .  t o determine d i r e c t l y  the impact of each of these  f a c t o r s on the magnitude of the e r r o r i n the I n t e n s i t y measurements. g e n e r a l , the best e s t i m a t e  of the e x p e r i m e n t a l  e r r o r would be o b t a i n e d by  r e p e a t i n g the complete s e t o f measurements u n t i l a s t a t i s t i c a l l y sample i s c o l l e c t e d , from which the standard each p o i n t c o u l d be c a l c u l a t e d .  In  meaningful  d e v i a t i o n i n the i n t e n s i t y at  T h i s approach was, however, unworkable i n  t h i s experiment because the time r e q u i r e d t o c o l l e c t a l a r g e sample would g r e a t l y exceed the time p e r i o d d u r i n g which the l i g h t reasonably drift  stable.  source  remained  I f the o n l y consequency of the lamp's i n s t a b i l i t y was a  i n i n t e n s i t y i t would have been p o s s i b l e to r e n o r m a l i z e  the r e s u l t s  -  before was  comparing them.  -  107  However, the observed change i n the  accompanied by a change i n the  l i n e shape of the s o u r c e .  a v a r i a t i o n i n the measurements which was imprecision  lamp's i n t e n s i t y  unrelated  of the e x p e r i m e n t a l t e c h n i q u e .  to the  This  produced  actual  Thus, to a c h i e v e a b e t t e r  e s t i m a t e of the a c t u a l e r r o r i n the e x p e r i m e n t a l measurements an a l t e r n a t i v e approach was  adopted.  each magnetic f i e l d o n l y a s m a l l but  Instead  of t a k i n g a l a r g e number of measurements at  s e t t i n g , a l a r g e number of measurements was  representative  group of f i e l d  settings.  complete set of measurements, these measurements c o u l d be s u f f i c i e n t l y short  p e r i o d of time to a v o i d  i n t e n s i t y and  shape of the  line  source.  Unlike  taken at the  taken i n a  the consequences of the  Thus, the  standard  i n t e n s i t y measurement at each of these r e p r e s e n t a t i v e  d e v i a t i o n In  s e t t i n g s can  as t r u l y i n d i c a t i v e of the magnitude of the e r r o r i n the  changing  be  the  taken  experimental  measurements. When the standard c a l c u l a t e d i t was the  light.  I t was  d e v i a t i o n i n the  found not  to be  simply  p r o p o r t i o n a l to the  i n s t e a d found to be v i r t u a l l y constant  taken at the same o s c i l l i s c o p e v o l t a g e of the  i n t e n s i t y measurements  trace varied considerably.  to observe l e s s I n t e n s e s i g n a l s ) .  of the  f o r measurements  s c a l e s e t t i n g was  height  d e v i a t i o n was decreased  found  ( i n order  T h i s r e s u l t suggests t h a t the p r i n c i p a l  cause of e r r o r i n the measurements was d e t e r m i n i n g the h e i g h t  i n t e n s i t y of  s c a l e s e t t i n g , even when the  However, the standard  g e n e r a l l y to i n c r e a s e when the v o l t a g e  was  the i n h e r e n t  imprecision  o s c i l l i s c o p e trace against  in  the background  -  noise,  since  108  -  the n o i s e becomes more pronounced as p r o g r e s s i v e l y  lower  scale  s e t t i n g s are used. S i n c e the magnitude of the e r r o r remained r e a s o n a b l y c o n s t a n t f o r a given voltage  s c a l e s e t t i n g , i t was p o s s i b l e  e s t i m a t e of the e r r o r  to p r o v i d e a r e a s o n a b l e  i n the i n t e n s i t y by a s c r i b i n g the same a b s o l u t e  error  a , to a l l measurements taken at the same s c a l e s e t t i n g , ( t h e s u b s c r i p t identifies  the i n t e n s i t y measurement a s s o c i a t e d  this ascribed the  w i t h the e r r o r a , ) .  e r r o r which was used i n a l l subsequent c a l c u l a t i o n s  i  I t was  involving  e x p e r i m e n t a l l y measured i n t e n s i t i e s . The e r r o r i n the i n t e n s i t y measurements a,, leads to an e r r o r i n a l l o f  the  q u a n t i t i e s , such as P, J , and J , that are c a l c u l a t e d ' xx yy  measured i n t e n s i t i e s . the  In g e n e r a l ,  each of these q u a n t i t i e s  from the  i s a function of  f o u r measured i n t e n s i t i e s g i v e n i n equations (3-1) and t h e r e f o r e  may be  r e p r e s e n t e d by: f(I  L  .Ij  ,13 ,1^ ) = f ( I )  Since there i s no c o r r e l a t i o n i n the e r r o r s a. I n t e n s i t y measurement I quantity  the  (3-2)  ±  associated  w i t h each  e r r o r i n the e x p e r i m e n t a l l y determined  a , i s g i v e n by:  4 °f  2  =  i =l Z  5f(I  *l  x  ,L, ,13  ,1^) (3-3)  - 109  (See B e v l n g t o n , 1969, The  p.  -  58.)  e r r o r i n each e x p e r i m e n t a l l y determined  next s e c t i o n was  found from t h i s e q u a t i o n .  