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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 in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1967 r In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . . I t i s u n d e r s t o o d t h a t c o p y i n g 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 g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa r tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada ABSTRACT The forbidden line (6sp*P0 - 6sa's6 ) 7^2656A i s forbidden for a l l multipole transitions i f the nucleus of Hg isotope has no magnetic moment. For Hg the selection rule i s broken by the interaction of the nuclear magnetic dipole moment with the atomic electrons. In this thesis the Zeeman effect of the line i s discussed in detai l . In particular, the intensity of the Zeeman components i s calculated as a function of the external magnetic f i e l d . It turns out that even for an external magnetic f i e l d sufficient to produce the Back-Goudsmit effect (30,000 gauss) the intensity of Zeeman components changes by less than one percent. Also the effect of the interaction of the nuclear magnetic moment and atomic electrons on the intensity of forbidden lines (6 *S0 - 63P,)X4618A and (6'S0 - 6 *Pa ) ^  5313A of Pb i s calculated (in absence of external magnetic f i e l d ) . The effect i s very small. I i i 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 . . .4 2:2 Calculation 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 o ) IN Hg ' " 24 4:1 General Expression for the Intensity of Multipole Radiation 24 4:2 Variation 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 'S0 - 6,*PZ) OF Pb*"* 33 5:1 Calculation of Perturbed Wave Functions. . . . 33 5:2 Relative Intensities of the Hyperfine Components. 38 BIBLIOGRAPHY 42 i v LISO? OP ILLUSTRATIONS Figure Page 1. The lowest five energy levels of Hg 11 2. Energy diagram of the perturbed levels of Hg,?' . .22: 3. Zeeman components of the forbidden electric dipole transition (6sp *P0 - 6s1 'S0) of Hg"* . . . 28 4. The lowest five energy levels of Pb 34 5. Hyperfine components of the levels % , *P, , X of Pb*** 37 V ACKNOWLEDGEMENTS I wish to express my gratitude to Professor W. Opechowski f o r suggesting t h i s problem and f o r h i s continued i n t e r e s t and f r u i t f u l discussions throughout; the performance of t h i s research. I wish to thank the Commonwealth Scholarship and Fellowship Committee f o r the f i n a n c i a l assistance i n the form of a scholarship. Also, to my fri e n d s Miss G l o r i a Chang and Miss Wat Chi- K i t I wish to express my gratitude f o r typing t h i s t h e s i s . CHAPTER 1. INTRODUCTION The forbidden l i n e (6sp*P0 - 6sa 'stt)A2656A of Hg ?* i s a pure e l e c t r i c dipole t r a n s i t i o n . Being a t r a n s i t i o n between two l e v e l s both with quantum numbers J=0, i t i s s t r i c t l y forbidden according to the usual s e l e c t i o n r u l e s . The f a c t that the l i n e may be due to the presence of the nuclear magnetic moment was suggested by several authors (Bowen and others). The weak i n t e r a c t i o n of the nuclear magnetic moment with the electrons causes a s l i g h t admixture of the *P, wave functions to the i P 0 wave function^and therefore there e x i s t s a small but f i n i t e t r a n s i t i o n p r o b a b i l i t y between the two perturbed l e v e l s ^P 0 and 'S0 . The r a t i o of the i n t e n s i t y of t h i s l i n e and the i n t e n s i t y of the l i n e ( S s p ^ - 6sx 'S6 )\2537A i n the absence of external f i e l d was calculated by ¥. Opechowski (1938) and was found i n agreement with experiment (Mrozowski 1937). We investigate the behaviour of the i n t e n s i t y of the forbidden l i n e under the influence of an external magnetic f i e l d by adding the e l e c t r o n i c and nuclear Zeeman i n t e r -actions to the hamiltonian. Only the mixing of the JP, and i P e wave functions i s considered. This approximation i s j u s t i f i e d as the energy