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Theoretical study of some forbidden spectral lines Wong, Kim Po 1967

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THEORETICAL STUDY OF  SOME FORBIDDEN SPECTRAL LINES  Kim Po Wong B. Sc. (Special Hons.), University of Hong Kong, 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of Physics  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1967  r  In  presenting  for  an a d v a n c e d  that  the  study. thesis  the  agree  that  freely  or  representatives..  of  my w r i t t e n  this  thesis  for  may be g r a n t e d  for  permission.  Depa r t m e n t The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  of  Columbia  It  of  British  available  permission  purposes  by h i s  fulfilment  University  scholarly  publication  without  at  in p a r t i a l  s h a l l make i t  I further for  thesis  degree  Library  Department or  this  for  the  requirements  Columbia,  I  reference  and  extensive  copying  this  by t h e Head o f my  is understood  financial  of  agree  gain  shall  that not  be  copying allowed  ABSTRACT  The forbidden l i n e (6sp*P  - 6s 's ) 7^2656A i s a  0  6  forbidden f o r a l l multipole t r a n s i t i o n s i f the nucleus of Hg isotope has no magnetic moment. For Hg  the s e l e c t i o n  rule i s broken by the i n t e r a c t i o n of the nuclear magnetic dipole moment with the atomic electrons.  In t h i s thesis  the Zeeman e f f e c t of the l i n e i s discussed i n d e t a i l . In p a r t i c u l a r , the i n t e n s i t y of the Zeeman components i s calculated  as a function  of the external  magnetic f i e l d .  It turns out that even f o r an external magnetic f i e l d s u f f i c i e n t to produce the Back-Goudsmit  effect  (30,000  gauss) the i n t e n s i t y of Zeeman components changes by l e s s than one percent. Also the e f f e c t of the i n t e r a c t i o n of the nuclear magnetic moment and atomic electrons  on the i n t e n s i t y of  forbidden l i n e s (6 *S - 6 P,)X4618A and (6'S - 6 *P ) ^ 3  0  5313A of Pb  i s calculated  0  ( i n absence of external  magnetic f i e l d ) . The e f f e c t i s very  small.  a  Iii  TABLE OP CONTENTS CHAPTER  Page  1. INTRODUCTION  1  2. HAM ILT ONI AN OP THE SYSTEM CONSIDERED  4  2:1 General Form of the Hamiltonian 2:2  . .  .4  C a l c u l a t i o n of Matrix Elements of the Stationary  Perturbation  .5  3. CALCULATION OF PERTURBED WAVE FUNCTIONS FOR H g  m  . .  10  4. THE INTENSITY OF ZEEMAN COMPONENTS OF THE FORBIDDEN LINE (&%  - 6 's ) IN H g ' "  24  o  4:1 General Expression f o r the Intensity of Multipole Radiation 4:2  24  V a r i a t i o n of the Intensity With External Magnetic F i e l d .  . .  27  5. THE RELATIVE INTENSITIES OF THE HYPERFINE COMPONENTS OF THE TV/0 FORBIDDEN LINES (6'S  0  -. 6%  ) AND (6 'S  0  - 6,*P ) OF Pb*"*  5:1  C a l c u l a t i o n of Perturbed Wave Functions.  5:2  Relative I n t e n s i t i e s of the Hyperfine Components.  BIBLIOGRAPHY  33  Z  . . .  33  38 42  iv  LISO? OP  ILLUSTRATIONS  Figure  Page  1. The lowest f i v e energy l e v e l s of Hg  11  2. Energy diagram of the perturbed l e v e l s of Hg ' . .22: ,?  3. Zeeman components of the forbidden e l e c t r i c dipole t r a n s i t i o n (6sp *P - 6s 'S ) of Hg"* . . . 28 1  0  0  4. The lowest f i v e energy l e v e l s of Pb  34  5. Hyperfine components of the l e v e l s %  , *P, , X  of Pb***  37  V  ACKNOWLEDGEMENTS  I w i s h t o express my g r a t i t u d e t o P r o f e s s o r Opechowski f o r s u g g e s t i n g interest of t h i s  W.  t h i s problem and f o r h i s continued  and f r u i t f u l d i s c u s s i o n s throughout; the performance research.  I w i s h t o thank t h e Commonwealth S c h o l a r s h i p and Fellowship  Committee f o r the f i n a n c i a l a s s i s t a n c e  i n the  form o f a s c h o l a r s h i p . A l s o , t o my f r i e n d s M i s s G l o r i a  Chang and M i s s Wat  C h i - K i t I wish t o express my g r a t i t u d e f o r t y p i n g thesis.  this  CHAPTER  1.  INTRODUCTION  f o r b i d d e n l i n e (6sp*P  The  6s 's )A2656A of a  -  0  i s a pure e l e c t r i c d i p o l e t r a n s i t i o n . B e i n g between two  l e v e l s b o t h w i t h quantum numbers J=0, i t i s  f a c t t h a t the l i n e may be due  o t h e r s ) . The  rules.  t o the presence o f the  n u c l e a r magnetic moment was suggested by s e v e r a l (Bowen and  ?  a transition  s t r i c t l y f o r b i d d e n a c c o r d i n g to the u s u a l s e l e c t i o n The  Hg *  tt  weak i n t e r a c t i o n  of the  authors nuclear  magnetic moment w i t h the e l e c t r o n s causes a s l i g h t admixture of the  *P, wave f u n c t i o n s t o the  i  P  t h e r e f o r e t h e r e e x i s t s a s m a l l but probability The intensity  wave  function^and  finite  transition  0  between the two p e r t u r b e d r a t i o o f the i n t e n s i t y  levels  ^ P and 'S . 