q u a n t i t y presented i n the  - 110 -  I I I . 4 EXPERIMENTAL RESULTS  In t h i s s e c t i o n  the e x p e r i m e n t a l r e s u l t s a r e p r e s e n t e d i n a form which  p e r m i t s a d i r e c t comparison  with the t h e o r e t i c a l  The e x p e r i m e n t a l coherency equations  results.  m a t r i x J , as i n i t i a l l y  determined  ( 3 - 1 ) , can not be d i r e c t l y compared to the t h e o r e t i c a l  from the coherency  m a t r i x because the e x p e r i m e n t a l x - a x i s i s not a l o n g the d i r e c t i o n of the magnetic  field.  However, the two m a t r i c e s can be made compatible by  performing a t r a n s f o r m a t i o n from the e x p e r i m e n t a l axes ( x ' , y ' ) . field,  The x' a x i s  i s aligned  (x,y) axes t o a new s e t o f  w i t h the d i r e c t i o n o f the magnetic  and forms an angle <)) w i t h r e s p e c t t o the x - a x i s .  m a t r i x J t r a n s f o r m s under t h i s r o t a t i o n expressed  The  coherency  i n t o the m a t r i x J ' which i s r e a d i l y  i n terms of the m a t r i x elements  o f J and the angle <j>, (see Born  and Wolf, 1975, P. 548).  J  xx  C  2  + J  yy  S  2  + (J + J )CS xy yx  (J  - J  yy  xx  )CS + J  C xy  2  - J S yx  2  (3-4) (J  yy  - J  xx  )CS + J  where C = cos $  The  elements  yx  C  2  - J s xy  2  J  xx  S  2  + J  C yy  2  - (J + J )CS xy yx  S = s i n <j>  of J ' a r e d i r e c t l y comparable to the t h e o r e t i c a l  m a t r i x once the a n g l e <j> i s determined.  coherency  - I l l-  The angle $ may be determined by n o t i n g that when the i n c i d e n t l i g h t i s u n p o l a r i z e d and the x ' - a x i s  i s aligned with  r e q u i r e s the coherency m a t r i x (see e q u a t i o n  the magnetic f i e l d  o f the t r a n s m i t t e d  light  the theory  t o be of the form  2-180) J' xx  0 (3-5) J' yy  That i s , the t r a n s m i t t e d  l i g h t may only c o n s i s t of an u n p o l a r i z e d  and/or a l i n e a r l y p o l a r i z e d component w i t h along e i t h e r the x' or y' a x i s . matrix  (3-4) by s e t t i n g  = J  When t h i s i s done i t i s found  <)> = a r c t a n {—)  the d i r e c t i o n o f p o l a r i z a t i o n  Thus, the angle $ may be found from the y  x  = 0 and s o l v i n g the two e q u a t i o n s  (2-174).  f o r <J>.  that:  o r <J> + n/2 = a r c t a n (—J  where B and C a r e c a l c u l a t e d from the e x p e r i m e n t a l (with unpolarized  component  (3-6)  i n t e n s i t y measurements,  i n c i d e n t l i g h t ) u s i n g the equations  (3-1),  (2-173) and  A c h o i c e can be made between the two p o s s i b l e s o l u t i o n s by  comparing them w i t h  the observed o r i e n t a t i o n o f the e x p e r i m e n t a l  r e s p e c t t o the magnetic  field.  x-axis  with  - 112 -  The f o r e g o i n g method f o r d e t e r m i n i n g only v a l i d  i f the t r a n s m i t t e d  light  <}>, (and hence J') i s of  course  i s found to c o n s i s t e n t i r e l y of an  u n p o l a r i z e d component and/or a l i n e a r l y p o l a r i z e d component as p r e d i c t e d by the  theoretical analysis.  F o r , i f the t r a n s m i t t e d  light  i s found to c o n t a i n  an e l l i p t i c a l l y  p o l a r i z e d component, t h a t i s i f Im(J ) * 0, then from the xy (3-4) J ' t 0 f o r a l l $ and the method i s i n a p p l i c a b l e . Thus, b e f o r e xy  matrix  the suggested technique p r e d i c t i o n that Im(J  xy  f o r f i n d i n g $ can be used, the t h e o r e t i c a l  ) = 0, ( f o r u n p o l a r i z e d  experimentally  verified.  magnetic f i e l d  s e t t i n g u s i n g the e x p e r i m e n t a l  equations Fig.  (3-1) and ( 3 - 3 ) .  To t h i s end, Im(J ) was c a l c u l a t e d a t each xy  The r e s u l t  i n t e n s i t y measurements and the  f o r a gas at 22.6°C i s p l o t t e d i n  12. It  i s evident  from F i g . 12 t h a t Im ( J ^ ) Is zero a t each magnetic  s e t t i n g to w i t h i n e x p e r i m e n t a l average value of I m ( J field  i n c i d e n t l i g h t ) , must be  x y  error.  However, i t Is a l s o e v i d e n t  ) i s not z e r o .  In f a c t i f  i s averaged over a l l  - 2  ± 1.7 x 1 0  - 2  Thus, i t would appear t h a t t h e r e i s a s m a l l e l l i p t i c a l t r a n s m i t t e d wave.  e r r o r which i s Introduced  c a l c u l a t i n g Im(J ), the measurement I xy the constant  component i n the  T h i s s m a l l component can, however, be  accounted f o r by the s y s t e m a t i c  [see equations  completely when, i n  (3-1)] i s m u l t i p l i e d  f a c t o r G = 1.14 I n o r d e r to compensate f o r the a t t e n u a t i o n  of the l i g h t by the quarter-wave p l a t e . ±  t h a t the  s e t t i n g s , i t i s found to be: -8.8 x 1 0  by  field  0.04 and, u s i n g e q u a t i o n  The e r r o r a  n  i n determining  ( 3 - 3 ) , t h i s produces a s y s t e m a t i c  G is  error i n  -1.2 -1.0 -i0.8 J  IM(JXY) IN RRBITRRRY UNITS -0.6 L _  -0.4 _!  -0.2 I  0.0 I  0.2 _i  Q.i  0.6 i  0.8  _j  ft)  CO  c 3 o o 3 2  CL rr r»  (0 3  OP  - £IT -  1.0  _i  - 114  I i n C J ^ ) of about elliptical  0.12  -  f o r a t y p i c a l value of 1^ .  Thus, the  apparent  component can be a t t r i b u t e d e n t i r e l y to t h i s s y s t e m a t i c e r r o r  the t h e o r e t i c a l p r e d i c t i o n t h a t Im(J  ) = 0 for a l l f i e l d  and  strengths i s  xy verified equations was  to w i t h i n e x p e r i m e n t a l e r r o r .  This result  j u s t i f i e s using  (3-6) to determine $ i n the experiment.  The average  v a l u e of $  found to be 86.2°±0.8°  For the purposes theoretical results unpolarized incident P,  of comparing the e x p e r i m e n t a l r e s u l t s w i t h  the q u a n t i t i e s of primary i n t e r e s t , light),  are  (where a g a i n the s u p e r s c r i p t »  q u a n t i t i e s were determined  , J  y  y  the  (when c o n s i d e r i n g  and the degree  of  has been omitted from P°°).  polarization These  from the e x p e r i m e n t a l measurements i n the  f o l l o w i n g ways. The and  degree  of p o l a r i z a t i o n P, i s independent  t h e r e f o r e was  coherency  s i m p l y determined  of the c h o i c e of x,y  from the elements  m a t r i x J , which were i n t u r n determined  From e q u a t i o n  axes  of the e x p e r i m e n t a l  from e q u a t i o n s  (3-1).  (2-195):  -ri P-Ll P  -  4(J  J' and J ' can be c a l c u l a t e d xx yy  J - J J ) xx yy xy y x (J—+ J )2 xx yy  y  1/2 -i  J  from the m a t r i x (3-5) by  s u b s t i t u t i o n , once the angle 4> and the m a t r i x J have been  simple  determined.  I n s t e a d , however, J ' and J ' were c a l c u l a t e d u s i n g a set of e q u a t i o n s i n ' ' xx yy n  - 115  which the v a l u e s of J  xx  -  ' and J ' a r e not d i r e c t l y dependent upon the yy  c a l c u l a t e d v a l u e o f ty. T h i s s e t o f equations i s a g a i n based on the f a c t that Im(J ) = 0 a t a l l f i e l d xy By  s t r e n g t h s which i m p l i e s t h a t J ' = J ' = 0. xy yx  f u r t h e r n o t i n g t h a t the t r a c e and the determinant  i n v a r i a n t under r o t a t i o n a l  of the m a t r i x J ' a r e  t r a n s f o r m a t i o n s the d e s i r e d s e t o f e q u a t i o n s can  be r e a d i l y d e r i v e d from the m a t r i x J ' .  J' = (J + J ) - A xx xx yy  and J ' = A yy  when  (3-7) C  and  = 0  1  and B'  * 0  that: J' = A xx  and J ' = ( J + J ) - A yy xx yy  when B  1  =0  and  0  where A i s g i v e n by e q u a t i o n (2-172) and B' and C  a r e found from  equations  (1-173) and (2-174) and the m a t r i x J ' which i s found from the c a l c u l a t e d values of ty and J . I t i s c l e a r from these equations that the e r r o r i n d e t e r m i n i n g ty o n l y affects  the p r e c i s i o n w i t h which B' and C  are determined  and these v a l u e s  are o n l y used  to choose between the two p o s s i b l e s o l u t i o n s f o r J ' and J ' . xx yy Thus, when e q u a t i o n s (3-7) were used t o c a l c u l a t e the e x p e r i m e n t a l v a l u e s o f J' and J ' , the u n c e r t a i n t y i n J ' and J ' was not a f f e c t e d by the xx yy xx yy J  u n c e r t a i n t y i n d e t e r m i n i n g ty .  - 116 -  The e x p e r i m e n t a l v a l u e s of J temperatures of 22.6°C In the next comared.  xx  J  yy  ' and P f o r the two gas  and 30°C are p l o t t e d  i n F i g . 13, 14, and  15.  c h a p t e r the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s are  o  MAGNETIC FIELD STRENGTH (KGRUSS)  - 120  -  CHAPTER IV COMPARING THEORY AND  When the 14,  15)  are  e x p e r i m e n t a l curves of  compared w i t h the  (given i n F i g s . differences  7,  8,  and  between the  certain characteristics  9)  This result the  single,  calculations  even i s o t o p e of mercury, H g  experiment was  an  incident  matching the calculation  o n l y be  to c a r r y  theoretical examined. f o r the  the  calculation  In p a r t i c u l a r ,  p r o p e r t i e s of  the  differences and  the  the  the  I = 0), while of  the  7 isotopes,  theoretical  an  theoretical  Incident l i n e  However, i n p r a c t i c e  performed w i t h l i m i t e d p r e c i s i o n  profile,  experiment are  and  calculation profile this no  complex  attempt  was  calculation. between the  characteristics  I t i s then suggested that  theory and  out  mercury and  made i n t h i s study to perform such a In t h i s s e c t i o n  certain  Furthermore, w h i l e the  e x p e r i m e n t a l source l i n e . can  since  a much more complex shape.  