separation between these two le v e l s i s much smaller than t h e i r separation from other l e v e l s . The range of the f i e l d f o r which our c a l c u l a t i o n i s v a l i d goes from zero to 10* gauss. I t i s well below the 2 value when the Paschen-Back effect, starts 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 for our case. The Back-Goudsmit decoupling of F into I and J in the JP, level occurs at field s around 3 X 10 gauss. At f i r s t sight one may think that the inten-sity of the forbidden line w i l l decrease considerably when the f i e l d reaches this Back-Goudsmit l i m i t . However the coefficients of the ?P, wave functions in the expression for the perturbed *P0 wave functions are mainly determined by the energy difference between the JP/ a n d ^ states. Now the energy difference varies only by a very small fraction even when the f i e l d i s as high as 10* gauss. The influence of the 3P, level on the levels *P0 and 'S0 in the presence of an external f i e l d is therefore nearly the same as that when no external f i e l d i s present and the two levels show normal Zeeman s p l i t t i n g . As a result, a normal Zeeman effect of the forbidden line (6sp JP 0 - 6s^ * S Q ) i s predicted and the intensity of the components changes only very slightly when an external magnetic f i e l d i s introduced even as high as 106 gauss. When the nuclear spin and external magnetic f i e l d are both absent, the two forbidden lines (6 *S0 - 6 iP, ) A 4618A and (6 'S0 - 6*P^)A.5313A of Pb are respectively pure magnetic dipole and pure electric quadrupole transitions. Their relative intensities 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 se to hyperfine structures i n the t ransi t ions , but also causes a mixing of wave functions between the *P, and *P 4 levels . Each hyperfine component w i l l no longer be a pure multipole t ransi t ion , but w i l l be a mixed t ransi t ion of both magnetic dipole and e lec t r ic quadrupole type. The relat ive intensi t ies for the hyper-f ine components of each forbidden l i n e are calculated from the perturbed wave functions. It i s found that they deviate only very s l i g h t l y from the relat ive s t a t i s t i c a l weights of the hyperfine components. Again this i s obvious i f one considers the fact that the hyperfine interaction i s extr-emely small when compared to the energy separation between the JP, and J P X l evels , 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 ines by Garstang (1962) contains a discussion of the case of the l ine (6sp*P6 - 6s* 'Se ) X2656A of Hg i n . absence of an external magnetic f i e l d . CHAPTER 2. HAMILTONIAN OP THE SYSTEM CONSIDERED general Form of the Hamiltonian 2:1 The hamiltonian of the atom of Pb or Hg i n the presence of an external f i e l d H consists of four terms: # = X e * + + 76' a). i s the unperturbed hamiltonian f o r the atom i n absence of nuclear spins and i n absence of the .interactions with the external magnetic f i e l d H and with the r a d i a t i o n f i e l d . 76» i s the i n t e r a c t i o n of the nuclear moments with the electrons. TCt i s the i n t e r a c t i o n of the atomic and nuclear magnetic moments with the external magnetic f i e l d H and 76' i s the i n t e r a c t i o n of the atom with the r a d i a t i o n f i e l d . Since nuclear e l e c t r i c quadrupole moment i s absent fo r both Pb-1*7 and Hg'?* and contributions due to higher moments are small compared to that due to the magnetic dipole moment, we can write where ^ i s the nuclear magnetic dipole moment and H 7 i s the magnetic f i e l d produced by the electrons at the nucleus. For our c a l c u l a t i o n , i t i s more convenient to express 76n as the scal a r product of two i r r e d u c i b l e tensors, each of which i s of rank 1. Xn = - I I * , ( » . where I , = - - £ ( X „ + i X > ) 5 and the l a t i n subscripts denote components i n a Cartesian coordinate system. To express i t i s convenient and without loss of gen e r a l i t y to