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On the theory of radiative electron capture Paquette, Guy 1956

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OK  THE  THEORY  OF  ELECTRON  RADIATIVE  CAPTURE  by GUY PAQUETTE . B.Sc,  Universite  M. A.,. U n i v e r s i t y  de M o n t r e a l , of British  1951 ••  Columbia,  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n t h e Department of Physi c s  We a c c e p t  this* t h e s i s as  to t h e r e q u i r e d  THE  conforming  standard  UNIVERSITY OF BRITISH August,  1956  COLUMBIA  1953  Faculty of Graduate Studies  PROGRAMME  FINAL  ORAL  FOR  OF T H E  EXAMINATION  T H E D E G R E E OF.  DOCTOR  OF  PHILOSOPHY of  GUY PAQUETTE B.Sc.  (Montreal)  M . A . (British Columbia)  M O N D A Y , SEPTEMBER 10th, 1956, at 10:30 a.m.  I N R O O M 301, PHYSICS B U I L D I N G  COMMITTEE IN CHARGE DEAN W.  F. H .  SOWARD,  OPECHOWSKI  K.  C.  G.  M.  R.  W.  Chairman  C.  A.  R.  D.  JAMES  VOLKOFF  T.  E.  H U L L  STEWART  J . G.  M A N N  MACDOWELL  ANDISON  External Examiner: R. T . McGill  University  SHARP  O N T H E THEORY OF RADIATIVE OF ORBITAL  CAPTURE  ELECTRONS  Abstract The continuous spectrum of gamma radiation which accompanies the capture of orbital electrons has been recently calculated independently by Glauber and Martin (1934) and: by Hess. (1955), Both calculations take into account the influence of the nuclear charge on the wave functions but otherwise involve different methods and approximations, the conclusions being also quite different: the intensity of the gamma radiation is an order of magnitude lower according to Hess than according to Glauber and Martin. The purpose of the calculations presented in this thesis has been to settle this disagreement and to explain its origin. To this effect, the high energy part of the gamma spectrum, which is almost entirely determined by the contributions of. the capture of the Is and 2s electrons, has been computed for the case of A " for which experimental data are available. In view of the low nuclear charge of A " (7.—18), the non-relativistic Coulomb wave functions could be used, and, apart from neglecting screening effects, the calculations are exact although partly numerical. In particular, the retardation effects which were neglected by Glauber and Martin have rigorously been taken into account. The conclusions are: first, approximations used by Hess were partly inconsistent, although the method was in principle correct; second, taking into account retardation effects results in a gamma spectrum whose intensity amounts to 0.8 of the intensity obtained by Glauber and Martin at 135 Kev. and to 0.2 at 675 Kev.  (the gamma spectrum limit being 816 Kev.). The gamma spectrum of A " determined by Lindqvist and W u (1955)  seems to agree quite well with Glauber and Martin's result. However, Lindqvist and W u measured only relative intensities and had to apply many instrumental corrections so that it is not yet clear whether the measured spectrum would not agree as well with the spectrum computed in this thesis.  PUBLICATIONS "Influence de la Resonance Paramagnetique sur I'Effet Faraday: Discussion d'un Exemple". Le Journal de Physique et le Radium, 15, 255 (1954). "Sur la Theprie de la Capture des Electrons Orbitaux Accompagnee de Rayonnement Gamma". Communication a 1'ACFAS, Novembre, 1955, Ottawa. (Co-authors: F. G . Hess and W . Opechowski).  GRADUATE STUDIES Field of Study: Physics Quantum Mechanics  ....G. M . Volkoff  Group Theory Methods in Quantum Mechanics  W. Opechowski  Electromagnetic Theory  W. Opechowski  Theoretical Nuclear Physics  W. Opechowski W. Opechowski G. M. Volkoff ! F. A. Kaempffer K. C. Mann  Nuclear Physics Theory of Measurements  A. M. Crooker  Special Theory of Relativity  W. W. Happ  General Theory of Relativity  F. A. Kaempffer  Quantum Theory of Radiation  H. Koppe i [ R Brown  Physics of the Solid State  "  Low Temperature Physics  J. M. Daniels  Chemical Physics  A. J. Dekker  Other Studies: Differential and Integral Equations Group Theory Theory of the Chemical Bond Network Theory  •.  T. E. Hull B. N. Moyls C. Reid A. D. Moore  Faculty of Graduate Studies  P R O G R A M M E OF FINAL ORAL FOR  THE  EXAMINATION  THE  DEGREE  DOCTOR OF  OF  PHILOSOPHY of  GUY  PAQUETTE  B.Sc. (Montreal) M . A . (British Columbia)  M O N D A Y , SEPTEMBER 10th, 1956, at 10:30 I N R O O M 301, PHYSICS B U I L D I N G  COMMITTEE IN CHARGE D E A N F. H . SOWARD, Chairman W.  OPECHOWSKI  K . C. M A N N G. M . V O L K O F F R. W .  C. A . M A C D O W E L L R. D. J A M E S  STEWART  T.  E.  HULL  J. G. A N D I S O N  External Examiner: R. T. SHARP McGill University  a.m.  O N T H E T H E O R Y OF RADIATIVE O'F O R B I T X L  CAPTURE  ELECTRONS  Abstract The continuous spectrum of gamma radiation which accompanies the capture of orbital electrons has been recently calculated independently by Glauber and Martin (1954) ahd'by Hess (1955). Both calculations take into account the influence of the nuclear charge oh the wave functions but otherwise involve different methods and approximations, the conclusions being also quite different: the intensity of the gamma radiation is an order of magnitude lower according to Hess than according to Glauber and Martin. The purpose of the calculations presented in this thesis has been to settle this disagreement and to explain its origin. T o this effect, the high energy part of the gamma spectrum, which is almost entirely determined by the contributions of the capture of the Is and 2s electrons, has been computed for the case "of A " for which experimental data are available. In view of the low nuclear^charge'of A " (Z—18), the non-relativistic Coulomb wave functions could be used, and, apart from neglecting screening effects, the calculations are exact although partly numerical. In particular, the retardation effects which were neglected by Glauber and Martin have rigorously been taken into account. The conclusions are: first, approximations used by Hess were partly inconsistent, although the method was in principle correct; second, taking into account retardation effects results in a gamma spectrum whose intensity amounts to 0.8 of the intensity obtained by Glauber and Martin at 135 Kev. and to 0.2 at 675 Kev.  (the gamma spectrum limit being 816 Kev.). The gamma spectrum'of A " determined by Lindqvist arid W u (1955)  seems to agree quite well with Glauber and Martin's result. However, Lindqvist and W u measured only relative intensities and had to apply many instrumental corrections so that it is not yet clear whether, the measured "spectrum would not agree as well with the spectrum computed in this thesis.  