In p r i n c i p a l i t i s p o s s i b l e p r o p e r t i e s of n a t u r a l  u s i n g the  yy  unexpected  accord with curves.  and  0 0  source l i n e w i t h a g u a s s i a n l i n e  e x p e r i m e n t a l source l i n e had  u s i n g the  not  J  13,  significant  mercury which c o n s i s t s  a unique spectrum.  curves were based on  xx  , and  , (with nuclear spin  2 0 2  performed u s i n g n a t u r a l  each of which has  the  out  r o  there are  theoretical  were c a r r i e d  (given i n F i g s .  i s , however, not  experiment do the  P,  curves f o r J  i t i s e v i d e n t that  two. of  and  theoretical  assumptions made i n c a l c u l a t i n g theoretical  EXPERIMENT  assumptions i n of  the  experiment  when these d i f f e r e n c e s  in qualitative  the  agreement.  are  are  accounted  - 121 -  IV.1  The E f f e c t o f U s i n g  N a t u r a l Mercury i n the A b s o r p t i o n  Cell  When the spectrum of n a t u r a l mercury i s examined the 253.7 nm l i n e i s found  t o c o n s i s t o f a s e r i e s of very c l o s e l y spaced l i n e s ,  1963).  T h i s complex l i n e  structure  df  (Schweitzer,  spectrum, r e s u l t s from superposing  the h y p e r f i n e  the v a r i o u s i s o t o p e s o f mercury which a r e present  i n natural  mercury. Different  i s o t o p e s of the same element have d i f f e r e n t  s p e c t r a because  the energy l e v e l s o f an atom a r e a f f e c t e d not only by the n u c l e a r charge but a l s o by other n u c l e a r c h a r a c t e r i s t i c s ,  such as the n u c l e a r a n g u l a r momentum  or s p i n , I , the n u c l e a r mass and the n u c l e a r charge d i s t r i b u t i o n .  The  e f f e c t of these n u c l e a r c h a r a c t e r i s t i c s i s examined i n d e t a i l i n Kuhn, 1969 at page 329 and i n Sobel'man, 1972 a t page 204. F o r the l i m i t e d purpose o f e x p l a i n i n g the r e s u l t s o f the experiment i t i s s u f f i c i e n t that the c h i e f e f f e c t  simply  t o note  o f the d i f f e r e n c e i n n u c l e a r mass and charge  d i s t r i b u t i o n between i s o t o p e s o f the same element i s a s m a l l s h i f t f r e q u e n c i e s o f the l i n e s p e c t r a . b r i e f l y considered  i n the  The e f f e c t o f the n u c l e a r s p i n I i s  below.  E f f e c t o f Nuclear  Spin  Isotopes w i t h an odd mass number have a h a l f - i n t e g r a l n u c l e a r s p i n , while  i s o t o p e s w i t h an even mass number have an i n t e g r a l n u c l e a r s p i n .  Isotopes w i t h an even number of protons  and neutons have zero n u c l e a r s p i n .  - 122 -  The n u c l e a r s p i n g i v e s r i s e  to a n u c l e a r magnetic moment, p.^, which i s  of the o r d e r of 1000 times s m a l l e r than the Bohr Magneton. magnetic moment i n t e r a c t s w i t h the magnetic f i e l d  The n u c l e a r  caused by the o r b i t a l  motion of the e l e c t r o n s , and t o a l e s s e r e x t e n t w i t h the magnetic generated by the e l e c t r o n s p i n .  field  T h i s i n t e r a c t i o n leads to a s h i f t  i n the  energy l e v e l s , A E ^ , which i s g i v e n by:  A E j = A/2 (F(F+1) - 1(1+1) - J(J+1))  (4-1)  where A i s the magnetic d i p o l e i n t e r a c t i o n c o n s t a n t , F i s the t o t a l a n g u l a r momentum of the atom and i s produced by t h e c o u p l i n g of I and J i n complete analogy w i t h the c o u p l i n g o f S and L t o produce J .  The quantum v a l u e s of F run from | j + l | to | j - l | .  A quadrupole c o u p l i n g which courses a s m a l l e r s h i f t l e v e l s has been i g n o r e d .  i n the energy  I t f o l l o w s from e q u a t i o n (4-1) that i s o t o p e s where  1*0 w i l l e x h i b i t a h y p e r f i n e s p l i t t i n g o f the s p e c t r a l l i n e s w h i l e i s o t o p e s where 1 = 0 , w i l l  e x h i b i t no h y p e r f i n e  splitting.  With these r e s u l t s , the s t r u c t u r e of the 253.7 nm l i n e d e p i c t e d i n S c h w e r i t z e r ' s paper can be u n d e r s t o o d . Hg  2 0 2  ,  Hg  2 0 0  and H g  1 9 8  The even i s o t o p e s of mercury  produce 253.7 nm l i n e s a t s l i g h t l y  Hg  2 0 4  ,  different  f r e q u e n c i e s because of the d i f f e r e n t n u c l e a r shapes, b u t , s i n c e 1=0 f o r these i s o t o p e s , t h e i r l i n e s do not e x h i b i t any h y p e r f i n e s p l i t t i n g . isotopes, H g  1 9 9  and H f  2 0 1  a l s o produces a frequency s h i f t e d  The odd  253.7 nm l i n e ,  - 123  but, s i n e 1*0  f o r these  d e s c r i b e d by e q u a t i o n  isotopes, their  Zeeman Components h a v i n g  AE^,  i s given  (8  of  the l i n e s It  field,  aj  The  into  first  order  ^0  V  (  4  _  2  )  i s no  splitting  longer v a l i d ,  and  , -F  exceeds the h y p e r f i n e the s p e c t r a l l i n e  (2-9)  and  I t i s important Is simply  (2-9)  and  Into the u  components i s i n  as a r e s u l t  to note that when 1=0,  governed by e q u a t i o n  splitting  splits  each of these  into several equidistant hyperfine lines  spin.  splits  values  and o-components g i v e n by e q u a t i o n  nuclear  which i s  by:  2F(F+1)  When the magnetic f i e l d  turn s p l i t  spliting  each h y p e r f i n e l i n e  m = F, F - l , ...  (4-2)  splitting  - F(F+1) + J(J+1) - 1(1+1) Z  equation  line  a d e f i n i t e s t a t e of p o l a r i z a t i o n .  change i n the energy l e v e l s ,  the  exhibit hyperfine  (those producing  s m a l l e r than the h y p e r f i n e s p l i t t i n g ) ,  where m has  lines  (4-1).  I n weak magnetics f i e l d s ,  AF  -  of  the Zeeman  the  splitting  not e q u a t i o n  (4-2).  f o l l o w s t h a t when n a t u r a l mercury i s immersed i n a weak magnetic the 253.7 nm  lines  more complex s p l i t t i n g  of the odd  isotopes w i l l independently  d e s c r i b e d by e q u a t i o n  s p l i t t i n g makes i t d i f f i c u l t  (4-2).  T h i s complex  exhibit  a  lines  to d e s c r i b e the r e s u l t i n g a b s o r p t i o n spectrum  - 124  with p r e c i s i o n , p a r t i c u l a r l y relatively  weak f i e l d  IV.2  The  light  source used argon.  The  J' i n qualitative yy  i n the experiment While  s p l i t t i n g of  was  a mercury lamp f i l l e d an i n t e n s e 253.7  some of the l i g h t  The  253.7 nm  l i n e produced  which i n the 5th o r d e r had Unfortunately, this  the lamp would be  with  nm  at the c e n t r e of the lamp's gas  to produce a h i g h l y s e l f - r e v e r s e d l i n e  the  Incident Line  to t r a v e r s e a volume of c o l d gas b e f o r e being e m i t t e d from  As the c o l d gas would r e - a b s o r b  the  terms.  the lamp produced  produced  at  to d e s c r i b e the shape of  Shape o f the E x p e r i m e n t a l  l i n e , most of the l i g h t was and had  i n s e c t i o n IV.3  f o r J ' and xx  The  n a t u r a l mercury and  s i n c e e q u a t i o n (4-2) becomes i n v a l i d  s t r e n g t h s of about 1 KGauss.  l i n e s w i l l , however, be used e x p e r i m e n t a l curves  -  bulb  the lamp. expected  profile.  by the lamp was  a reciprocal linear  examined i n a  spectrograph  d i s p e r s i o n of 400 mA/mm.  d i d not p r o v i d e s u f f i c i e n t l y h i g h r e s o l u t i o n t o  the h y p e r f i n e s t r u c t u r e or the d e t a i l e d shape of the l i n e . p o s s i b l e to e s t i m a t e t h a t the width of the l i n e was  I t was,  between 70 and  observe however, 85  mA.  S i n c e Schweitzer's work shows t h a t the width of the h y p e r f i n e s t r u c t u r e i n n a t u r a l mercury i s l e s s than 50 mA measured l i n e width  i t may  be assumed from the s i z e of  t h a t each componet l i n e was  sufficiently  a c o n t i n u i o u s l i n e p r o f i l e marked by narrow troughs produced r e - a b s o r p t i o n of the c o l d In curves  the next  by  to c r e a t e  the  gas.  s e c t i o n t h e f f e c t of t h i s  f o r J ' and xx  broad  the  J ' i s examined, yy  line profile  on the  experimental  - 125  IV.3  The  e x p e r i m e n t a l and  The  -  Experimental  Results  t h e o r e t i c a l r e s u l t s are i n broad agreement when  the d i f f e r e n c e s between the t h e o r e t i c a l assumptions and  the  experimental  c o n d i t i o n s are c o n s i d e r e d . In  the t h e o r e t i c a l c u r v e s , F i g s . 7, 8 and  magnetic f i e l d J  x x  by  9  i s independent  of  s t r e n g t h w h i l e i n the e x p e r i m e n t a l c u r v e s , F i g s . 13 and  v a r i e s with f i e l d the presence  strength.  of odd  14,  T h i s d i f f e r e n c e i s , however accounted  for  i s o t o p e s of mercury i n the e x p e r i m e n t a l but not  the  t h e o r e t i c a l a b s o r p t i o n gas. I t was  shown i n m a t r i x  a f f e c t e d by the s t r e n g t h and  (2-180) t h a t the magnitude of J  y  y  i s only  frequency of the a-components of the a b s o r b i n g  gas, w h i l e the magnitude of  i s o n l y a f f e c t e d by the s t r e n g t h and  frequency  The  of the n-components.  nuclear spin,  (see the appendix) and a non-degenerates ground s t a t e  t h e r e f o r e , from e q u a t i o n s 253.7 nm  even i s o t o p e s of mercury have z e r o  l i n e are f i e l d  (2-129) and  dependent.  (2-130),  o n l y the o-components of  Thus, when the a b s o r p t i o n gas  e n t i r e l y of even i s o t o p e s of mercury, as i n the t h e o r e t i c a l o n l y J ' should be f i e l d yy In  c o n t r a s t , odd  mercury used field  dependent f o r the 253.7 nm  the  consist  calculation,  line.  i s o t o p e s of mercury which were present i n the n a t u r a l  i n the experiment,  dependent.  and  produce both tt and a-components which are  T h i s i s because odd  i s o t o p e s of mercury have a mon-zero  n u c l e a r s p i n , I , (see the appendix) and  t h i s produces a degenerate  s t a t e w i t h a non-zero a n g u l a r momentum, F.  The  Zeeman s p l i t t i n g of  ground the  - 126 -  253.7 nm l i n e  i s governed by e q u a t i o n  examined f o r a l l a l l o w e d  transitions,  it-components a r e f i e l d dependent. odd  (4-2).  When e q u a t i o n  i t i s found t h a t some of the  Thus, when the a b s o r p t i o n gas c o n t a i n s  i s o t o p e s both J ' and J ' a r e f i e l d dependent. xx yy While i t i s c l e a r  t h a t the presence o f odd i s o t o p e s i n the e x p e r i m e n t a l  a b s o r p t i o n gas l e a d s t o the f i e l d dependence o f for  i n Figs.  13 and 14 can not be v e r i f i e d  i s because the dependence of J ^ the  (4-2) i s  line  shape of the i n c i d e n t  x  calculation.  This  source  line,  and i n the experiment t h i s  line  The g e n e r a l shape o f the J ' xx  can, however, be e x p l a i n e d .  Referring 1963)  by d i r e c t  on f i e l d s t r e n g t h depends c r i t i c a l l y upon  shape c o u l d not be determined w i t h p r e c i s i o n .  curves  the shape of the curves  t o the e m i s s i o n  and F i g . 13, the i n i t i a l  spectrum of n a t u r a l mercury decrease  in  with  (Schweitzer,  increasing  field  s t r e n g t h i s caused by the s p l i t t i n g  o f the it-component a b s o r p t i o n away from  the c e n t r e s of the a b s o r p t i o n l i n e s  of the odd i s o t o p e s , H g  This s p l i t t i n g source J  x  x  line  2 0 1  and hence a decrease  due t o t h r e e e f f e c t s .  absorption lines reducing  i n the magnitude o f  First,  a b s o r p t i o n o f the it-component o f the source  again reducing  line.  Finally,  line, line.  t o the same  a b s o r p t i o n of the it-component o f the  the t o t a l  i n stronger  i n F i g . 13 i s  some of the it-components of the  Second, some of the it-component a b s o r p t i o n l i nes w i l l s h i f t  source  .  The i n c r e a s e i n  w i l l be s p l i t beyond the width of the source  the t o t a l  frequency,  1 9 9  l e a d s t o an i n c r e a s e d a b s o r p t i o n of the n-component of the  a t a f i e l d s t r e n g t h of about 2.4 K Gauss which i s e v i d e n t  likely  and H g  fields  breaks down and the Zeeman s p l i t t i n g  the c o u p l i n g between J and I  i s no l o n g e r d e s c r i b e d by e q u a t i o n  - 127  (4-2).  At these h i g h e r  field  -  s t r e n g t h s the f i e l d  dependence of some of  TI-components i s r e v e r s e d and  there l i n e s  incident  l i n e s which reduces  the t o t a l a b s o r p t i o n of the n-component of  incident  source  The curves  a g a i n approach the c e n t r e s of  y  y  even i s o t o p e s i n the a b s o r p t i o n gas From F i g . 13, K Gauss and  the magnitude of J  2.5  K Gauss.  y  peaks at f i e l d  y  Using e q u a t i o n  - 3  to the e m i s s i o n  curves. a number of  i s examined.  from the l i n e c e n t r e by 86.75 x 1 0 c m  it  the  experimental  of h a v i n g  (2-9)  s t r e n g t h s of about  these f i e l d  l e a d t o a s p l i t t i n g of the a-component a b s o r p t i o n l i n e s  By r e f e r r i n g  and  i s the o s c i l l a t o r y appearance of the e x p e r i m e n t a l  T h i s f e a t u r e can, however, be e x p l a i n e d i f the e f f e c t  1.25  the  line.  most s t r i k i n g d i f f e r e n c e between the t h e o r e t i c a l  for J  the  - 1  and  173.5  x 10  strengths  of the even i s o t o p e s - 3  cm  - 1  respectively.  spectrum of n a t u r a l mercury ( S c h w e i t z e r ,  can be seen t h a t the e m i s s i o n  lines  of the even i s o t o p e s are  1963),  separated  from each o t h e r as shown below  a)  between 202  and  204  174  x 10"  3  cm  -1  b)  between 200  and  202  177  x 10"  3  cm  -1  c)  between 198  and  200  160 x 1 0  cm  -1  Thus, a frequency  -3  s h i f t of about 86.75 x 1 0 c m - 3  o-component a b s o r p t i o n l i n e s  approximately  c e n t r e s , w h i l e a s h i f t of about 173.5 absorption l i n e s line.  approximately  - 1  would p l a c e  half-way  x 10 cm - 3  - 1  the  between the  line  would p l a c e the o-component  at the c e n t r e of the a d j a c e n t  absorption  I t t h e r e f o r e appears t h a t the o s c i l l a t o r y appearance of J '  is  caused  - 128 -  by the c r o s s i n g of the a-component a b s o r p t i o n l i n e s which r e s u l t s i n a decrease  i n the t o t a l a b s o r p t i o n of the a-component of the i n c i d e n t  source  line. In c o n c l u s i o n , the d i f f e r e n c e s between the t h e o r e t i c a l assumptions the e x p e r i m e n t a l  and  r e s u l t s p r e c l u d e u s i n g the l a t t e r to v e r i f y d i r e c t l y a l l of  the t h e o r e t i c a l p r e d i c t i o n s .  