assume that the external magnetic f i e l d H i s p a r a l l e l to the z-axis. Denoting by j^the magnetic dipole moment of the electrons, we have While # n and are stationary perturbations, i s time dependent. Its magnitude i s much smaller than^nd #x The energy s h i f t of the atom caused by #'is n e g l i g i b l e as compared to that caused by e i t h e r Hnor by K^* Therefore we treat X,' only as an agent that causes t r a n s i t i o n s between d i f f e r e n t states. C a l c u l a t i o n of Matrix Elements of the Stationary Perturba- t i o n 2:2 In the c a l c u l a t i o n of the i n t e n s i t y of the forbidden l i n e s , one needs to know the energy and the perturbed wave functions of the states between which the t r a n s i t i o n takes 6 place. For such purpose, we derive an expression f o r the matrix elements of the perturbation which w i l l be denoted by V: V * Hn + fa «). I f we denote by I the nuclear spin quantum number, J the t o t a l e l e c t r o n i c angular momentum, F the t o t a l atomic angular momentum, the atomic magnetic quantum number, and «t a set of eigenvalues of other observables which are mutually commutative and which also commute with l \ T\ f l and the z-component of F, then £ 1**7ra^j i s a set of eigenstates of which i s the hamiltonian of a completely i s o l a t e d atom without any nuclear spin where Ej&o) i s the corresponding unperturbed eigen-energy. being s c a l a r product of two vectors i s invariant under r o t a t i o n and therefore commutes with F A and F:^ . Consequently, i s diagonal i n quantum numbers F and »y. Using theorems on additi o n of angular momenta, we have Applying Wigner-Erckart theorem to matrix elements of H J and I, i t follows that 7 and < I ^ | ^ | I W , > ~ ^i^+M < x I >»SM I X 7nti-M> where :r> i s a reduced matrix 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 properties of the Glebsch-Gordan c o e f f i c -i e n t s , t h e i r indices can be rearranged as follows: < X / y*>x / x >y,'x > _ (_0>»»i-»«z < x t ^ W2-**»x / X > » r > and < 7 / > » f - > » 9 x " i - ^ / J' w^-^i > 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 obtain (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 matrix elements of 3^ , we calculate those of Is and J> f i r s t . 8 But = - / iTxTo < i <*>x I x *w,> Also, symmetry properties 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 ' Therefore we have: ? ' ¥ / 1 ^ f <*a 5 p l>y > S i m i l a r c a l c u l a t i o n s give:: Using the above two expressions, we can write down the expression f o r the matrix elements of Hk, . = $r'j ^ > » f / 2 f ' + j ' < » >»if 0 | f" wf > Wj. * + t H ) l x o - U ? + T 0 W ( m - t ' } i x ) j F i n a l l y , the expression f o r the matrix elements of V can 9 be written as CHAPTER 3. CALCULATION OP PERTURBED WAVE FUNCTIONS FOR Hg' f f The lowest energy l e v e l s of Hg are shown schemat-i c a l l y i n fi g u r e 1. i'rom the fi g u r e we see that energy diffe r e n c e between l e v e l s *P and *Pe are small while energy differences between other l e v e l s are much l a r g e r . This f a c t suggests that to a good approximation we may consider only the mixing of the iP, and sP e wave functions, neglecting the admixture of other wave functions to these two l e v e l s . The nuclear spin of Eg"* i s £ . From:eq.(3), i t follows immediately that the matrix elements of V f o r the ground state *S0 i n representation j I^FH^J are given by: = - X ™f/Ux ?-v f », p ( «>»»£ £ ) (of course thetsame"expression follows from eq.(7)) This l e v e l therefore s p l i t s into two Zeeman Comp-onents <K'S. i-Mf) * \ f S . ± *>f> (9) with energy 11 Figure 1. The lowest five energy levels of Hg (Baeher and Goudsmit 1932). The P levels in the figure belong to configuration 6s6p and the 'Se level belongs to Ss*con-figuration. The excited levels are drawn to some arbitrary scale but the ground level i s not to scale. 