0  0  o f t h i s l i n e and  - 6s 'S )\2537A i n the  o f the l i n e ( S s p ^  x  6  was found i n agreement w i t h experiment We i n v e s t i g a t e the behaviour  absence  (1938)  o f e x t e r n a l f i e l d was c a l c u l a t e d by ¥. Opechowski and  the  (Mrozowski 1937).  o f the i n t e n s i t y  o f the  f o r b i d d e n l i n e under the i n f l u e n c e of an e x t e r n a l magnetic field  by adding  the  electronic  a c t i o n s t o the h a m i l t o n i a n . and is  i  P  e  and  n u c l e a r Zeeman i n t e r -  Only the m i x i n g  wave f u n c t i o n s i s c o n s i d e r e d . T h i s  justified  o f the P , J  approximation  a s the energy s e p a r a t i o n between these two  l e v e l s i s much s m a l l e r than t h e i r l e v e l s . The range o f the f i e l d  s e p a r a t i o n from  f o r which our  other  calculation  i s v a l i d goes from zero t o 10* gauss. I t i s w e l l below the  2  value when the Paschen-Back effect,  s t a r t s to decouple J  into L and S which i s estimated to occur at a f i e l d of order 10  7  gauss f o r our case. The Back-Goudsmit  decoupling  of F into I and J i n the P, l e v e l occurs at f i e l d s around J  3 X 10  gauss. At f i r s t sight one may think that the i n t e n -  s i t y of the forbidden l i n e w i l l decrease considerably when the f i e l d reaches t h i s Back-Goudsmit  l i m i t . However the  c o e f f i c i e n t s of the P, wave functions i n the expression ?  for  the perturbed  *P wave functions are mainly determined 0  by the energy difference between the P J  /  a n d ^ s t a t e s . Now  the energy difference varies only by a very small f r a c t i o n even when the f i e l d i s as high as 10* gauss. The influence of the P, l e v e l on the l e v e l s 3  *P and 'S i n the presence 0  0  of an external f i e l d i s therefore nearly the same as that when no external f i e l d i s present and the two l e v e l s show normal Zeeman s p l i t t i n g . As a r e s u l t , a normal Zeeman e f f e c t of the forbidden l i n e (6sp P J  0  - 6s^ * S ) Q  i s predicted and  the i n t e n s i t y of the components changes only very s l i g h t l y when an external magnetic f i e l d i s introduced even as high as 10  6  gauss. When the nuclear spin and external magnetic  field  are both absent, the two forbidden l i n e s (6 *S - 6P, ) A i  0  4618A and (6 'S - 6*P^)A.5313A of Pb are r e s p e c t i v e l y pure 0  magnetic dipole and pure e l e c t r i c quadrupole t r a n s i t i o n s . Their r e l a t i v e i n t e n s i t i e s were studied experimentally by Mrozowski  (1940). The isotope Pb*  07  has a nuclear spin  and the perturbation produced by the nuclear magnetic  3  moment not only gives r i s e to hyperfine structures i n the transitions,  but also causes a mixing of wave functions  between the  *P, and * P l e v e l s . Each hyperfine component 4  w i l l no longer be a pure multipole t r a n s i t i o n , but w i l l be a mixed t r a n s i t i o n of both magnetic dipole and e l e c t r i c quadrupole type. The r e l a t i v e i n t e n s i t i e s f o r the hyperf i n e components of each forbidden l i n e are calculated from the perturbed wave f u n c t i o n s . It i s found that they deviate only very s l i g h t l y from the r e l a t i v e s t a t i s t i c a l  weights  of the hyperfine components. Again t h i s i s obvious i f one considers the fact that the hyperfine i n t e r a c t i o n i s extremely small when compared to the energy separation between the  J  P,  and P J  X  l e v e l s , causing only very small admixture  to the states considered. We should l i k e to mention that a recent review of the present  state of research concerning forbidden l i n e s  by Garstang (1962) contains a discussion of the case of the l i n e (6sp*P  6  - 6s* 'S  external magnetic  e  ) X2656A of Hg  field.  i n . absence of an  CHAPTER 2. HAMILTONIAN OP THE SYSTEM CONSIDERED  g e n e r a l Form o f the H a m i l t o n i a n 2:1 The h a m i l t o n i a n o f the atom o f Pb the  or Hg  in  presence o f an e x t e r n a l f i e l d H c o n s i s t s o f f o u r terms: #  = X  *  e  +  +  76'  a).  i s the unperturbed h a m i l t o n i a n f o r the atom i n absence of the  n u c l e a r s p i n s and i n absence o f the . i n t e r a c t i o n s w i t h e x t e r n a l magnetic f i e l d H and w i t h the r a d i a t i o n  field.  76» i s the i n t e r a c t i o n o f the n u c l e a r moments w i t h the TCt i s the i n t e r a c t i o n o f the atomic and n u c l e a r  electrons.  magnetic moments w i t h the e x t e r n a l magnetic f i e l d H and 76' i s the i n t e r a c t i o n o f the atom w i t h the r a d i a t i o n Since n u c l e a r e l e c t r i c for  b o t h Pb * -1  and Hg' *  7  ?  field.  quadrupole moment i s absent  and c o n t r i b u t i o n s due t o h i g h e r  moments are s m a l l compared t o t h a t due t o the magnetic d i p o l e moment, we can w r i t e where ^  i s the n u c l e a r magnetic d i p o l e moment and H 7 i s the  magnetic f i e l d For  produced by the e l e c t r o n s at the n u c l e u s .  