However, when the t h e o r e t i c a l p r e d i c t i o n s  c o u l d be t e s t e d d i r e c t l y as i n the case of 1^ ( J ment were i n good agreement.  1  xx l e a s t c o n s i s t e n t w i t h the e x p e r i m e n t a l  and  J ' , the theory was yy  be  shown t o be at  results.  the c o n c l u s i o n which may  are examined i n more d e t a i l .  ) the theory and e x p e r i -  F u r t h e r , even when the theory c o u l d not  t e s t e d d i r e c t l y as i n the case of J  In the next Chapter  x y  be drawn from t h i s  study  - 129 -  CHAPTER V  Concluding D i s c u s s i o n  The  o b j e c t of t h i s study was t o examine the f e a s i b i l i t y  t r a v e r s e Zeeman e f f e c t ultra-violet  region.  t o produce a narrow band p o l a r i z e r i n the The t h e o r e t i c a l model i n d i c a t e d t h a t an a b s o r p t i o n gas  c o n s i s t i n g of the even i s o t o p e of mercury, H g  2 0 2  , would a c t as a h i g h  q u a l i t y p o l a r i z e r f o r narrow 253.7 nm e m i s s i o n l i n e s produced source.  of u s i n g the  by an H g  2 0 2  The e x p e r i m e n t a l r e s u l t s , w h i l e not i n c o n s i s t e n t w i t h the t h e o r y ,  were i n s u f f i c i e n t  to v e r i f y  the t h e o r e t i c a l p r e d i c t i o n s because of the  c o m p l i c a t i o n s I n t r o d u c e d by the h y p e r f i n e s t r u c t u r e of n a t u r a l mercury i n the a b s o r p t i o n gas, and by the broad The of  light  self-reversed  source  g e n e r a l t h e o r e t i c a l model which was developed I n a gas immersed i n a magnetic f i e l d  spectroscopic studies.  Equation  line.  f o r the t r a n s m i s s i o n  i s a p p l i c a b l e to a v a r i e t y of  (2-112) p r o v i d e s a complete d e s c r i p t i o n of  the p r o p a g a t i o n of an EM-wave i n a gas w i t h a non-degenerate ground  state  where the magnetic f i e l d may assume any o r i e n t a t i o n w i t h r e s p e c t t o the d i r e c t i o n of propagation. extended  A demonstration  o f how e q u a t i o n (2-112) c o u l d be  t o the case o f a gas w i t h a degenerate  i n equations  ground s t a t e was p r e s e n t e d  (2-131) t o (2-134).  A g e n e r a l d e s c r i p t i o n o f the s t a t e of p o l a r i z a t i o n o f an i n c i d e n t wave a f t e r t r a v e r s i n g a gas i n a t r a n s v e r s e magnetic f i e l d was p r o v i d e d by m a t r i x  - 130 -  (2-179).  The m a t r i x i s v a l i d  f o r i n c i d e n t waves w i t h any i n i t i a l  s t a t e of  polarization. The  t h e o r e t i c a l c a l c u l a t i o n of the p o l a r i z i n g e f f e c t  a b s o r p t i o n gas on H g l i n e widths  2 0 2  indicated  of an H g  2 0 2  c r e a t e d 253.7 nm e m i s s i o n l i n e s w i t h a v a r i e t y of  t h a t the technique should produce  h i g h q u a l i t y narrow  band p o l a r i z e r s i n the u l t r a - v i o l e t r e g i o n . The  experimental r e s u l t s  supported some of the t h e o r e t i c a l  For example, the t h e o r e t i c a l p r e d i c t i o n t h t I m  r  incident  source l i n e was v e r i f i e d  predictions.  ( J ) = 0 f o r an u n p o l a r i z e d xy r  to w i t h i n experimental e r r o r .  However,  the h y p e r f i n e s t r u c t u r e o f the n a t u r a l mercury used i n the e x p e r i m e n t a l a b s o r p t i o n gas, and the broad comparison  between t h e o r y and  source l i n e g e n e r a l l y p r e c l u d e d a d i r e c t experiment.  The width of the source l i n e l i m i t e d  the v a r i a t i o n of J ' t o a s i n g l e yy order of magnitude, w h i l e the t h e o r e t i c a l curves of J ' v a r i e d over s e v e r a l yy o r d e r s of magnitude.  I t was f o r t h i s reason t h a t the t h e o r e t i c a l and  e x p e r i m e n t a l p l o t s of J ' and J ' yy x  were not presented i n the same  The h y p e r f i n e s t r u c t u r e caused the t h e o r e t i c a l model. the presence  JP^ t o be f i e l d  dependent c o n t r a r y t o  F u r t h e r , the i s o t o p e s h i f t o f the 253.7 nm l i n e and  of a number of even i s o t o p e s i n n a t u r a l mercury l e a d to l i n e  c r o s s i n g which i n t u r n caused  the p l o t s of J '  s t r e n g t h t o assume on o s c i l l a t o r y The  graph.  