PUBLICATIONS "Influence de la Resonance Paramagnetique sur l'Effet Faraday: Discussion d'un Exemple". lie Journal de Physique et le Radium, 15, 255 (1954). "Sur la Theorie de la Capture des Electrons Orbitaux Accompa'gnee de'Rayonnement Gamma". Communication a 1'ACFAS, Novembre, 1955, Ottawa. (Co-authors: F. G . Hess and W . Opechowski):  G R A D U A T E STUDIES Field of Study: Physics Quantum Mechanics  G . M . Volkoff  Group Theory Methods in Quantum Mechanics  W . Opechowski  Electromagnetic  W . Opechowski  Theory  Theoretical Nuclear Physics  W . Opechowski  ! Nuclear Physics Theory of Measurements Special Theory of Relativity General Theory of Relativity Quantum Theory of Radiation...., Physics of the Solid State....: Low Temperature Physics Chemical Physics  W . Opechowski G. M . Volkoff F. A . Kaempffer K . C. Mann A . M . Crooker W . W . Happ F. A . Kaempffer H . Koppe { ] J. ™%™  O T E  ..]. M . Daniels .....A. J. Dekker  Other Studies: Differential and Integral Equations Group Theory Theory of the Chemical Bond Network Theory  T. E . Hull B. N . Moyls C. Reid A . D . Moore  ACKNOWLEDGEMENTS  I w i s h t o e x p r e s s my g r a t i t u d e t o P r o f e s s o r W. Opechowski f o r s u g g e s t i n g .for the  h i s continued performance  interest  problem and  and v a l u a b l e a d v i c e  throughout  o f the research. I wish a l s o  t o thank the N a t i o n a l  C o u n c i l o f Canada f o r f i n a n c i a l Studentship  this  and a F e l l o w s h i p .  Research  h e l p i n t h e form o f a  ABSTRACT  The continuous spectrum of gamma radiation which  accompanies  the capture of o r b i t a l electrons has been recently calculated i n dependently by Glauber and Martin (1954), and by Hess (1955). Both calculations take into account the influence of the nuclear charge on the wave functions but otherwise involve d i f f e r e n t methods and approximations,. the conclusions.being also quite d i f f e r e n t : the i n t e n s i t y of the gamma radiation is'an order o f magnitude lower according to Hess than according to Glauber and Martin".  The purpose of the calculations presented i n this  thesis has been to s e t t l e t h i s disagreement and to explain i t s origin.  To this e f f e c t the high energy part o f the gamma  spectrum, which i s almost e n t i r e l y determined by the contributions of the capture of the Is and 2s electrons, has been com37 < puted f o r the case o f A f o r which experimental data are a v a i l 37 able.  In view of the low nuclear charge of A-  (z =  the  n o n - r e l a t i v i s t i c Coulomb wave functions could be used, and, apart from neglecting screening e f f e c t s , the calculations are exact although partly numerical.  In p a r t i c u l a r , the retardation e f f e c t s  which were neglected by Glauber and Martin have rigorously been taken into account. The conclusions are: f i r s t , approximations used by Hess were partly inconsistent, although the method was i n p r i n c i p l e correct; second, taking into account retardation e f f e c t s results i n a gamma spectrum whose i n t e n s i t y amounts to 0.81 of the i n t e n s i t y obtained by Glauber and Martin at 135 Kev, and to 0.24 at 675 KeV (the gamma spectrum l i m i t being 6*16 KeV).  37  The gamma spectrum o f A  determined by L i n d q v i s t and'.Wu  (1955) seems to agree q u i t e w e l l with Glauber and M a r t i n ' s r e s u l t . However, L i n d q v i s t and Wu measured o n l y r e l a t i v e i n t e n s i t i e s and had t o a p p l y many i n s t r u m e n t a l c o r r e c t i o n s so that i t i s not yet c l e a r whether the measured spectrum would not agree as w e l l w i t h the spectrum computed i n t h i s  thesis.  V  Table o f Contents. Page ACKNOWLEDGEMENTS  i i  ABSTRACT  i i i  Table o f Contents  v  I n t r o d u c t i o n and Summary  1  Chapter I  -  General Formalism o f the Theory o f E l e c t r o n Capture  8 %'  A  -  R a d i a t i o n l e s s K-Capture  B  -  R a d i a t i v e .K-Capture  C  -  The M a t r i x Element o f E l e c t r o m a g n e t i c Interaction  let  The E x p r e s s i o n f o r t h e P r o b a b i l i t y o f E l e c t r o n Capture i n Terms o f Approximate Wave F u n c t i o n s .  26  Wave F u n c t i o n s and the General Expression f o r A V f t i f y ^ ^  27  Chapter I I -  A  -  14  B  -  Non-Relativistic Expression f o r f ^ ^ ^ j ^  C  -  The Approximation o f Hess ( 5 5 )  Chapter I I I -  R a d i a t i v e K-Capture  i n a Non-Relativistic  Approximation A  -  Method o f Glauber .and M a r t i n ( 5 4 )  B  -  Evaluation o f ^ V A * * C A  39 40  in a  N o n - R e l a t i v i s t i c Approximation f o r t h e Case o f I s and 2s E l e c t r o n s i n the I n i t i a l State 1 ° Discrete States (a) F o r the case o f I s e l e c t r o n s (b) For the case o f 2s e l e c t r o n s 0  37  44 44 49 54  Table of Contents (Cont'd) Page 2° Continuous States  56  (a) For the case of Is electrons (b) For the case of 2s electrons  57 62  3 0 Results and Conclusions  66  a) The Case of A ?  66  3  b) The Case of C s  1 3 1  Tables I  -  Various Contributions to the Integral L , (11.30 with n . 1)  II  -  -  70  Various Contributions to the Integral (11,30 with n = 2)  III  s  71  Values f o r the Probability Ratio 72  Appendices A  On the neglecting of the second term of II.13a  B C  73  On the expression II.15b  74  On the expression III.52  75  Figures  Facing Page l S (III.68b) f o r a photon energy of 135 Kev  1  -  Integrand  2  -  Integrand Xis f o r a photon energy of 269 Kev  70  3  -  Integrand I|s f o r a photon energy of 404 Kev  70  4  -  Integrand lis for a photon energy of 673 Kev  70  r  70  vii  Table o f Contents  (Cont'd)  F a c i n g Page  5  -  The t h e o r e t i c a l gamma spectrum o f t h i s t h e s i s as compared w i t h t h a t o f Glauber and M a r t i n (54) .  72 Pafte  References  77  1  Introduction and Summary  This thesis i s concerned with the theory of the "radiative capture" of an o r b i t a l electron by the nucleus.  We  c a l l the process of capture " r a d i a t i v e " when i t i s accompanied by the emission o f a gamma radiation.  When no gamma emission  takes place, the process i s called " r a d i a t i o n l e s s " . 3  The radiative capture, although a f a c t o r 10 less probable than the radiationless capture, has been observed f o r several elements, and the corresponding continuous gamma spectrum has been determined  (See, for instance, Lindqvist and Wu (55),  where also references to e a r l i e r papers are given).  