our c a l c u l a t i o n , i t i s more c o n v e n i e n t t o express  76n  as the s c a l a r product o f two i r r e d u c i b l e t e n s o r s , each o f which i s o f rank 1.  where  I  Xn  =  -  I,  =  - - £ ( X„  I*  +  i X>)  ,  (».  5  and  the l a t i n s u b s c r i p t s denote components i n a C a r t e s i a n  coordinate  system.  To express  i t i s c o n v e n i e n t and w i t h o u t l o s s of  g e n e r a l i t y t o assume t h a t the e x t e r n a l magnetic f i e l d H i s p a r a l l e l t o the z - a x i s . Denoting by j ^ t h e magnetic  dipole  moment o f the e l e c t r o n s , we have  While #  n  and  are s t a t i o n a r y perturbations,  time dependent. I t s magnitude i s much s m a l l e r The  is  than^nd  #x  energy s h i f t o f the atom caused by # ' i s n e g l i g i b l e as  compared t o t h a t caused by e i t h e r H or n  treat  by K^* T h e r e f o r e  we  X,' o n l y as an agent t h a t causes t r a n s i t i o n s between  different  states.  C a l c u l a t i o n of Matrix  Elements o f the S t a t i o n a r y  Perturba-  t i o n 2:2 In the c a l c u l a t i o n o f the i n t e n s i t y o f the f o r b i d d e n lines,  one needs t o know the energy and the perturbed  wave  f u n c t i o n s o f the s t a t e s between which the t r a n s i t i o n takes  6  p l a c e . F o r such purpose,  we d e r i v e an e x p r e s s i o n f o r the  m a t r i x elements o f t h e p e r t u r b a t i o n which w i l l be denoted by V: V  *  Hn  «).  fa  +  I f we denote by I the n u c l e a r s p i n quantum number, J  the t o t a l e l e c t r o n i c  a n g u l a r momentum,  a n g u l a r momentum, F the t o t a l  the atomic  atomic  magnetic quantum number, and  «t a s e t o f e i g e n v a l u e s o f o t h e r o b s e r v a b l e s which a r e m u t u a l l y commutative and which a l s o commute w i t h and  the z-component o f F, then  eigenstates of  i s a set of  any n u c l e a r s p i n  Ej&o) i s the c o r r e s p o n d i n g unperturbed b e i n g s c a l a r product  eigen-energy.  o f two v e c t o r s i s i n v a r i a n t  under r o t a t i o n and t h e r e f o r e commutes w i t h F Consequently,  A  and F:^.  i s d i a g o n a l i n quantum numbers F and »y.  U s i n g theorems on a d d i t i o n o f a n g u l a r momenta, we have  Applying Wigner-Erckart and  l  which i s the h a m i l t o n i a n o f a c o m p l e t e l y  i s o l a t e d atom without  where  £ 1**7ra^j  l \ T\ f  I, i t follows that  theorem t o m a t r i x elements o f H  J  7  and  <I^|^|IW,> ~ ^i^+M  <  where  :r>  I >» M  x  S  I X  7n i-M> t  i s a reduced m a t r i x element of H  7  which i s independent of M and other magnetic quantum numbers. By summing over M f i r s t , we have <*'J 7 V  77^ J  / ofx j  >  Using symmetry p r o p e r t i e s of the Glebsch-Gordan c o e f f i c i e n t s , t h e i r i n d i c e s can be rearranged as f o l l o w s : < X  / y*>  x  _ _>»»i-»«z ( 0  and  <  7  /  <  x  >» ->»9x f  t  "  i  / x  >y,' >  ^  W2-**»x  -  ^  x  / J'  / X  w^-^i  >»r>  >  Then the summation can be expressed i n terms of the Racah c o e f f i c i e n t s ( Biedenharn,Blatt,Rose 1952 ) and we o b t a i n (see Biedenharn, B l a t t , Rose, eq.(19) )  where W(IIJJ';1F) i s the Racah c o e f f i c i e n t as defined i n the above paper. To f i n d m a t r i x elements of 3 ^ , of Is and J> f i r s t .  we c a l c u l a t e those  8  But  = - /iTxTo  < i  <*> I x x  *w,>  A l s o , symmetry p r o p e r t i e s of Racah and Clebsch-Gordan c o e f f i c i e n t s give:  and  i v d n j i i ^ s - (-o'^^'wdi  T h e r e f o r e we  '  have: ? ' ¥ / 1 ^ f <*a 5 p l>y >  Similar calculations give::  U s i n g the above two e x p r e s s i o n s , we can w r i t e down the  Hk, .  e x p r e s s i o n f o r the m a t r i x elements o f  =  $r'j ^ > » f / 2 f ' + j  +  Finally,  t  H  )  l x o - U  '<  ?  +  T  » >»i 0 | f" w > Wj. * f  0  W  f  ( m - t ' } i x ) j  the e x p r e s s i o n f o r the m a t r i x elements o f V can  9  be w r i t t e n as  CHAPTER 3. CALCULATION OP PERTURBED WAVE FUNCTIONS FOR Hg'  The  lowest energy l e v e l s o f Hg  ff  a r e shown schemat-  i c a l l y i n f i g u r e 1. i'rom the f i g u r e we see t h a t  energy  d i f f e r e n c e between l e v e l s *P  while  and *P  e  are small  energy d i f f e r e n c e s between other l e v e l s a r e much l a r g e r . This fact  suggests t h a t t o a good a p p r o x i m a t i o n we may  consider  o n l y t h e mixing o f t h e P, and P i  s  e  wave f u n c t i o n s ,  n e g l e c t i n g t h e admixture o f other wave f u n c t i o n s to these two  levels. The  nuclear  s p i n o f E g " * i s £ . From:eq.(3), i t  f o l l o w s immediately t h a t the m a t r i x elements o f V f o r the ground s t a t e  *S  0  i n representation  j I^FH^J  are  g i v e n by:  =  - X ™ /U f  x  ?-v », f  ( «>»»£  p  £ )  (of course t h e t s a m e " e x p r e s s i o n f o l l o w s from eq.(7)) T h i s l e v e l t h e r e f o r e s p l i t s i n t o two Zeeman Components <K'S.  with  energy  i-Mf)  * \  f  S . ±  *> > f  (9)  11  Figure 1.  