x  as a f u n c t i o n of f i e l d  form.  p r e c i s e shapes of the e x p e r i m e n t a l curves of J ' and J ' c o u l d not xx yy  - 131  be t h e o r e t i c a l l y c a l c u l a t e d calculation resolution  critically  -  because the source l i n e p r o f i l e , upon which the  depend, c o u l d not be measured w i t h the l i m i t e d  of the a v a i l a b l e  spectrograph.  Thus, w h i l e the e x p e r i m e n t a l r e s u l t s c o n f i r m the theory i n some r e s p e c t s , the experiment theory.  To t e s t  the t h e o r e t i c a l p r e d i c t i o n  e f f e c t can be used single  t o produce a h i g h q u a l i t y  i s o t o p e w i t h zero n u c l e a r s p i n  F u r t h e r the source its  c o u l d not p r o v i d e a q u a n t i t a t i v e that  t e s t of the  the t r a n s v e r s e Zeeman-  polarizer  should be used  a gas c o n s i s t i n g  of a  i n the experiment.  l i n e should be r e a s o n a b l y narrow; i f i t i s broad,  l i n e p r o f i l e s h o u l d be known.  limited  then  - 132 -  References  1.  Abramowitz,  M. and Stegun, I.A.,  Dover P u b l i c a t i o n s , New  2.  A l l , A.W.  York,  and Griem, H.R.,  B e v i n g t o n , P.R., Sciences,  4.  Born, M.  Functions,  1972.  P h y s i c a l Review, 140, A1044 (1965) and  Erratum, P h y s i c a l Review, 144,  3.  Handbook of M a t h e m a t i c a l  366  (1966).  Data R e d u c t i o n and E r r o r A n a l y s i s f o r the P h y s i c a l  McGraw-Hill Book Co., New  York,  1969.  and Wolf, E., P r i n c i p l e s o f O p t i c s , Pergamon P r e s s ,  Oxford,  1975.  5.  Camm D.M.  and Curzon, F.L., Canadian J o u r n a l of P h y s i c s ,  50, 2866  (1972).  6.  Condon, E.V., Cambridge  7.  and S h o r t l e y , G.H.,  U n i v e r s i t y P r e s s , New  Corney, A., K i b b l e , B.P. (1966).  The Theory o f Atomic York,  Spectra,  1964.  and S e r i e s , G.W.,  Proc. R. S o c ,  A293, 70  - 133 _  8.  D'Yakonov, M.I. and P e r e l , V . I . , S o v i e t P h y s i c s JETP, 21_, 227 (1965).  9.  Dodd, J.N. and S e r i e s , G.W.,  10.  H e i t l e r , W.,  P r o c . R. S o c , A263, 353 (1961).  The Quantum Theory o f R a d i a t i o n , Oxford U n i v e r s i t y  Press,  Oxford, 1954.  11.  Jackson, W.D.,  C l a s s i c a l E l e c t r o d y n a m i c s , John W i l e y & Sons, New  York,  1975.  12.  K l e i n , M.V.,  O p t i c s , John W i l e y & Sons, I n c . , New York, 1970.  13.  Kuhn, H.G.,  14.  L u r i o , A., P h y s i c a l Review, 140, A1505 (1965).  15.  Merzbacher, E., Quantum Mechanics, John W i l e y & Sons, I n c . , New York,  Atomic S p e c t r a , Academic P r e s s , New York, 1969.  1970.  16.  M e s s i a h , A., Quantum Mechanics Volumes I and I I , John W i l e y & Sons, I n c . , New York, 1962.  17.  R a b i n o v i t c h , K., C a n f i e l d , L.R. and Madden, R.P., A p p l i e d O p t i c s , 4, 1005  (1965).  - 134  18.  Rose, M.E., Inc., New  19.  -  Elementary Theory o f Angular Momentum, John W i l e y & Sons, York,  S c h w e i t z e r , W.G.,  1957.  J o u r n a l of the O p t i c a l S o c i e t y of America,  53_»  1055  (1963).  20.  Sobel'man, I . I . , I n t r o d u c t i o n to the Theory o f Atomic Press, Oxford,  21.  S t a n z e l , G.,  22.  S t r o k e , G.W.,  1972.  Z. P h y s i k , 270,  361  Physics Letters,  (1974).  5, 45  (1963).  S p e c t r a , Pergamon  - 135 -  APPENDIX  P r o p e r t i e s o f the 253.7 nm L i n e  A term diagram o f the Atomic S p e c t r a o f Hg can be found i n Condon and Odishaw ( e d . ) , Handbook of P h y s i c s , McGraw-Hill Book Company, New 196, at page 7-45.  L i f e t i m e of the s t a t e T  3  Pj^ , from L u r i o , 1965  , ( P , ) = 1.14 x 1 0 aj 3  - 7  1  which g i v e s an a b s o r p t i o n  sec.  o s c i l l a t o r strength, f a , of f a = 2.54 x 1 0  - 2  Lande g , ( P , ) , from L u r i o , 1965 aj ! 3  g  (^  ) = 1.486094(8)  York,  -  d)  N a t u r a l Isotopes  ISOTOPE  NUCLEAR SPIN, I  0.146  H  -  o f Mercury  % NATURAL ABUNDANCE  Ha* 9 6 80 8  1 3 6  ATOMIC MASS  0  195.9658  Hal 9 8 80 8  10.02  0  197.9668  Hal 9 9 S  16.84  1/2  198.9683  ,Hg200  23.13  0  199.9683  201  13.22  3/2  200.9703  202  29.80  0  201-9706  204  6.85  0  203-9735  H  80 80  H  1  80 § H  80  H g  80 Hg'  (Taken from Handbook of Chemistry  and P h y s i c s , The Chemical Rubber Co.,  C l e v e l a n d , 1969 on page B-491 t o B-494)  e)  FREQUENCY AND WAVELENGTH OF THE 253.7 nm LINE  X0  = 2536.519 A  (Taken from, Handbook o f Chemistry  f)  u 0 = 7.42613 x 1 0 1 5 Hz.  and P h y s i c s Op. C i t . Page E-222)  The P r o p e r t i e s of the H y p e r f i n e  S t r u c t u r e o f the 253.7 nm L i n e  S t r u c t u r e of the 253.7 nm L i n e can be found  i n Schweitzer,  1963.  

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