The most  important radiative capture process i s that accompanying the r a d i a t i o n l e s s capture of a K-electron.  The theory f o r t h i s  case has been developed by several authors (Morrison and S c h i f f (40), Glauber and Martin (-54) and Hess ( 5 5 ) ) .  In order  to describe b r i e f l y what these authors have done, and to i n d i cate the contribution to the theory, presented i n t h i s thesis, we s h a l l have f i r s t to sketch the common t h e o r e t i c a l basis of a l l these calculations. The expression f o r the probability o f the radiative capture i s given by a standard formula of the second order time-dependent perturbation theory.  In the product of the matrix elements  entering t h i s formula, one factor a r i s e s from the electromagnetic interaction Hy °o t ^ r from the Fermi i n t e r a c t i o n Hft The electromagnetic interaction induces a t r a n s i t i o n between an a n  t  n  e  i n i t i a l state and an intermediate state.  For. instance, the  i n i t i a l state may correspond to an electron characterized by  2 the p r i n c i p a l and The  intermediate  o r b i t a l quantum numbers n and  s t a t e then corresponds to an e l e c t r o n with the  quantum numbers The  JC"  and  =0  (an s - e l e c t r o n ) , and  t o a photon.  Fermi i n t e r a c t i o n g i v e s r i s e to the capture of the  i n that intermediate The  respectively.  assumption t h a t  s t a t e and JL*"  to the c r e a t i o n o f a  neutrino.  - 0 (but n* a r b i t r a r y ) i n the  i a t e s t a t e s , or, i n o t h e r words, t h a t the  electron  intermed-  captured, e l e c t r o n i s  an s - e l e c t r o n , d e f i n e s the r a d i a t i v e process corresponding to the "allowed" r a d i a t i o n l e s s capture o f an s - e l e c t r o n from the K-shell.  We  s h a l l c a l l t h i s process the " r a d i a t i v e K-capture",  although, i n view of the remark above, t h i s phrase should be i n t e r p r e t e d i n too l i t e r a l a sense.  We  hot  denote the p r o b a b i l i t y  o f r a d i a t i v e K-capture, i . e . the p r o b a b i l i t y per second t h a t a photon of energy #ck  i n the range d(/ick) i s emitted  K-capture process by /uT^dLM. , and  ratio  with theory.  ^^rV^j^r »  a s  A. out  f o r Ajj^Jlrl  .  o f the p r o b a b i l i t i e s obtained  and JL s 1; and n • 3, X  f o r a l l values of n  and  However, i t t u r n s ^  :  0;  n » 2,  appreciable  •  are f i v e d i f f e r e n t i n t e r a c t i o n s  p o s s i b l e i n the Fermi theory forbidden  ob-  contribute  = 1, c o n t r i b u t e an  ^ K ^ ^ / M J ^  As i s w e l l known, there  t o allowed and  I t i s the  T h i s means t h a t AAr^<Lr\  t h a t o n l y the o r b i t s c h a r a c t e r i z e d by n = 1,  amount to the r a t i o  the  .  f u n c t i o n of k, t h a t i s compared  t h a t correspond to the occupied o r b i t s .  X> • 0  Hp  c  In p r i n c i p l e , a l l o r b i t a l e l e c t r o n s  to the experimental value i s a sum  a  the  the p r o b a b i l i t y t h a t a r a d i a -  t i o n l e s s K-capture occurs per second by /ur served  during  spectra.  o f beta decay, g i v i n g r i s e In t h i s t h e s i s , as w e l l as i n  p u b l i c a t i o n s quoted above, o n l y allowed t r a n s i t i o n s have been  considered. The  first  t h e o r e t i c a l e v a l u a t i o n o f AAT^dLK was made by  M o r r i s o n and S c h i f f  (40) who considered o n l y the case o f t h e  K - s h e l l e l e c t r o n s , i . e . the case i n which n s 1 and JL s 0. These authors made, among o t h e r s , two important  simplifying  assumptions:  i n t e r m e d i a t e and  initial  first,  they represented the f i n a l ,  s t a t e s by plane waves, thereby n e g l e c t i n g the Coulomb  i n t e r a c t i o n between the e l e c t r o n and the n u c l e u s ; second, took i n t o account /UJ£  o n l y the v e c t o r i n t e r a c t i o n .  i n the same approximation  f o l l o w i n g simple  they  They c a l c u l a t e d .  and they f i n a l l y obtained the  formula:  O f i ^ where  ft  and  aC  £  =  ~  K(I  -.-Jf—  J^rtr\/c}~  (dimenslonless)  i s the f i n e s t r u c t u r e c o n s t a n t . ...  T h i s formula more o r l e s s agrees w i t h experimental data f o r photons o f l a r g e energy first  (i.e.1 -  shown by S a r a f ' s experiments  formula f a i l s  'K^^^I) .  However, as was  (54), the M o r r i s o n - S c h i f f  completely t o e x p l a i n t h e l a r g e r a t i o * \ ^^/MJ" A  J  (  t h a t one o b t a i n s e x p e r i m e n t a l l y f o r low e n e r g i e s o f the emitted photon. In  o r d e r t o e x p l a i n t h i s breakdown o f the theory a t low  photon e n e r g i e s , one may extend Morrison and S c h i f f s f  in  s e v e r a l d i r e c t i o n s , by t a k i n g i n t o 1  s t  t  n  e  calculations  account:  Coulomb i n t e r a c t i o n between t h e e l e c t r o n and t h e  nucleus, 2nd an a r b i t r a r y mixture o f the f i v e beta i n t e r a c t i o n s ,  3 r d the c o n t r i b u t i o n t o r a d i a t i v e capture o f e l e c t r o n s with higher v a l u e s o f n and JL  .  Such an e x t e n s i o n o f the theory has been c a r r i e d out by Glauber and M a r t i n (54)  and by Hess (55)  independently,  the l a t t e r  author r e s t r i c t i n g h i m s e l f , however, t o t a k i n g i n t o account the c o n t r i b u t i o n o f the I s e l e c t r o n s o n l y .  The c a l c u l a t i o n s a r e  approximate i n both cases, but the method a p p l i e d by Glauber and M a r t i n i s quite d i f f e r e n t from t h a t a p p l i e d by. Hess.  The r e s u l t s  do not agree: the i n t e n s i t y o f gamma r a d i a t i o n i s an o r d e r o f magnitude s m a l l e r a c c o r d i n g t o Hess than i t i s a c c o r d i n g t o Glauber  and M a r t i n .  Glauber and M a r t i n ' s  (54)  calculation i s non-relativistic  as f a r as wave f u n c t i o n s o f the e l e c t r o n s a r e concerned; i n o t h e r words, they used the Schroedinger wave f u n c t i o n s o f an e l e c t r o n i n the Coulomb f i e l d o f the n u c l e u s . mation,, they c o u l d f i n d a r e l a t i v e l y simple  In t h a t a p p r o x i -  closed expression  f o r the sum over the i n t e r m e d i a t e energy s t a t e s , u s i n g , i n an i n g e n i o u s way, the p r o p e r t i e s o f the Green's f u n c t i o n .  