The lowest f i v e energy l e v e l s of Hg (Baeher and Goudsmit 1932). The P l e v e l s i n the f i g u r e belong to configuration 6s6p and the 'S l e v e l belongs to Ss*conf i g u r a t i o n . The excited l e v e l s are drawn to some a r b i t r a r y scale but the ground l e v e l i s not to scale. e  12  To c a l c u l a t e the p e r t u r b e d  s t a t e s of l e v e l  have to s o l v e the e i g e n v a l u e problem f o r the ( b e l o n g i n g to l e v e l s  *P  0  and  *P,  P , we 0  submatrix  ) o f the m a t r i x f o r the  t o t a l s t a t i o n a r y h a m i l t o n i a n of the system, namely K  «  Ke  +  V  The r e q u i r e d m a t r i x  and  elements are .of the'.form  a r e g i v e n by e q u a t i o n  (7). We w r i t e them down e x p l i c i t -  l y as f o l l o w s :  <*P, i  t  / W*P, i  i >  13  B = <*P, w i v| r,t-i> J  = <V,f i|v|*f,i  i>  = <f, * i I v p r . i i > ?  14  where  <f> = <^  P(  \ and  =  J|H || P,> J  3  < F,l|H l| r6> ?  :r  i  a l l o t h e r m a t r i x elements v a n i s h . The  z e r o t h order h a m i t l o n i a n # i s d e f i n e d as unique e  o n l y up to an a d d i t i v e c o n s t a n t and we  can, f o r  convenience  i n the f o l l o w i n g c a l c u l a t i o n s , choose E „ ( P , ) = 0 . Then 5  The  symmetry group f o r the h a m i l t o n i a n i s  and  m a t r i x elements of the h a m i l t o n i a n are d i a g o n a l i n ») . f  Taking  t h i s i n t o account,  i n the f o l l o w i n g reduced  the submatrix form:  of ~H can be w r i t t e n  15  T, 3.  'f,  j_  X 2  v,  'f,  X  2.  2  A  Jk X  1  3. ~a  0  0  e  0  _ J.  a-  B  F  _ X X  B  0  F  J.  T*  0  %  6 *»  From t h i s submatrix, i t i s immediately > the p e r t u r b e d wave f u n c t i o n s are  w i t h c o r r e s p o n d i n g energy  and  EC P, i J  I)  * ~£  <M  JL 2-  JL A.  »f ' i  »P F  ,  »P • <  5.  ~  A.  3  "tof*±j[  *  1  0  for  i  ' l  "  A  i  obvious  i f  that  16  The  problem i s now  reduced  t o the s o l u t i o n of the  e i g e n v a l u e problem o f the f o l l o w i n g two matrices:-  0  ?*  't.  B  V, a i  B  T  To  * l  *p,  f-i  i  t  0  F* B  0  T,  T. J.  1  X  We  s h a l l f i n d the e i g e n v a l u e s of 3l{  f i r s t . Assuming  t h a t X i s the e i g e n v a l u e , the s e c u l a r e q u a t i o n f o r JCt  A* + where  <*, V + A , .  A •+  Aj  =  0  is:  0  o.).  17  Since the hamiltonian i s hermitian i t s eigenvalues are a l l r e a l . The cubic equation with three r e a l roots can be solved by standard algebraic methods. I f we define (See, f o r example, J-. W. Arehbold 1958, Chapter 12)  <t> = W  ( z £ L )  ©< * *  7 1  then i n our case both G and H. are r e a l and furthermore  h <o  and  Cf < o . The l a s t two i n e q u a l i t i e s are not  obvious a n a l y t i c a l l y ,  but t h e i r v a l i d i t y has been checked  numerically using an IBM ?040 computer f o r magnetic f i e l d s up to 10" gauss which i s much higher than the range of magnetic f i e l d we are interested i n . When the above condi t i o n s are s a t i s f i e d , the three roots are given by the following expressions:-  When we put we have  X,-*o  yU —* o  and  x  ^ - A  -  *  and  H| X  °  -» ©  3  Therefore from c o n t i n u i t y the eigenstate belonging to X  x  becomes  \ ?t>t~i> l  i n the above l i m i t s , and we have \  as the energy of the perturbed state  ^( Po * ~£) . J  x  18/  t  fC P -t-i)= ft, l*P t-£> i  b  c, = £  where  + «*  0  J  A , ( A , - A  (  4  )  £ - * > * S IV. i-i>  -B } X  and The  secular  A  where  +  1  ^  e q u a t i o n f o r m a t r i x ^ i s as f o l l o w s : -  +  + ^ U j  U  %  = - ( E  * x ~  + O  QL' f  A  *  0  \ -6  Here we observe^, t h a t tion  1  i s obvious t h a t reversing  =  {  A )(B  4  i  ) l  + A )  >  4  -  /fi*  e q u a t i o n (13) i s o b t a i n e d from  (10) simply by r e p l a c i n g  From t h e e x p l i c i t  i°  A  and A  I + J  equa-  ( i= 1, 2, 3 ) •  e x p r e s s i o n s f o r t h e m a t r i x elements i t  each A  X + I  can be o b t a i n e d from A ^ by  t h e s i g n o f H.^, and t h a t B " i s an even f u n c t i o n 1  of H j w h i l e F i s independent i n equation  o f HJ;. S i n c e o n l y B* o c c u r s  (10) and (13)> i t i s c l e a r t h a t  o b t a i n e d from  (13) can be  (10) by simply r e v e r s i n g t h e s i g n o f H^.  T h e r e f o r e t h e r o o t s A* (•£=',*,*) o f (13) a r e t h e c o r r e s p o n d i n g r o o t s o f (10) w i t h t h e s i g n o f H^ r e v e r s e d . c o n t i n u i t y arguments, A ^ ( w p =Ai(-Wj) i s t h e energy to t h e p e r t u r b e d s t a t e  From belonging  f ( Po t i) . 3  E<*P. £ 1 ) = A ; c n p = A,C-H^  04).  19  where  ( A '  <^=^{  x  In o r d e r t o f i n d  - A  4  ) -  B} x  out i f we can expand the e x p r e s s i o n s  f o r the e i g e n v a l u e s and wave f u n c t i o n s i n t o and r e t a i n i n g  powers of H.