12 To ca l c u l a t e the perturbed states of l e v e l P0 , we have to solve the eigenvalue problem f o r the submatrix (belonging to l e v e l s *P0 and *P, ) of the matrix f o r the t o t a l stationary hamiltonian of the system, namely K « Ke + V The required matrix elements are .of the'.form and are given by equation (7). We write them down e x p l i c i t -l y as follows: <*P, i t / W * P , i i > 13 B = <*P, w i v| Jr ,t-i> = <V,f i|v|*f, i i> = <?f, * i I v p r . i i > 14 where <f> = <^ P( J|HJ||3P,> \ = <?F,l|H:rl|ir6> and a l l other matrix elements vanish. The zeroth order hamitlonian # e i s defined as unique only up to an additive constant and we can, f o r convenience i n the following c a l c u l a t i o n s , choose E„( 5P, )=0. Then The symmetry group f o r the hamiltonian i s and matrix elements of the hamiltonian are diagonal i n »)f . Taking t h i s into account, the submatrix of ~H can be written i n the following reduced form: 15 T , ' f , v , ' f , 3 . j _ 2. X 2 3. X 2 A 1 Jk X ~ a 0 0 e 0 _ J . a- ' l F B _ X X i »f * ' i 0 T* J. J L »P 2 - F , 0 B % JL » P A. • < F 6 1 0 *» 5. ~ A. A 3 i i f From t h i s submatrix, i t i s immediately obvious that f o r "tof*±j[ > the perturbed wave functions are with corresponding energy and EC JP, i I ) * ~ £ < M " 16 The problem i s now reduced to the s o l u t i o n of the eigenvalue problem of the following two matrices:-0 ?* 't. B V, T B a i T, To * l *p, f-i i t 0 F* T. 0 B J . 1 X We s h a l l f i n d the eigenvalues of 3l{ f i r s t . Assuming that X i s the eigenvalue, the secular equation f o r JCt i s : A* + <*, V + A , . A •+ A j = 0 0 o.). where 17 Since the hamiltonian i s hermitian i t s eigenvalues are a l l r e a l . The cubic equation with three real roots can be solved by standard algebraic methods. If we define (See, for example, J-. W. Arehbold 1958, Chapter 12) <t> = W ( z £ L ) © < * * 7 1 then in our case both G and H. are real and furthermore h < o and Cf < o . The last two inequalities are not obvious analytically, but their v a l i d i t y has been checked numerically using an IBM ?040 computer for magnetic fields 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 cond-itions are satisfied, the three roots are given by the following expressions:-When we put yUx —* o and H| ° we have X,-*o ^ - A - * and X3 -» © Therefore from continuity the eigenstate belonging to X x becomes \l?t>t~i> in the above limits, and we have \ x as the energy of the perturbed state ^(JPo * ~£) . 18/ tfC iP b-t-i)= ft, l*P 0t-£> + «* £ - * > * S IV. i-i> where c, = £ J ( A , ( A , - A 4 ) - B X } and The secular equation f o r matrix ^ i s as follows:-A 1 + QL'% f + O + A i = ° where ^ = - ( E 0 + ^ U j U { ) * x ~ A * \ - 6 1 4 A i ) ( B l > + A 4 ) - / f i * Here we observe^, that equation (13) i s obtained from equa-t i o n (10) simply by rep l a c i n g and A I + J ( i= 1, 2, 3)• From the e x p l i c i t expressions f o r the matrix elements i t i s obvious that each A X + I can be obtained from A^ by reversing the sign of H.^ , and that B1" i s an even function of Hj while F i s independent of HJ;. Since only B* occurs i n equation (10) and (13)> i t i s cl e a r that (13) can be obtained from (10) by simply reversing the sign of H^. Therefore the roots A* (•£=',*,*) of (13) are the correspond-ing roots of (10) with the sign of H^ reversed. From continuity arguments, A ^ ( w p =Ai(-Wj) i s the energy belonging to the perturbed state f (3Po t i) . E<*P. £ 1 ) = A ; c n p = A , C - H ^ 0 4 ) . 19 where <^=^{ ( A ' x - A 4 ) - Bx} In order to f i n d out i f we can expand the expressions f o r the eigenvalues and wave functions into powers of H.^  and r e t a i n i n g only terms l i n e a r and quadratic i n H^ ., we have to estimate the numerical values of such quantities as^u^, yUj , f ,^and E e . 