However,  i n o r d e r t o evaluate the r e s u l t i n g e x p r e s s i o n f o r the p r o b a b i l i t y /or^ d l K  » they s i m p l i f i e d the e x p r e s s i o n f o r the.matrix  element  o f the e l e c t r o m a g n e t i c i n t e r a c t i o n by n e g l e c t i n g the r e t a r d a t i o n factor.  On t h i s s i m p l i f y i n g assumption they came t o the r a t h e r  s t a r t l i n g c o n c l u s i o n t h a t the (n s 1, JL s 0) - c o n t r i b u t i o n to the photon spectrum i s e x a c t l y the same as t h a t found by M o r r i s o n and  Schiff  ( 4 0 ) , who, as w i l l be r e c a l l e d , completely  the Coulomb i n t e r a c t i o n .  neglected  T h i s c o n c l u s i o n seems, on the o t h e r  hand, to be i n agreement w i t h the h i g h energy p a r t o f the measured spectrum, s i n c e t h i s p a r t , as i t was mentioned b e f o r e , i s w e l l  5  represented  by the M o r r i s o n - S c h i f f formula.  Glauber and M a r t i n . .  a l s o showed (using, the same s i m p l i f y i n g assumption) t h a t the observed sudden r i s e i n the gamma spectrum a t low e n e r g i e s i s (i.e. X  = 1, n • 2, 3  due  t o the c o n t r i b u t i o n o f p - e l e c t r o n s  and  t h a t t h i s c o n t r i b u t i o n becomes n e g l i g i b l e a t h i g h photon  ...)  energies. H e s s s c a l c u l a t i o n s , on the o t h e r hand, s t a r t from g e n e r a l . T  formulae i n which the Dirae r e l a t i v i s t i c wave f u n c t i o n s a r e used. However, the a c t u a l computation o f the e x p r e s s i o n f o r /\xr^<i-K so o b t a i n e d ,  turned out t o be i m p r a c t i c a b l e even f o r the s i m p l e s t  case o f the I s e l e c t r o n .  Consequently, number o f s i m p l i f y i n g  assumptions were made: the sum over the i n t e r m e d i a t e d i s c r e t e s t a t e s was assumed t o be s m a l l compared with the sum over the continuous  s t a t e s , then a " s e m i - r e l a t i v i s t i c " approximation f o r  t h e . r a d i a l p a r t o f the i n t e r m e d i a t e  s t a t e wave f u n c t i o n was i n t r o -  duced, and the subsequent c a l c u l a t i o n was c a r r i e d out n u m e r i c a l l y  131 f o r the case o f  Cs  f o r which, a t t h a t time, t h e best e x p e r i -  mental data were a v a i l a b l e (Saraf In view o f the d i s c r e p a n c y  (54)).  between Hess's r e s u l t s and those  o f Glauber and M a r t i n , and o f the n e g l e c t i n g o f the r e t a r d a t i o n e f f e c t s by the l a t t e r authors, AAr^d-k  the s i t u a t i o n . =  i t was f e l t t h a t an e v a l u a t i o n o f  » i n which the n o n - r e l a t i v i s t i c Coulomb wave f u n c t i o n s a r e  used, but which i s otherwise  and n  '  rigourous  , would g r e a t l y c l a r i f y  Such, an e v a l u a t i o n o f the c o n t r i b u t i o n o f  1 and 2 (the c o n t r i b u t i o n i n case n^'3  the c o n t r i b u t i o n t h a t determines completely  JL  =  0  is-negligible), i.e.  the photon energy  spectrum not v e r y f a r from i t s l i m i t , i s presented i n t h i s t h e s i s 1) I t i s t r u e t h a t the f i n a l p a r t o f these c a l c u l a t i o n s has been done n u m e r i c a l l y . However, no a r b i t r a r y approximation has been i n t r o d u c e d i n those numerical c a l c u l a t i o n s . On the other hand, t h e s c r e e n i n g e f f e c t s a r e e n t i r e l y d i s r e g a r d e d , which f o r a l i g h t atom l i k e ^37 i s probably not too s e r i o u s .  37 A and  f o r the case of 13  Cs  131  1)  55  Our calculations s e t t l e the disagreement  between Glauber  and M a r t i n i results and those of Hess e s s e n t i a l l y i n favour of the former authors. The discrepancy i s traced down to a rather well hidden i n consistency i n the argument which leads from the general r e l a t i v i s t i c formulae to Hess's " s e m i - r e l a t i v i s t i c " expressions. question i s discussed i n d e t a i l i n Chapter I I , Section C  f  This  of t h i s  thesis, as i t i s of some general i n t e r e s t . Although our results e s s e n t i a l l y confirm those of Glauber ..' and Martin, the fact that we did not neglect the-retardation < effects has some important quantitative consequences.  I t turns  out that the taking into account of these e f f e c t s leads to a decrease of the i n t e n s i t y of gamma radiation accompanying the 37 K-capture i n A  9  the decrease increasing with the increasing  photon energy, as can be seen i n F i g . 5, i n Chapter I I I (the corresponding numerical data are summarized i n Table I I I , i n Chapter I I I ) . 2) In an unpublished paper (56)  , Glauber and Martin discuss,  among other things, some r e l a t i v i s t i c corrections to t h e i r e a r l i e r r e s u l t s , and also announce that they have applied a more f u l l y 11  r e l a t i v i s t i c treatment of the process which takes account of screening'*.  A comparison of the r e s u l t s of t h i s treatment with 37 Lindqvist and Wu's experimental data f o r A i s ^ i v e n i n a recent 1) Only one value of the energy spectrum f o r Cs i s evaluated, to make a comparison with Hess's result possible. On the other hand, the case of ^37, f o r which we have the most recent and most r e l i a b l e experimental data, (Lindqvist and Wu (55)) i s considered in detail. 2) A copy of the paper has kindly been made available by Professor Glauber to Professor Opechowski.  note  b y Wu  and  al.  (-56).  The  d o e s n o t seem t o be a f f e c t e d lation;  p a r t o f the  spectrum  by t h e s e r e f i n e m e n t s i n t h e  however, i t i s not c l e a r w h e t h e r t h e r e t a r d a t i o n  have b e e n t a k e n i n t o The  high energy  problem  calcueffects  account.  of a detailed  comparison  o f our  r e s u l t s w i t h the experimental data o f L i n d q v i s t b r i e f l y discussed i n Chapter  I I I , Section  B.  theoretical and  wu  (55) i s  Chapter I  General Formalism o f the Theory o f E l e c t r o n Capture.  In  t h i s Chapter, we  s h a l l be concerned w i t h the g e n e r a l  t h e o r y u n d e r l y i n g the c a l c u l a t i o n s o f Chapter I I I .  In Chapter I I ,  we c o n s i d e r , i n more d e t a i l s , the wave f u n c t i o n s to be used, and we d i s c u s s the passage from the g e n e r a l t h e o r y to the  approxima-  t i o n o f Hess ( 5 5 ) .  