^  o n l y terms l i n e a r and q u a d r a t i c i n H^., we  have t o e s t i m a t e the n u m e r i c a l v a l u e s o f such  quantities  as^u^, yUj , f ,^and E . e  191  i s 0.49926  The n u c l e a r magnetic moment o f Hg  n u c l e a r magneton (H. Kopfermann 1958) which i s approximately 1.27 * 10  tm.'; gauss  . S i n c e our c a l c u l a t i o n  approximate and i s expected  i s only  o n l y t o g i v e v a l u e s to w i t h i n  an o r d e r o f magnitude, we can assume the K u s s e l l - S a u n d e r s c o u p l i n g t o c a l c u l a t e ^ ^ t j . . The Lande g - f a c t o r , which i s given t h e o r e t i c a l l y 5 =  /  +  as _TC7+Q  + S(Stl)  -L(L+l)  i s 1.5 f o r the l e v e l P . T h i s g i v e s a v a l u e o f 7.01x10 * Orr>~' gauss"' f o r yUy . i  l  To estimate v a l u e s ofandyu^f, m a t r i x elements of ^£,for the l e v e l s  we assume t h a t the i  P  and *P,  e  have the  same o r d e r o f magnitude. T h i s i m p l i e s that_/*a|> and /**\\\ have the same order o f magnitude. A l s o when the e x t e r n a l field  i s absent  i s o f the order  the h y p e r f i n e s p l i t t i n g of the l e v e l *P,  < P f *> \3t»\ P, ? f > J  i  (  f  M  20  and  i s found  experimentally  to have a value  o f 0.4  frrf'  ( S c h u l e r and Jones 1932), t h e r e f o r e we can assume t h a t  y^ipcr  0«4 fc*»v' . F o r convenience,  we r e w r i t e the n u m e r i c a l  v a l u e s we have d i s c u s s e d :  Using  the above v a l u e s we f i n d t h a t f o r an e x t e r n a l  f i e l d which does n o t exceed 1 0  6  gauss, the f o l l o w i n g  i n e q u a l i t y always h o l d s : 0  $  <  0- I 0 A  To a good approximation,  we can w r i t e  In a l l the q u a n t i t i e s G, H and D, the terms depending on  a r e much s m a l l e r than the c o n s t a n t  We can t h e r e f o r e approximate the e x p r e s s i o n f o r a polynomial  terms. by  i n H^. In f a c t , a f t e r expanding the f u n c t i o n s  i n G, H and D i n powers o f H£ i n the u s u a l manner, we have  Expressions  f o r the o t h e r q u a n t i t i e s r e q u i r e d are determined  i n the s i m i l a r way. We j u s t w r i t e them down as f o l l o w s :  2-1  Here we see that the wave function and the eigen-energies ... for  the P 3  e  l e v e l change extremely slowly with changing  external magnetic f i e l d . The closeness of the expansions to the exact expressions have checked numerically using the IBM 7040 computer and are found to be remarkably good. The perturbed energies f o r the *P, wave functions have also been c a l c u l a t e d numerically using the computer to get an idea about the influence of t h i s l e v e l on the wave functions of the P 3  0  l e v e l . The energy diagrams of the three l e v e l s  are shown schematically i n figure 2. The energy of the hyperfine components of the *P, l e v e l changes greatly i n pattern at a f i e l d around 3 * 1 0  4  gauss. This i s due to the onset of the Back-Goudsmit e f f e c t on the hyperfine components. However, the change i n magnitude of the energies i s s t i l l extremely small i n comparison to the energy separation between the P, and *P l e v e l s . This 3  0  explains q u a l i t a t i v e l y the fact that the wave functions of  F i g u r e 2.  Energy diagram of the p e r t u r b e d of  Hg . m  levels  23  the  3  P  0  l e v e l changes o n l y v e r y s l i g h t l y even f o r v a l u e s of  magnetic f i e l d may  such t h a t P i s decoupled  i n t o I and J .  a l s o mention t h a t i n our case the e l e c t r o n i c  splitting  i s of order E  has v a l u e .  The  S needs a f i e l d v a l u e s below 10^  0  and  We  fine  the e l e c t r o n i c magnetic moment  Paschen-Back d e c o u p l i n g of J i n t o L and  of order 3 X 10 gauss as we  ?  gauss and f o r f i e l d s  are now  with  c o n s i d e r i n g , J remains  a good quantum number. As  the wave f u n c t i o n s f o r l e v e l  v e r y s l i g h t l y i n the presence  *P  0  change;, o n l y  of e x t e r n a l f i e l d , we  expect  t h a t the i n t e n s i t i e s of the Zeeman components w i l l a l s o change v e r y l i t t l e . However, t o prove t h a t we  s h a l l derive  a n a l y t i c e x p r e s s i o n s f o r the i n t e n s i t i e s i n the f o l l o w i n g chapter.  4.  CHAPTER  THE INTENSITY OP ZEEMAN COMPONENTS OP THE FORBIDDEN LINE (6 *P  D  - 6 'S ) 0  IN Hg'"  General Expression  f o r the I n t e n s i t y of M u l t i p o l e R a d i a t i o n  4:1 The  forbidden l i n e  an e l e c t r i c  (6sp*P  0  - 6s" 's ) o f Hg'  ??  e  is  d i p o l e r a d i a t i o n . However, s i n c e we are going  to c o n s i d e r the case and e l e c t r i c  of mixed magnetic d i p o l e r a d i a t i o n  quadrupole r a d i a t i o n i n the f o l l o w i n g c h a p t e r ,  we s h a l l w r i t e down a g e n e r a l e x p r e s s i o n f o r m u l t i p o l e radiation i n this section. If  H/I**»,>  and KjUw^) are two e i g e n s t a t e s of a  r a d i a t i n g system w i t h a n g u l a r momentum quantum numbers and  and w i t h e n e r g i e s E*,^  jim*.  