191 The nuclear magnetic moment of Hg i s 0.49926 nuclear magneton (H. Kopfermann 1958) which i s approximately 1.27 * 10 tm.'; gauss . Since our c a l c u l a t i o n i s only approximate and i s expected only to give values to within an order of magnitude, we can assume the Kussell-Saunders coupling to calculate^^tj.. The Lande g-factor, which i s given t h e o r e t i c a l l y as 5 = / + _TC7+Q + S(Stl) -L(L+l) i s 1.5 f o r the l e v e l iP l . This gives a value of 7.01x10 * Orr>~' gauss"' f o r yUy . To estimate values ofandyu^f, we assume that the matrix elements of ^ £,for the l e v e l s iP e and *P, have the same order of magnitude. This implies that_/*a|> and /**\\\ have the same order of magnitude. Also when the external f i e l d i s absent the hyperfine s p l i t t i n g of the l e v e l *P, i s of the order <JP( f *>f\3t»\ iP, ? M f > 20 and i s found experimentally to have a value of 0.4 frrf' (Schuler and Jones 1932), therefore we can assume that y^ipcr 0«4 fc*»v' . For convenience, we rewrite the numerical values we have discussed: Using the above values we f i n d that f o r an external f i e l d which does not exceed 10 6 gauss, the following i n e q u a l i t y always holds: 0 $ < 0 - I 0 A To a good approximation, we can write In a l l the quantities G, H and D, the terms depending on are much smaller than the constant terms. We can therefore approximate the expression f o r by a polynomial i n H^. In f a c t , a f t e r expanding the functions i n G, H and D i n powers of H£ i n the usual manner, we have Expressions f o r the other quantities required are determined i n the s i m i l a r way. We just write them down as follows: 2-1 Here we see that the wave function and the eigen-energies .. for the 3P e level 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 for the *P, wave functions have also been calculated numerically using the computer to get an idea about the influence of this level on the wave functions of the 3P 0 l e v e l . The energy diagrams of the three levels are shown schematically in figure 2. The energy of the hyperfine components of the *P, level changes greatly in pattern at a f i e l d around 3 *10 4 gauss. This i s due to the onset of the Back-Goudsmit effect on the hyperfine components. However, the change in magnitude of the energies i s s t i l l extremely small in comparison to the energy separation between the 3P, and *P0 levels. This explains qualitatively the fact that the wave functions of Figure 2. Energy diagram of the perturbed l e v e l s of H g m . 23 the 3P 0 l e v e l changes only very s l i g h t l y even f o r values of magnetic f i e l d such that P i s decoupled into I and J . We may also mention that i n our case the e l e c t r o n i c f i n e s p l i t t i n g i s of order E 0 and the e l e c t r o n i c magnetic moment has v a l u e . The Paschen-Back decoupling of J into L and S needs a f i e l d of order 3 X 10 ? gauss and f o r f i e l d s with values below 10^ gauss as we are now considering, J remains a good quantum number. As the wave functions f o r l e v e l *P0 change;, only very s l i g h t l y i n the presence of external f i e l d , we expect that the i n t e n s i t i e s of the Zeeman components w i l l also change very l i t t l e . However, to prove that we s h a l l derive a n a l y t i c expressions f o r the i n t e n s i t i e s i n the following chapter. CHAPTER 4. THE INTENSITY OP ZEEMAN COMPONENTS OP THE FORBIDDEN LINE (6 *PD - 6 'S0) IN Hg'" General Expression for the Intensity of Multipole Radiation  4:1 The forbidden l i n e (6sp*P0 - 6s" 'se ) of Hg' ? ? i s an e l e c t r i c dipole r a d i a t i o n . However, since we are going to consider the case of mixed magnetic dipole r a d i a t i o n and e l e c t r i c quadrupole r a d i a t i o n i n the following chapter, we s h a l l write down a general expression f o r multipole r a d i a t i o n i n t h i s section. I f H / I * * » , > and KjUw^) are two eigenstates of a r a d i a t i n g system with angular momentum quantum numbers and jim*. and with energies E*,^ and E<*^ \_ re s p e c t i v e l y , then the expression f o r the i n t e n s i t y of the mM. component of an e l e c t r i c (or magnetic; 2^ -pole t r a n s i t i o n between the two states i s given by (Blatt and weisskopf 1952J where 1 ^ (or ) i s the » A component of