calculations  In Chapter I I I we present our own  and c o n c l u s i o n s , and compare them with those o f Glauber and M a r t i n (54). The n o t a t i o n t h a t we use i n s k e t c h i n g the g e n e r a l t h e o r y of e l e c t r o n capture i s e s s e n t i a l l y that used by De Groot and (50).  Tolhoek  In S e c t i o n A o f t h i s Chapter, we set up the g e n e r a l e x p r e s -  s i o n f o r the p r o b a b i l i t y Mtf^  o f r a d i a t i o n l e s s K-capture.  A more,  complete treatment o f the s u b j e c t can be found i n the review a r t i c l e s o f Rose (55), Konopinski ( 5 5 ) , and Konopinski and (53).  In S e c t i o n B, we  p r o b a b i l i t y /U^olK  .  Langer  c o n s i d e r the case o f the r a d i a t i v e K-capture  F i n a l l y , i n S e c t i o n C, we d e r i v e an a l t e r n a -  t i v e e x p r e s s i o n f o r t h e e l e c t r o m a g n e t i c i n t e r a c t i o n matrix element. T h i s e x p r e s s i o n i s e q u i v a l e n t to the standard one, but i s more convenient when one c a r r i e s out the passage to the n o n - r e l a t i v i s t i c approximation.  A - R a d i a t i o n l e s s K-capture The p r o b a b i l i t y per second o f a r a d i a t i o n l e s s K-capture depends on a matrix element d e s c r i b i n g a t r a n s i t i o n between an i n i t i a l and a f i n a l s t a t e .  The i n i t i a l  state  ( r e p r e s e n t e d by the  symbol 0) c o n s i s t s o f a n u c l e u s o f charge Z i n an energy s t a t e o  VV^  The f i n a l s t a t e ( r e p r e s e n t e d by t h e symbol. F) c o n s i s t s  o f a n u c l e u s o f charge Z - l i n an energy s t a t e \V^_j, o f an e m i t t e d n e u t r i n o o f energy ELy> and o f a h o l e i n t h e K- s h e l l . if E  j s  Therefore,  i s t h e energy o f t h e e l e c t r o n i n t h e K - - s h e l l ,  W-z. t- E . , = s  W _ , -+- E x  1.1  v  e x p r e s s e s the law o f t h e c o n s e r v a t i o n o f energy.  W^j-V/  i s t h e energy a v a i l a b l e t o the t r a n s i t i o n , , On assuming t h e s t a n d a r d Fermi t h e o r y o f b e t a p r o c e s s e s , t h e m a t r i x element which d e t e r m i n e s t h e p r o b a b i l i t y o f e l e c t r o n capt u r e can be w r i t t e n i n t h e f o l l o w i n g form:  (F|H |o) p  =  where  l j  Hp  (^t.sUCfg^TOAV is.the i n t e r a c t i o n Hamiltonian i s t h e wave f u n c t i o n f o r t h e e m i t t e d neutrino  ^  s  i s the wave f u n c t i o n f o r t h e e l e c t r o n . i n the K - s h e l l  "Vjr^ and T|7^ a r e t h e i n i t i a l and f i n a l nuclear states t  respectively,  i s t h e symbol f o r t h e a d j o i n t , i . e . t h e complex conjugate and t r a n s p o s e , o f a matrix  i\  and -O,^ a r e t h e i n t e r a c t i o n o p e r a t o r s o p e r a t i n g r e s p e c t i v e l y on t h e l e p t o n and  10  the }  nucleon wave f u n c t i o n s  means that, the f u n c t i o n  i n the b r a c k e t s  i s evaluated a t the p o s i t i o n o f the n^*  1  nucleon ^dL ^ means t h a t the. i n t e g r a t i o n 1  i s carried  over the volume c o n t a i n i n g the n  nucleon,  i . e . the whole n u c l e a r volume  A <^  i s a summation over a l l the nucleons.  By r e s t r i c t i n g the i n t e r a c t i o n matrix elements t o be r e l a t i v i s t i c a l l y invariant, possible  c h o i c e s f o r S\  one can show t h a t  there a r e o n l y f i v e  , and t h i s , on v e r y g e n e r a l assumptions.  XL. , i n g e n e r a l , c o n s i s t s  o f two components, one o f which i s  " l a r g e " o r n o n - r e l a t i v i s t i c and the o t h e r i s " s m a l l "  or r e l a t i v i s -  tic  ( o f o r d e r v / c where v i s the n u c l e o n v e l o c i t y ) .  The " l a r g e "  and  "small"  components g i v e r i s e t o d i f f e r e n t s e l e c t i o n  rules.  In t h i s t h e s i s , we o n l y c o n s i d e r allowed beta t r a n s i t i o n s . T h i s means, as i s w e l l known, 1 s t , t h a t we n e g l e c t the " s m a l l " 1)  components o f the f i v e Fermi i n t e r a c t i o n s ,  and 2nd,  that  we  assume t h a t the emitted n e u t r i n o c a r r i e s away zero o r b i t a l a n g u l a r momentum. The l a r g e the  components o f the f i v e i n t e r a c t i o n o p e r a t o r s a r e  following: The S c a l a r  interaction  ^  I.3a  The V e c t o r i n t e r a c t i o n  1  I.3b  The Tensor i n t e r a c t i o n  per  1.3c  The A x i a l Vector i n t e r a c t i o n  <?  I.3d  1) I f one n e g l e c t s the small components, the r e s u l t i n g e x p r e s s i o n i s o f course no more r e l a t i v i s t i c a l l y . i n v a r i a n t .  11 The  Pseudoscalar  " i , ^ i CT  yy  and  1  ^fifs  interaction  I.3e  a r e f o u r by f o u r m a t r i c e s d e f i n e d as f o l l o w s :  1.4a  »  1.4b A  where  , ^  and  are u n i t v e c t o r s and  are matrices d e f i n e d as follows: / ©  o o  o  o  o o ©  |\  o o  o I o 1 o o  JL  / O  \  O  j  0  o o o -l  o  I o o o  y 1 o o oj  JL  0  I.4d  o -1 o o,  o o/  The f i v e i n t e r a c t i o n o p e r a t o r s thus d e f i n e d are H e r m i t i a n . The  pseudoscalar  interaction  t h e r e i s no " l a r g e "  -<!.(&^5 i s r e l a t i v i s t i c o f o r d e r v / c ;  component i n t h i s  Since the o p e r a t o r fl.  case.  o f the g e n e r a l e x p r e s s i o n 1.2  c o n t a i n an a r b i t r a r y l i n e a r combination we i n t r o d u c e f i v e corresponding  o f these f i v e  may  interactions,  constants  1.5 The  s u b s c r i p t s r e f e r t o the s c a l a r , v e c t o r , t e n s o r , a x i a l v e c t o r  and pseudoscalar i n t e r a c t i o n s  respectively.  However, the f i v e i n t e r a c t i o n s 1.3 a c t u a l l y c o n t a i n nine .A.  d i f f e r e n t matrices nine c o e f f i c i e n t s  /\  , and i t i s convenient t o i n t r o d u c e the  C„  d e f i n e d as f o l l o w s :  A.  l  2  p  1  C  C  r B  s  3  -5  V  6  7  .8  fo; h i P<r a;  03 i$y  3  v  C  c  T  c  T  9  c  C  A  A  C  5  P  With the n o t a t i o n 1.6, the i n t e r a c t i o n matrix element 1.3  takes  the form:  (F) H I o) = GjL p  c £ A  ((^VT f ,  s  L  A ^ A T *  i.7  Here we have supposed t h a t the i n t e r a c t i o n constants a r e normalized a c c o r d i n g t o  C  c  - r C + C + C + C „ = i  S and we have denoted by G.  V  T  rS  the i n t e n s i t y f a c t o r  P  (the "Fermi  The C'^.'s can always be chosen r e a l .  Weisskopf  ' constant")  (See B l a t t and  (52)).  