and E<*^\_  r e s p e c t i v e l y , then the e x p r e s s i o n f o r the i n t e n s i t y o f the mM. component o f an e l e c t r i c  ( o r magnetic; 2^  -pole  t r a n s i t i o n between the two s t a t e s i s g i v e n by ( B l a t t and weisskopf  where  1952J  1 ^ (or  ) i s the » A component o f the e l e c t r i c  ( o r magnetic) 2  l  -pole moment,  between the two s t a t e s , k  i s the energy s e p a r a t i o n  i s a c o n s t a n t which has the  25  and c i s the v e l o c i t y of l i g h t i n vacuo. It has been shown (Ling and F a l k o f f 1949) that the t o t a l i n t e n s i t y f o r the  component of a mixed e l e c t r i c  2 -pole r a d i a t i o n with magnetic 2  -pole r a d i a t i o n i s  obtained by just adding up the two i n d i v i d u a l i n t e n s i t i e s .  where "J^and and magnetic  are the corresponding e l e c t r i c (4*t )-pole (/^rw)-pole  moment.  I f the measuring apparatus are i n s e n s i t i v e to the p o l a r i s a t i o n of the incident the  l i g h t as i s assumed here, then  i n t e n s i t y measured w i l l be the contribution  possible  due to a l l  multipole components and the expression f o r the  t o t a l i n t e n s i t y f o r a mixed e l e c t r i c 2^-pole and magnetic 2 -pole t r a n s i t i o n between the states  l o t , a n d  /<^»/>>>^>  The i n t e n s i t y f o r a pure e l e c t r i c or magnetic 2^-pole r a d i a tion i s  For convenience i n l a t e r c a l c u l a t i o n s ,  i t i s useful  here to derive an e x p l i c i t expression f o r the matrix elements  :2S  of the m u l t i p o l e moment Tf^ In t h i s d e r i v a t i o n , does not concern  and a sum r u l e f o r them. the e x p l i c i t e x p r e s s i o n f o r  7^  us because we o n l y use i t s t r a n s f o r m a t i o n  p r o p e r t i e s under r o t a t i o n s .  I t i s well-known t h a t the m u l t i -  (^ = -^.-, A ) transforms  pole moments  as an i r r e d u c i b l e  t e n s o r of rank 1 and we can a g a i n a p p l y Wigner-Eckart theorem to s i m p l i f y  our c a l c u l a t i o n s .  - Z < xj'm, y - ^ t j T W > < I T >*>* wy W i ' * *V *' ' <£  =• < J JL m -™ f  where  x  <<K'J'I|T^|I«<T>  >» / -j' ^-m,  7  Ttr* I * *>~'*' 7  > <*'J' H 1*H #J>  i s the reduced  m a t r i x element o f  T /  M  ,  T h e r e f o r e we have  where  #=.2 <* 3 X  > » > t - ^ |  ^  ^  I  ^  V  ^  Prom symmetry p r o p e r t i e s of the Clebsch-Gordan we have  The  sum i n the e x p r e s s i o n f o r S i s e q u a l t o  ^-^/Tw^-"^ coefficients,  C J>  27  The expression f o r the matrix elements can therefore be rewritten as follows:  VaF )<3Lj' ,) +  +  W(jj'ff  (20).  J**)<4'J'|(T£/|VJ>  (except f o r notation, t h i s expression i s the same as eq.(6.24) i n E. M. fiose (1957) ) using eq.(20) and the orthogonalityr e l a t i o n of the Glebsch-Gordan c o e f f i c i e n t s  We can write down the following sum-rule f o r the sum of absolute squares of matrix elements:  V a r i a t i o n of the Intensity with External Magnetic F i e l d 4:2 The perturbed  states V ( i p 0 t ™ j : ) and  ^('S^t^pj  between which the forbidden t r a n s i t i o n takes  place,are  shown schematically i n figure J>. Prom equation  (19) » the  t o t a l i n t e n s i t y of the t r a n s i t i o n between states T"( *P £ -£) 0  and  f( 'S ttop)i s given by: 0  28  F i g u r e 3.  Zeeman components of the F o r b i d d e n electric  6s*  'S,  dipole t r a n s i t i o n  ) of  Hg" . f  (SsTp^-  29  T r a n s i t i o n s between unperturbed  levels  *P  e  and  *S  are  0  forbidden. This implies that lei-  Prom eq.  «nv<|- »>  - E  ^  o  Prom e q u a t i o n  ( 1 1 ) , we  (X)  0  (^)  s  f  >  ( 8 ) and eq.  =  "Here  >|  ( 2 0 ) , we  - £  have  + -I  ('s >  0  o  have  and  A  * •> / "ft I X t V t - i | T,U *ri? < 3 , v * * > / < P. IIT*iiX> p  s  £  Xt v  J  I f we  and into  denote  4  x  s u b s t i t u t e v a l u e s of ( 2 2 ) , we  c, , C  if  C± from e q u a t i o n s  o b t a i n ah e x p l i c i t  (11), (12)  e x p r e s s i o n f o r the  f o r the t r a n s i t i o n between p e r t u r b e d s t a t e s ^ ( P t * i ) 3  0  1 < Po-i?3 J  X V  intensity and  3 0  Similar c a l c u l a t i o n s also give the i n t e n s i t i e s of the other two components. Prom eq. (19)» we have X 0?  o  $ i '5. ™ f )  Again we have  =t  ^  c t -»- > I <pU TNI ' >I" , i  f  l  So  I I <*P,ii|T,*J*.tvr and  <r.*11 r t i x t > v  =2 <»f, i t (T.S, ix t  J  t  "*f**> 4-"r> I <P. HT^Ii X>|* L  i  By s u b s t i t u t i n g values of c' , c'^ and 6^ from equations (11), f  (12) into (24), we obtain an e x p l i c i t expression f o r the i n t e n s i t y of the t r a n s i t i o n between perturbed states  <K*P.tt) and lf( S t»»y). ,  e  In our case, the external magnetic f i e l d never exceeds 10  6  gauss. The expressions (23) and (25) can be expanded  into powers of H:^.  31  1'  The t o t a l i n t e n s i t y  Go  I  a  E„  Go*  I  of the forbidden l i n e has been  measured i n the absence of an external magnetic f i e l d . The value of the normalisation factor K i n terms of I to  J V I , . ^ f o r Hs = 0) :  0  i s equal  K=  The i n t e n s i t i e s f o r the Zeeman components °of the forbidden l i n e (6sp *P - 6s* 's ) of Eg' 0  I i s therefore given  e  by the f o l l o w i n g expressions:-  i -2>r46  Xio""  ttj.  -1.0.7  fto''*/^}  The f a c t that the above approximate expressions are very close to the exact expressions  (23) and (25) have been  checked numerically by means of the computer. From equations  (8), (16) and (17), we see that the  two Zeeman components a and b have the same energy C.C'M-E.C'W  •  ^  32  The hyperfine energy s h i f t which i s represented by the term  r  * " i s extremely small. I t s value i s about  - ^ 'OX \  - A.  0  _ 1  C~*v .  Therefore i n the presence  of external magnetic f i e l d which  does not exceed 10^ gauss, the system exhibits normal Zeeman e f f e c t with one s l i g h t l y s h i f t e d  c e n t r a l l i n e and two  lines  symmetrically displaced from the centre. The c e n t r a l line consists of two components a and b and has i n t e n s i t y  given  by  The  intensity  of the c e n t r a l l i n e decreases extremely  with the i n c r e a s i n g f i e l d , but that of the other two  slowly lines  varies l i n e a r l y , l i n e d increases with the f i e l d while l i n e c decreases. However, the change i n i n t e n s i t y  for a l l three  l i n e s i s s t i l l very small, as we have expected i n the previous chapter. When the f i e l d increases from zero to 10  fc  gaussy the c e n t r a l l i n e decreases by only 10*%,  decreases by 8$ and l i n e d increases by  line c  8$.  The separation between l i n e s c and d increases l i n e a r l y with the f i e l d . For a f i e l d of 10 separation i s 0.0254 cm"'.  6  gauss, the  For f i e l d s available i n laboratory,  the three Zeeman l i n e s cannot be resolved by conventional spectroscopic methods and the unresolved l i n e s w i l l have a total intensity  I = I* * l + c  CHAPTER  5.  THE RELATIVE INTENSITIES OP THE HYPERFINE COMPONENTS OP TWO FORBIDDEN LINES (6 's^ - 6 *P) (  AND  (6 'S. - 6 * P J OF P b *  C a l c u l a t i o n of Perturbed  e7  5:1  Wave F u n c t i o n s  For those i s o t o p e s o f Pb which do not have a n u c l e a r magnetic moment, the two f o r b i d d e n l i n e s  (6 'S - 6 P ) and l  0  (  (6 'S - 6 *Pj.) are due t o pure magnetic d i p o l e and pure 0  electric  quadrupole t r a n s i t i o n s . However, as we have mention-  ed i n the i n t r o d u c t i o n , the i s o t o p e Pb^" * has a n u c l e a r 0  s p i n £ , the p e r t u r b a t i o n due t o the presence o f the n u c l e a r magnetic d i p o l e moment causes a m i x i n g o f the ;P, and P J  X  wave f u n c t i o n s . As a r e s u l t , the two f o r b i d d e n l i n e s become mixed m u l t i p o l e  t r a n s i t i o n l i n e s . To c a l c u l a t e the i n t e n s i -  t i e s of t h e i r hyperfine perturbed  components, we have t o f i n d the  wave f u n c t i o n s o f the l e v e l s P, and P . The 3  J  Z  lowest f i v e energy l e v e l s of Pb  are shown s c h e m a t i c a l l y  i n f i g u r e 4 . To f i n d the p e r t u r b e d  wave f u n c t i o n s we c o n s i -  der o n l y the case when t h e r e i s no e x t e r n a l magnetic  field.  The p e r t u r b a t i o n c o n s i s t s o f o n l y one term which i s the hyperfine  interaction  g i v e n by eq. ( 5 ) .  elements of the p e r t u r b a t i o n #  | (ote?F»Y> J  is  h  The m a t r i x  i n the r e p r e s e n t a t i o n  34  Figure 4.  The lowest f i v e energy l e v e l s of Pb. A l l of them belong to configuration  6s*-6v (Mrozowski 1940) x  35  i s d i a g o n a l i n quantum numbers F and >tfp. The s t a t e s j)o<i3-pWp>!  a  r  e  t h e r e f o r e c o r r e c t z e r o t h order s t a t e s f o r  each l e v e l . The  s e p a r a t i o n between the l e v e l s T  and *P i s  i  (  s m a l l when compared t o t h e i r s e p a r a t i o n from other To a f i r s t  approximation  f u n c t i o n s of l e v e l s mixing while  S  P  (  i n calculating and *P  levels.  p e r t u r b e d wave  , we can c o n s i d e r only the  X  o f the wave f u n c t i o n s o f the l e v e l s P, and *P^ , J  c o n t r i b u t i o n s due t o wave f u n c t i o n s o f other  are n e g l e c t e d . With t h i s i n mind and u s i n g standard  levels first  o r d e r p e r t u r b a t i o n t h e o r y , we f i n d t h a t the p e r t u r b e d wave f u n c t i o n s of the l e v e l s  *S  , *P, and P J  0  X  and t h e i r  energies  are as f o l l o w s : -  •fe( ' S o *  <  ,  ; ^ i % i  l  r , i v  I  (14)  where E (o() (o< = ! s , P J  e  6  of the c o r r e s p o n d i n g figure  4.  (  , *P ) a r e the unperturbed  levels  X  energies  t h e i r v a l u e s are g i v e n i n  36  The r e q u i r e d m a t r i x elements can be w r i t t e n e x p l i c i t l y u s i n g e q u a t i o n (32)  <'*o i * I X I X i > » > ^ f  f  where  T>, - < P, l|H li r,> J  K =  3  J  < pxiin n r»> J  3  5  The n o r m a l i s e d p e r t u r b e d s t a t e s and t h e i r e n e r g i e s t h e r e f o r e have the f o l l o w i n g v a l u e s : •  £<*P, i ^  w  n  e  r  e  fcB  "  * ^ P'^ + ^ ^ P ' C?  