the e l e c t r i c (or magnetic) 2l -pole moment, i s the energy separation between the two states, k i s a constant which has the 25 and c is the velocity of light in vacuo. It has been shown (Ling and Falkoff 1949) that the total intensity for the component of a mixed electric 2 -pole radiation with magnetic 2 -pole radiation i s obtained by just adding up the two individual intensities. where "J^and are the corresponding electric (4*t )-pole and magnetic (/^rw)-pole moment. If the measuring apparatus are insensitive to the polarisation of the incident light as is assumed here, then the intensity measured w i l l be the contribution due to a l l possible multipole components and the expression for the total intensity for a mixed electric 2^-pole and magnetic 2 -pole transition between the states l o t , a n d /<^ »/>>>^ > The intensity for a pure electric or magnetic 2^-pole radia-tion is For convenience in later calculations, i t i s useful here to derive an explicit expression for the matrix elements :2S of the multipole moment Tf^ and a sum rule f o r them. In t h i s d e r i v a t i o n , the e x p l i c i t expression f o r 7 ^ does not concern us because we only use i t s transformation properties under r o t a t i o n s . I t i s well-known that the mu l t i -pole moments (^ = -^ .-, A ) transforms as an i r r e d u c i b l e tensor of rank 1 and we can again apply 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 < x j 'm, y - ^ t j T W > < I T >*>* wy W i ' * *V <£*'7' Ttr* I * 7 *>~'*'JC > =• < J JL mf-™x >» / -j' ^ -m, > <*'J' H 1*H #J> where <<K'J'I|T^ |I«<T> i s the reduced matrix element of T / M , Therefore we have where #=.2 <* 3 X > » > t - ^ | ^ ^ I ^ V ^ ^-^/Tw^-"^ Prom symmetry properties of the Clebsch-Gordan c o e f f i c i e n t s , we have The sum i n the expression f o r S i s equal to 27 The expression for the matrix elements can therefore be rewritten as follows: VaF +)<3Lj' +,) W ( j j ' f f J * * ) < 4 ' J ' | ( T £ / | V J > (20). (except for notation, this expression i s the same as eq.(6.24) in E. M. fiose (1957) ) using eq.(20) and the orthogonality-relation of the Glebsch-Gordan coefficients We can write down the following sum-rule for the sum of absolute squares of matrix elements: Variation of the Intensity with External Magnetic Field  4:2 The perturbed states V ( i p 0 t ™ j : ) and ^('S^t^ p j between which the forbidden transition takes place,are shown schematically in figure J>. Prom equation (19) » the total intensity of the transition between states T"( *P0£ -£) and f( 'S0t top) is given by: 28 Figure 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 (SsTp^-6s* ' S , ) of Hg" f. 2 9 Transitions between unperturbed l e v e l s *Pe and *S0 are forbidden. This implies that l e i - >| «nv<|- >»> f Prom eq. ( 8 ) and eq. ( 1 1 ) , we have = - E 0 (X) + -I "Here ^ o s (^) - £ 0 ('so> Prom equation ( 2 0 ) , we have and A s £ * •> / "ft I X t V t - i | T,U X t v *ri? < 3 , v * * > / <JP. IIT*iiX> p I f we denote x 4 and substitute values of c, , Cif C± from equations ( 1 1 ) , ( 1 2 ) i n t o ( 2 2 ) , we obtain ah e x p l i c i t expression for the i n t e n s i t y for the t r a n s i t i o n between perturbed states ^ ( 3 P 0 t * i ) and 1 <JPo-i?3 X V 3 0 and Similar calculations also give the intensities of the other two components. Prom eq. (19)» we have X 0?o $ i '5. ™ f ) Again we have = t ^ c t -»-f> I <iplU TNI 'So>I" , I I < * P , i i | T , * J * . t v r =2 <»f, i t (T.S, ix t <Jr.*11 r t t i x t>v "*f**>L4-"r> I <iP. HT I^i X>|* By substituting values of c'f , c'^  and 6^  from equations (11), (12) into (24), we obtain an expli c i t expression for the intensity of the transition between perturbed states <K*P.tt) and lf( ,Set»»y). 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 ' Go E „ G o * I The total intensity I a of the forbidden line has been measured in the absence of an external magnetic f i e l d . The value of the normalisation factor K in terms of I 0 i s equal to J V I , . ^ for Hs = 0) : K = The intensities for the Zeeman components °of the forbidden line (6sp *P0 - 6s* 'se) of Eg' I i s therefore given by the following expressions:-i -2>r46 X i o " " ttj. - 1.0 . 