The assumption  t h a t the n e u t r i n o c a r r i e s no o r b i t a l  momentum means t h a t the n e u t r i n o wave f u n c t i o n CO  angular  i n the express-  i o n 1.7 i s c h a r a c t e r i s e d by a t o t a l a n g u l a r momentum quantum number j a \ .  One can show t h a t the e x p r e s s i o n 1.7 i s much  s m a l l e r when <J^ corresponds ("Forbidden"  transitions).  t o v a l u e s o f j h i g h e r than  |  13  Finally, (Cj>  A  we make t h e u s u a l  ^fjsj/Yt  o  slowly varying matrix  t  h  e  F  e  r  m  i n t e r a c t i o n matrix  i  function i n s i d e the nucleus.  e l e m e n t c a n be w r i t t e n o u t s i d e  outside  part  e l e m e n t 1.7  i sa  Hence, t h i s  lepton  the i n t e g r a l  s i g n and, a l s o ,  t h e s i g n o f summation o v e r a l l t h e n u c l e o n s i n t h e e x -  p r e s s i o n 1.7. at  f  assumption that t h e l e p t o n  The l e p t o n m a t r i x  a distance  element i s then u s u a l l y  R from t h e o r i g i n ,  R being  the nuclear  evaluated  radius..  One  obtains  &i WAV) (lA*) c  (F|H |0)=  R  p  where  i s a nuclear matrix o n l y when a p r o t o n  element t h a t  changes i n t o a n e u t r o n .  therefore the matrix capture.  i s assumed d i f f e r e n t  e l e m e n t f o r an a l l o w e d  The p r o b a b i l i t y  given by t h e f o l l o w i n g  Expression  from  zero  1.8 i s  r a d i a t i o n l e s s K-  p e r s e c o n d o f s u c h an e v e n t  i s then  formula:  r s Sctn s |(F|Hp|o) v  where  S  v  means t h a t we  v  1.9  o  sum o v e r t h e s p i n s o f t h e e m i t t e d  neutrino, ^JXL,  t h a t we i n t e g r a t e o v e r t h e a n g l e s  o f emission  of the neutrino, and  S  0  >  t  n  a  t  w  e  s  u  m  electron.  over the spins o f the i n i t i a l  K-  14 The  l e p t o n wave f u n c t i o n s t h a t we use i n t h i s t h e s i s are normal-  i s e d per u n i t energy i n t e r v a l ; hence, the n e u t r i n o i m p l i c i t e l y c o n t a i n s the d e n s i t y o f f i n a l  s t a t e s that otherwise  would appear as a f a c t o r i n the e x p r e s s i o n •  1.9.  In t h i s t h e s i s , we t r e a t the nucleons and t h e i r wave f u n c t i o n s  as n o n - r e l a t i v i s t i c .  I t f o l l o w s t h a t f o r the S c a l a r and Tensor  i n t e r a c t i o n s I . 3 a and I.3b i n the n u c l e a r matrix one  elements I.8b,  can w r i t e  •  |P  =  J l  One can thus r e p l a c e  ^>  relativistic  1.10  'a J ?  and  by the u n i t matrix,  formalism, the +1 components o f ^  the  wave f u n c t i o n  will  since, i n our  connect t h e " l a r g e " non-  components o f the nucleon wave f u n c t i o n s whereas  -1 components o f ^  will  connect the " s m a l l " components,  which we are n e g l e c t i n g .  B -  Radiative  K - capture  In the case i n which the e l e c t r o n capture by the emission intermediate evaluate theory,  i s accompanied  o f a gamma photon, one has t o c o n s i d e r  all.the  s t a t e s that the system may occupy, i n o r d e r t o  the p r o b a b i l i t y AAJ^OLK . two types o f processes  from the i n i t i a l  to the f i n a l  In accordance with the hole  are p o s s i b l e i n the t r a n s i t i o n state.  The i n i t i a l  s t a t e o f the  15 . system c o n s i s t s o f the nucleus all  the n e g a t i v e energy  c o n s i s t s o f a nucleus in  N j ^ i n the energy  s t a t e s are o c c u p i e d .  N z _ | i n the energy  state  Wz.  The f i n a l  s t a t e W^-j  ;  state , a hole  place o f the e l e c t r o n c h a r a c t e r i z e d by the quantum numbers n  and JL  , a photon of energy tfck, and a n e u t r i n o o f energy  a l l the n e g a t i v e energy  s t a t e s are again o c c u p i e d .  E. > v  The pro-cesses  t h a t may take place are o f the two f o l l o w i n g t y p e s : I.  -  The atomic  electron  ( c h a r a c t e r i z e d by n and JL  ) makes  a t r a n s i t i o n to either a) an unoccupied  d i s c r e t e state or  b) an unoccupied  p o s i t i v e continuum s t a t e with the  emission of a photon.  The e l e c t r o n i s then  captured  by the- nucleus with the emission o f a. n e u t r i n o . II.  -  An e l e c t r o n i n e i t h e r a) an occupied d i s c r e t e s t a t e o r b) an occupied negative continuum s t a t e is. captured  •  by the n u c l e u s with the emission o f a neutrino/.. Another e l e c t r o n jumps i n t o the remaining hole w i t h the emission of a photon The  c o n s e r v a t i o n o f energy  (between the i n i t i a l and the f i n a l ,  s t a t e s ) i s expressed by the r e l a t i o n \A/  Z  1- E ^ j ^ =  W^_ -r- £ f  v  *r *<,k  .1.11  where E. j_ i s the energy o f the e l e c t r o n i n the s h e l l c h a r a c t e r i z e d by n.  Therefore,  W  x  t-  - W^,  -  Ey + ^  k  i s the energy a v a i l a b l e to the t r a n s i t i o n . The matrix element e n t e r i n g the e x p r e s s i o n f o r the p r o b a b i l i t y , AA/t d_K  should i n v o l v e a sum over a l l the i n t e r m e d i a t e s t a t e s  16 o f t y p e I p l u s a sum o v e r t h e i n t e r m e d i a t e s t a t e s o f type I I . However, i t can be shown (see, f o r example, a s i m i l a r p r o o f i n H e i t l e r ( 5 0 ) , p. 147) t h a t t h e two sums can be e v a l u a t e d  together  as one sum taken o v e r a l l t h e i n t e r m e d i a t e s t a t e s , p o s i t i v e and n e g a t i v e , o c c u p i e d and n o t o c c u p i e d .  From t h e well-known f o r m u l a  o f time-dependent p e r t u r b a t i o n t h e o r y , t h e m a t r i x element i s o f the form ^  CF|ryU)UjHxLgl  £  Ex-Eo  1.12  where F, I and 0 a r e symbols f o r t h e f i n a l , i n t e r m e d i a t e and i n i t i a l states respectively.  ( I | Hy ] 0) i s t h e e l e c t r o m a g n e t i c  i n t e r a c t i o n m a t r i x element and ( F ( J ~ | ^ j l ) i s t h e same b e t a  inter-  a c t i o n m a t r i x element as t h e one used i n t h e r a d i a t i o n l e s s case E-<^. I . \0  T  e x c e p t t h a t t h e l s - e l e c t r o n wave f u n c t i o n i s r e p l a c e d  by t h e i n t e r m e d i a t e s t a t e e l e c t r o n wave f u n c t i o n Vj/ ^  (which i s  a l s o an s - f u n c t i o n because we suppose t h e t r a n s i t i o n t o be a l l o w e d ) Elj  and E- a r e t h e energy v a l u e s c o r r e s p o n d i n g t o t h e i n t e r 0  mediate and t h e i n i t i a l s t a t e s .  I n terras o f the. q u a n t i t i e s de-  f i n e d p r e v i o u s l y , and o f E, t h e energy o f t h e e l e c t r o n i n t h e i n t e r m e d i a t e s t a t e , we have  El The  - E  x  0  =  E ~ E„JL  1- Ack  i . i 3  e x p r e s s i o n f o r w dk, s i m i l a r t o t h a t f o r w , E q . I . 