I  ^t p >B ( fl3 J  v  0  J  The p e r t u r b e d s t a t e s and p o s s i b l e o p t i c a l between them are shown s c h e m a t i c a l l y i n f i g u r e 5.  transitions  F i g u r e 5.  H y p e r f i n e components of the levels  'S , \ 4  , *P, of Pb***-  38  Relative Intensities  of the Hyperfine  Because some of the i n the p r e v i o u s unperturbed  section  Components  p e r t u r b e d wave f u n c t i o n s  obtained  are l i n e a r combinations of the  wave f u n c t i o n s of both *P,  forbidden transitions  5:2  ((>'S  0  -t*P, ) and  and  *P  (b's  o  l e v e l s , the  t  - 6*P ) are X  no  l o n g e r pure m u l t i p o l e t r a n s i t i o n s , each b e i n g a mixed magnetic d i p o l e and  e l e c t r i c quadrupole t r a n s i t i o n . Since  t h e r e i s no e x t e r n a l magnetic f i e l d , are s t i l l  states  degenerate i n the magnetic quantum number Yn . f  Prom e q u a t i o n given  the p e r t u r b e d  ( 1 8 ) , the i n t e n s i t y f o r the component 1 i s  by  Since e l e c t r i c  quadrupole t r a n s i t i o n s between  levels  'S  *P  and  Prom e q u a t i o n  0  are s t r i c t l y f o r b i d d e n , we  ( 2 1 ) , we  unperturbed have  have  = -I I < ' U l T * l l f i > f J  Denot i n g  we  obtain We  calculate  s i m i l a r way,  expressions  using also,  f o r 1^,  i f necessary,  1^ and  1^ i n a  the f a c t t h a t  magnetic d i p o l e t r a n s i t i o n s between unperturbed  *P  A  and  39  'S  e  levels  are s t r i c t l y f o r b i d d e n , i . e.  where  ( ,  The unperturbed U*SP  -&P,  Jf  Mzt>f  j < \ || J | J */> > P  ^  T  total intensity  of the  line  ) X.46I8A can be o b t a i n e d by adding I, and 1^ and  then p u t t i n g  Similarly,  +  which g i v e s the value  by p u t t i n g  the unperturbed t o t a l  /»x.*e  and adding I  intensity  $  and  of the l i n e  I  4  (4 's  6  we o b t a i n -4*P ) X  A5313A  -^2>  I <X»Tfii'p >l  Prom e x p r e s s i o n s (39)  (  t o (44),  x  the r e l a t i v e  t i e s of the two h y p e r f i n e components of each l i n e w r i t t e n as f o l l o w s : -  -4  intensican be  40  To o b t a i n a n u m e r i c a l value to f i n d the v a l u e s f o r /hf, , From e q u a t i o n s splitting  f o r 4? » and ~ , we have  /**fx  ,^  I /«x<TY/l HYfi >| .  and  (37) and (38), the h y p e r f i n e energy  o f the l e v e l s P, and ^Pj, i s g i v e n r e s p e c t i v e l y by J  A E, m E C r . t ^ ) - ^ ^ f  »^  and  The  experimental  A  £  A  t  (  - 0• M7 -  x  v a l u e s are (Mrozowski 1940)  -  0 '2.1 ^  = r«o *  and Therefore  0.056 cm"',  the v a l u e s o f  and  are r e s p e c t i v e l y  and 0.109 cm""' .  I f we assume t h a t a l l reduced connecting l e v e l s  i  P  i  Therefore  m a t r i x elements o f  HA  and *P, have the same order o f  JA  magnitude, then we can put  The  /*Jp».  • «l*«s 5 *'  x  I  0  <'P> H H7H P, > | ^ ?  C-lt^[  .  r e l a t i v e i n t e n s i t i e s f o r the h y p e r f i n e components  t h e r e f o r e have the v a l u e s X  The effect  I  second  term: i n the b r a c k e t s r e p r e s e n t s the  o f the presence  o f the n u c l e a r d i p o l e moment. The  41  f a c t o r s 2 and }/ axe the r a t i o s of the s t a t i s t i c a l x  weights o f  the h y p e r f i n e components of the two f o r b i d d e n l i n e s . The s t a t i s t i c a l weight of the h y p e r f i n e component which i s caused by the t r a n s i t i o n F - » F ' i s g i v e n by (i"p + 0 ( * Using t h i s e x p r e s s i o n the s t a t i s t i c a l weight of the h y p e r f i n e components  ( 'S6T--Z-+ *P,  and ( *Se  £ -» V,  l y 4 and 8, and t h a t of the h y p e r f i n e ( ' S ^ i ^ ' P , f » i:  8 and 12.  are r e s p e c t i v e -  components  ) and ( 'S, ? * i -* *P> F = £) are r e s p e c t i v e l y  42  BIBLIOGRAPHY 1. Archbold, J . W. 1958. Sons, L t d . ) .  Algebra  ( S i r Isaae Pitman and  2. Baeher, R. P. and Goudsmit, S. 1932. Atomic Energy States (McGraw H i l l Book Company, New York and London). 3. Biedenharn, L. C , B l a t t , J . M., and Rose, M. E. Rev. of Mod. Phys. 24, 249.  1952.  4. B l a t t , J . M. and Weisskopf, V. P. 1952. Theoretical Nuclear Physics (John Wiley and Sons, New York). 5. Garstang, R. H. 1962. (Chapter 1) Atomic and Molecular Processes (Edited by Bates, Academic Press). 6. Kopfermann, H. 1958. Nuclear Moments (English Version prepared from the second German e d i t i o n by Schneider, E. E., Academic Press). 7. Ling, D. S. and Palkoff, D. L.  1949.  Phys. Rev. 76, 1639.  8. Mrozowski, S.  1938.  Z e i t s . f . Physik 108, 204.  9. Mrozowski, S.  1940.  Phys. Rev. 58, 1086.  10. Opechowski, W.  1938.  Z e i t s . f . Physik 109, 485.  11. Rose, M. E. 1957. Elementary Theory of Angluar Momentum (John Wiley and Sons, Inc., New York). 12. Schttler, H. and Jones, E . G. 77, 801.  1932.  Z e i t s . f . Physik  

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