7 fto''*/^} ^ The fact 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 shift which is represented by the term r *"is extremely small. Its value is about - A. _ 1 - ^  'OX \ 0 C~*v . Therefore in the presence of external magnetic f i e l d which does not exceed 10^ gauss, the system exhibits normal Zeeman effect with one sl i g h t l y shifted central line and two lines symmetrically displaced from the centre. The central line consists of two components a and b and has intensity given by The intensity of the central line decreases extremely slowly with the increasing f i e l d , but that of the other two lines varies linearly, line d increases with the f i e l d while line c decreases. However, the change in intensity for a l l three lines is s t i l l very small, as we have expected in the previous chapter. When the f i e l d increases from zero to 10fc gaussy the central line decreases by only 10*%, line c decreases by 8$ and line d increases by 8$. The separation between lines c and d increases linearly with the f i e l d . For a f i e l d of 10 6 gauss, the separation i s 0.0254 cm"'. For f i e l d s available in laboratory, the three Zeeman lines cannot be resolved by conventional spectroscopic methods and the unresolved lines w i l l have a total intensity I = I* * lc + CHAPTER 5 . THE RELATIVE INTENSITIES OP THE HYPERFINE COMPONENTS OP TWO FORBIDDEN LINES (6 's^  - 6 *P() AND (6 'S. - 6 *PJ OF Pb* e 7 C a l c u l a t i o n of Perturbed Wave Functions 5 :1 For those isotopes of Pb which do not have a nuclear magnetic moment, the two forbidden l i n e s (6 'S0 - 6 lP( ) and (6 'S0 - 6 *Pj.) are due to pure magnetic dipole and pure e l e c t r i c quadrupole t r a n s i t i o n s . However, as we have mention-ed i n the introduction, the isotope Pb^"0* has a nuclear spin £ , the perturbation due to the presence of the nuclear magnetic dipole moment causes a mixing of the ;P, and J P X wave functions. As a r e s u l t , the two forbidden l i n e s become mixed multipole t r a n s i t i o n l i n e s . To calc u l a t e the i n t e n s i -t i e s of t h e i r hyperfine components, we have to f i n d the perturbed wave functions of the l e v e l s 3P, and JP Z. The lowest f i v e energy l e v e l s of Pb are shown schematically i n figure 4 . To f i n d the perturbed wave functions we consi-der only the case when there i s no external magnetic f i e l d . The perturbation consists of only one term which i s the hyperfine i n t e r a c t i o n given by eq. ( 5 ) . The matrix elements of the perturbation # h i n the representation | (ote?F»Y> J i s 34 Figure 4. The lowest five energy levels of Pb. A l l of them belong to configuration 6s*-6vx (Mrozowski 1940) 35 i s diagonal i n quantum numbers F and >tfp. The states j)o<i3-pWp>! a r e therefore correct zeroth order states f o r each l e v e l . The separation between the l e v e l s iT( and *P i s small when compared to t h e i r separation from other l e v e l s . To a f i r s t approximation i n c a l c u l a t i n g perturbed wave functions of l e v e l s SP ( and *P X , we can consider only the mixing of the wave functions of the l e v e l s JP, and *P^ , while contributions due to wave functions of other l e v e l s are neglected. With t h i s i n mind and using standard f i r s t order perturbation theory, we f i n d that the perturbed wave functions of the l e v e l s *S0 , *P, and JP X and t h e i r energies are as follows:-• fe ( ' S o * < , ; ^ i % i l r , i v I (14) where E 6(o() (o< = !s e, JP ( , *P X) are the unperturbed energies of the corresponding l e v e l s t h e i r values are given i n figure 4 . 