1 0 , k  c  i s as f o l l o w s :  ****  f  ••5& rf V 'K ->.  where  s  Ar  S  ,  1.14  Sy- i n d i c a t e s summation o v e r t h e two d i r e c t i o n s o f p o l a r i z a t i o n o f t h e photon  17 Sv  and S  i n d i c a t e s summation o v e r t h e  Q  spins o f the neutrino i n i t i a l electron $o\fy and  .^dLfXv  and those o f t h e  respectively  mean, t h a t we i n t e g r a t e  over  the a n g l e s o f e m i s s i o n o f t h e photon and of the neutrino  respectively  i s t h e photon c o n t r i b u t i o n t o t h e d e n s i t y of f i n a l The  states.  wave f u n c t i o n s a r e supposed t o be n o r m a l i z e d per. u n i t energy  interval.  The m a t r i x elements i n 1.14 a r e g i v e n by:  (F I H^li)  =  GlLc^A^iM)  1.15  (the symbols have t h e same meaning as i n S e c t i o n A) and by (See H e i t l e r (50), p.95)  (n.H|o) r  where  e Jr^-K ?  = =  ^  — JL  Q  '  e. ( JIT**. \  , the retardation  factor  —^  ji^ i s t h e p o l a r i z a t i o n v e c t o r o f t h e photon whose wave v e c t o r  is K  } |rt] —  oT i s t h e D i r a c m a t r i x d e f i n e d  K  i n 1.4  JL i s t h e charge o f t h e e l e c t r o n Y  B  and  a r e t h e wave' f u n c t i o n s  e l e c t r o n i n the intermediate  f o r the  and t h e  I>16  18 i n i t i a l states respectively, v"^" means t h e H e r m i t i a n conjugate  of  ^  The i n t e g r a t i o n i s over a l l space.  In  the f o l l o w i n g S e c t i o n , we d e r i v e a n o t h e r e x p r e s s i o n f o r •  •  the e l e c t r o m a g n e t i c i n t e r a c t i o n m a t r i x element 1.16, n e s s o f which w i l l be apparent i n the n e x t  •  the  useful-  Chapter.  C - The M a t r i x Element o f E l e c t r o m a g n e t i c I n t e r a c t i o n  In t h i s S e c t i o n , we  c o n s i d e r the g e n e r a l case o f an  electro-  magnetic t r a n s i t i o n between a s t a t e o f energy E d e s c r i b e d by e l e c t r o n wave f u n c t i o n by  ^E  1  •  vL»^  and  the  and a s t a t e o f energy E^" d e s c r i b e d a r e D i r a c e i g e n f u n c t i o n s f o r an  e l e c t r o n i n the presence o f an e l e c t r o m a g n e t i c f i e l d c h a r a c t e r i z e d by a v e c t o r p o t e n t i a l f\  and a s c a l a r p o t e n t i a l  Cij>  .  The m a t r i x element o f the e l e c t r o m a g n e t i c i n t e r a c t i o n duced i n S e c t i o n B, Eq. 1.16 ing  intro-  i s t h e n a s p e c i a l case o f the f o l l o w -  m a t r i x element:  M = C E|« /e*) r  We a r e now  = C •Jf/flf-^hQvfE'* ?  1.17  3  g o i n g t o put t h i s m a t r i x element i n t o  another  form by a p p l y i n g a t r a n s f o r m a t i o n i n t r o d u c e d by Gordon (28) t o decompose the D i r a c p r o b a b i l i t y c u r r e n t i n t o an " o r b i t a l "  part  »  19 and  a "spin"  part.  To t h i s purpose i t w i l l be convenient t o use a c o v a r i a n t tensor notation Somraerfeld  ( i n t h i s , we more o r l e s s f o l l o w P a u l i  (33) and  i n t h e i r p r e s e n t a t i o n o f Gordon's method).  We f i r s t  introduce  the c o n t r a v a r i a n t  four-vector i.id  0 , 1 , 2 , 3  of which the'components are chosen t o be  y ^  0  -  OC.^^ 0 ^ " d a  }  y^y^and y  3  a  -1.  ,  y  0C.3  a  .  . .2.  '•= P * i r  e  ,  y  =• £ < * ,  d e f i n e d i n Eq. 1.4.  y = J  a  The  are thus a n t i - H e r m i t i a n :  3 (<* 3  components  - y"  K  ,  K~ i , 2 , 3  The y/*" ' s thus chosen obey the f o l l o w i n g commutation where 0/  i s the well-known c o n t r a v a r i a n t metric  1.19  r u l1.20 e:  tensor:  1.21  and where I i s the u n i t m a t r i x .  The c o v a r i a n t  i s defined  such t h a t 1.22 where  i s the Kronecker t e n s o r .  Therefore, /1 o 0-1 Q  o o 0 0  O -(  \ 0 o  .0  O -|>  .  1.23  20 The  position four-vector i s defined OC^  as  -  1.24  such t h a t the energy - momentum f o u r - v e c t o r  i s r e l a t e d t o the  operator  It a. -  ^  V«.  =  1.26  by the r u l e V  3  ^ =Z Therefore,  from  j  r  f>  1.27  1.23 1.28  The  electromagnetic  field  can a l s o be e x p r e s s e d as  a  four-vector  A^= The  ( q > X)  covariant four-vector  A^c  J- ' 2  i s given  by  I n the n o t a t i o n i n t r o d u c e d above, the e q u a t i o n by  vp ,  and  £  ^j,!  s a t i s f y the a d j o i n t  % {h.^t 0  -  i s the f o l l o w i n g  I n t r o d u c i n g the b a r f u n c t i o n and  satisfied  =  T  »  w e  s e e  yV  that  £  equation  V ) T Y  -±f*f  = 0  '  i.3ib  21 Eqs. 1.31  a) and b) are equivalent to the equations used by Pauli  (33), p.232. By solving Eqs. 1.31a one  and 1.31b  for  a n c  i  y  E  respectively,  obtains  1.32a  1.32b  If we write f o r s i m p l i c i t y component  JL ~ 0 Q  = JL  and introduce the  , the matrix element M, Eq. 1.17,  can be  written i n the form  We now  replace Vj^t and ^  i n  I  «33 each i n turn by the  expression I.32a and I.32b respectively, which y i e l d s :  M - " ztifc^T^V&V)*-* 1. and  Writing again  = ^""vf^  » adding the two expressions  1.34  and dividing the r e s u l t by 2, we obtain f o r M the following expression:  22  where we have separated the f i e l d  - dependent terms from the  others. Using the f a c t t h a t terms i n which  y »yU.  "f^Y*  Y^=  t  2^we  separate the  V^yu.  from those i n which  to obtain  1.36  We now use the r e l a t i o n s for  f<+  Vf^-,  O  .  ^ y ^ /  (See Egs. 1.21  e  terms i n which  —  f  and 1.30).  * -^cA/V:,  v  Dirac spin operator.  l i n e o f 1.36.  JH - O  rr^  v  v>  In t h i s way,  n  d  Ay^. = ""A  A l s o we note t h a t when  , where  We a l s o separate the ^ s Q and  a  CT^  V  i s the  term from the  are d i f f e r e n t from zero i n the l a s t 1.36  becomes the sum o f f o u r terms: 1.37  where  1.37a  MJL-  $ X  Q. Y  E  Y > E  I.37b  23 3 3 Je,m--i  1.37c  We now make the assumption that one o f the energy l e v e l s E, E"" ( l e t us say, E ) i s a d i s c r e t e l e v e l . 1  vpe*  vanishes  Then the f u n c t i o n  a t i n f i n i t y and the f i r s t term o f M , Eq. 1.37a.  can be i n t e g r a t e d by p a r t s to y i e l d :  ~ -  ( Ve.  Q *  V  YE!  s i n c e w = ^_  .  The l a s t term o f I . 3 S i s zero s i n c e the  v e c t o r s JL and  a r e p e r p e n d i c u l a r t o each o t h e r .  