36 The required matrix elements can be written e x p l i c i t l y -using equation (32) <'*o i * f I X I X i >» f>^ where T>, - <JP, l|H3liJr,> K = < Jpxiin3n5r»> The normalised perturbed states and t h e i r energies there-fore have the following values: • £<*P, i ^  * ^ C ?P'^ + ^ ^ P ' I w n e r e fcB" ^t Jp v>B0( Jfl3 The perturbed states and possible o p t i c a l t r a n s i t i o n s between them are shown schematically i n figu r e 5. Figure 5. Hyperfine components of the l e v e l s 'S4 , \ , *P, of Pb***-38 Relative I n t e n s i t i e s of the Hyperfine Components 5:2 Because some of the perturbed wave functions obtained i n the previous section are l i n e a r combinations of the unperturbed wave functions of both *P, and *Pt l e v e l s , the forbidden t r a n s i t i o n s ((>'S0 -t*P, ) and (b'so - 6*PX) are no longer pure multipole t r a n s i t i o n s , each being a mixed magnetic dipole and e l e c t r i c quadrupole t r a n s i t i o n . Since there i s no external magnetic f i e l d , the perturbed states are s t i l l degenerate i n the magnetic quantum number Ynf . Prom equation (18), the i n t e n s i t y for the component 1 i s given by Since e l e c t r i c quadrupole t r a n s i t i o n s between unperturbed l e v e l s *P and 'S0 are s t r i c t l y forbidden, we have Prom equation (21), we have = -I I <'UlT*ll Jfi>f Denot ing we obtain We cal c u l a t e expressions f o r 1^, 1^ and 1^ i n a s i m i l a r way, using also, i f necessary, the f a c t that magnetic dipole t r a n s i t i o n s between unperturbed *PA and 39 'Se l e v e l s are s t r i c t l y forbidden, i . e. where ( , + Jf Mzt>f j < \ || T J | J */> > P ^ The unperturbed t o t a l i n t e n s i t y of the l i n e U*SP -&P, ) X.46I8A can be obtained by adding I, and 1^ and then putting which gives the value S i m i l a r l y , by putting /»x.*e and adding I $ and I 4 we obtain the unperturbed t o t a l i n t e n s i t y of the l i n e (4 's6 -4*P X) A5313A - ^ 2 > I <X»Tfii 'p ( >l x Prom expressions (39) to (44), the r e l a t i v e i n t e n s i -t i e s of the two hyperfine components of each l i n e can be written as follows:-- 4 40 To obtain a numerical value for 4? » and ~ , we have to f i n d the values f o r /hf, , /**fx , ^  and I /«x<TY/l HYfi >| . From equations (37) and (38), the hyperfine energy s p l i t t i n g of the l e v e l s JP, and P^j, i s given r e s p e c t i v e l y by A E, m E C r . t ^ ) - ^ ^ f »^ and The experimental values are (Mrozowski 1940) A £ ( - 0 • M 7 A t x - - 0 '2.1 ^  and = r«o * Therefore the values of and / * J p » . are re s p e c t i v e l y 0.056 cm"', and 0.109 cm""' . I f we assume that a l l reduced matrix elements of HA connecting l e v e l s i P i and *P, have the same order of magnitude, then we can put J A <'P> H H7H?P, > | ^ C-lt^[ Therefore • «l*«s 5 *' x I 0 . The r e l a t i v e i n t e n s i t i e s f o r the hyperfine components therefore have the values X I The second term: i n the brackets represents the e f f e c t of the presence of the nuclear dipole moment. The 41 f a c t o r s 2 and }/xaxe the r a t i o s of the s t a t i s t i c a l weights of the hyperfine components of the two forbidden l i n e s . The s t a t i s t i c a l weight of the hyperfine component which i s caused by the t r a n s i t i o n F - » F ' i s given by (i"p + 0(* Using t h i s expression the s t a t i s t i c a l weight of the hyperfine components ( 'S6T--Z-+ *P, and ( *Se £ -» V, are respective-l y 4 and 8, and that of the hyperfine components ( ' S ^ i ^ ' P , f » i : ) and ( 'S, ? * i -* *P> F = £) are r e s p e c t i v e l y 8 and 12. 42 BIBLIOGRAPHY 1. Archbold, J. W. 1958. Algebra (Sir Isaae Pitman and Sons, Ltd.). 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 , Blatt, J . M., and Rose, M. E. 1952. Rev. of Mod. Phys. 24, 249. 4. Blatt, 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 edition by Schneider, E. E., Academic Press). 7. Ling, D. S. and Palkoff, D. L. 1949. Phys. Rev. 76, 1639. 8. Mrozowski, S. 1938. Zeits. f. Physik 108, 204. 9. Mrozowski, S. 1940. Phys. Rev. 58, 1086. 10. Opechowski, W. 1938. Zeits. 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. 1932. Zeits. f. Physik 77, 801. 

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