Therefore,  becomes  M, The writing  * A (r 3 e  integrand of  E  v  «**  Eq. I.37c can be expressed  B * ^ f o r the q u a n t i t y  ^V^g*  "  1.39 as a c u r l :  , ( fc"** ^ - B * ^ ,  M3  becomes  1.40  where  P  (Ti. = c r  3  =  1  Rearranging  Y E °" Y e ' ,  cr = c r 3  » , a  -  a  n  d  °"  n  a  s  t  h  components  e  cr, - fl" ,  1.41  .  and i n t e g r a t i n g 1.40 by p a r t s , one o b t a i n s Ma  =  1=^  ( ^ x B )  •  1.42  24 since  and  consequently  bounding s u r f a c e .  B  become zero on the  *?C^ = - A R Q  Since  1.42  infinite  becomes 1.43  In 1.43 The  i <1«  expression  and A From Eqs.  equivalent  *  .  I.37d, can be w r i t t e n as  E  since' the time-dependence of y X  by AZ  we have denoted the v e c t o r  =  and  follows  M ' I i s ^ i v e n by the f a c t o r s  respectively. 1.37,  t o Eq. 1.17  1.39,  1*43  i s the  and  1.44,  the e x p r e s s i o n  for M  following:  -r 1.45  where  -t  >u.  =  —*  -2-^ X  K  For the case o f an e l e c t r o n i n Coulomb f i e l d 1.45  becomes  (A"  ^  ),  25 If  we  interaction expression in  the  compare t h e matrix just  matrix  e x p r e s s i o n f o r the  e l e m e n t o f S e c t i o n B,  g i v e n , we  e l e m e n t may  can be  conclude  electromagnetic Eq.  1.16  with  the  t h a t the o p e r a t o r  r e p l a c e d by  the  Z.^  operator  1.47  in  this  sense, these  In our replace by  the  the  general  two  operators  are  equivalent.  e x p r e s s i o n o f S e c t i o n B,  electromagnetic  expression 1.46  with  interaction \^^\  Eq.  matrix  r e p l a c e d by  1.42,  we  thus  element o f Eq. and  1.16  replaced  26  Chapter  The  Expression  II  f o r t h e P r o b a b i l i t y o f E l e c t r o n Capture  i n Terms o f Approximate Wave  Functions  In S e c t i o n A o f t h i s Chapter, we i n t r o d u c e t h e wave f u n c t i o n s used i n o u r c a l c u l a t i o n s and we g i v e the g e n e r a l p r e s s i o n f o rf*rk^//u*^  t  t u t e d i n the expressions  1.9 f o r fuJ^  r e s u l t i n g ex-  when these wave f u n c t i o n s are s u b s t i -  S e c t i o n B, t h i s g e n e r a l e x p r e s s i o n  and 1.14 f o r AAJ^dLK • In  i s s i m p l i f i e d by i n t r o d u c i n g  the n o n - r e l a t i v i s t i c approximation f o r the e l e c t r o n wave f u n c t i o n s ) , and  put i n t o the form which w i l l serve as a s t a r t i n g p o i n t i n  our c a l c u l a t i o n s o f Chapter I I I .  In S e c t i o n C, we d i s c u s s the  passage from the g e n e r a l e x p r e s s i o n by Hess.  o f S e c t i o n A t o t h a t used  27 A -  Wave F u n c t i o n s  for  We s t a r t Dirac  t h e s p h e r i c a l wave s o l u t i o n t o t h e  for a particle  functions are given, They a r e n o r m a l i z e d  Expression  Axr^dK/^  by c o n s i d e r i n g  equation  ized  and t h e G e n e r a l  i n a Coulomb f i e l d .  f o rinstance,  (37)  i n Rose  o r i n H e s s (55).  p e r u n i t energy i n t e r v a l and a r e c h a r a c t e r -  by t h e f o l l o w i n g quantum numbers: j , t h e t o t a l  momentum quantum number, w h i c h t a k e s h a l f - i n t e g r a l which takes a l l h a l f - i n t e g r a l takes the  the values  i  wave f u n c t i o n s  angular  ( j +• J ) .  Since  the  wave f u n c t i o n s a r e :  '«  t h e wave f u n c t i o n s  angular v a l u e s ; jx  i n this  t h e s i s we o n l y case:  i n their  general  form.  need  no o r b i t a l we  shall  For j  :  J,  A.  o 3  ,  f r o m - j t o + j ; a n d K, w h i c h  momentum i s c a r r i e d away b y t h e p a r t i c l e s ) ,  write  v  values  f o r which j = * ( a l l o w e d  not  /  These wave  IIo  l i "AT  03^ The  functions F  G  t o t h e D i r a c wave e q u a t i o n : in  this  for  thesis.  instance, Since  zero  i n Blatt  we c o n s i d e r  we s h a l l  not write  them  solutions explicitly  's a r e t h e s p h e r i c a l h a r m o n i c s (52), p . 7#3.  and W e i s s k o p f only  allowed  defined  t r a n s i t i o n s , we s e t e q u a l t o  t h e components o f t h e wave f u n c t i o n I I . 1 w h i c h c o n t a i n a  spherical in  The \i  a n d G~ a r e t h e r a d i a l  that  harmonics d i f f e r e n t  approximation,  from  J  0  .  We have  therefore,  26*  K---1  r-i 1 o \  / •  .\  / o 0  0  1 0' j '°  w  II. 2  o  vf  0  \  A.  V°  * /  I f one sets Z, the nuclear charge, and m, the electron mass, equal to zero i n II.2, one obtains the neutrino wave functions (S)  which w i l l be used i n the expressions 1.8a and 1.45. I f  one denotes by ^*  and  the r a d i a l parts of the non zero corn-  ponents of these wave functions, evaluated at the nuclear surface, one e a s i l y shows that JL  II.3  where Ey i s the energy of the emitted neutrino. In the case of the Is electron, the two functions with K=*[in II.2 are i d e n t i c a l l y zero.  In the case of s electrons i n  higher discrete l e v e l s , these two functions no longer vanish but are small and w i l l be neglected i n our calculations.  The two  remaining wave functions are  H<'- */* I Y ° 5SA \  \  II.4  o 0 0  With the help o f II.4 (with n s 1), and of II.3, one r e a d i l y evaluates the expression 1.9 f o r AJLT^  .  The absolute square o f  29 the matrix element i n 1.9 i s ( F|  o) '  z G  S%  (see Eq. I . £ ) .  ^  ()(J/f f (  Operating w i t h S  G  V  tis t i l A"' ^ )  R  on t h e product  I  ^ ^ i l  .  2  5  yields,  because o f I I . 4 , I±j3  a.  where ^ Sy  i s d e f i n e d i n Io4.  II.6  I t i s a l s o e a s i l y seen t h a t the sum  over the f o u r n e u t r i n o s t a t e s y i e l d s  where the o p e r a t o r T r means that we take the t r a c e of t h e matrix on the r i g h t o f i t . f a c t that  E  v  In d e r i v i n g I I . 7 , we used I I . 2 , I I . 3 and the  = W, the a v a i l a b l e energy  (See E q . I . l ) .  from I I . 5 , 6 and 7, S , S o | ( F l H . | 0 ) | . =  Thus,  « Yo  x  ^ 3 7 3  J  G  *  c^, (tfXS/OV*  A ' A  N  -*  The matrices o f which we have to e v a l u a t e the t r a c e s , i n Eq.II.o*, are of the form 1 . 6 ,  A^A*  and A ^ A *  , where the A ^ . a s d e f i n e d In .  a r e products o f the D i r a c matrices  p** ^ 0  and  0^3  ."  Such t r a c e s o f products o f D i r a c matrices a r e g i v e n , f o r i n s t a n c e , in Heitler  (50), p.87. When they are e v a l u a t e d i n II.6*, t h e  30 double  sum o v e r  r and r * becomes  S S c e o (VflCStfT  =  T\ A *  A*  a.T  Remembering t h a t t h e o p e r a t o r  1.9  by a f a c t o r  1.9,  V  4 TT  and t h a t  m u l t i p l i e s the expression  Yo  s  Vj^TT  »  w  e  o  b  t  a  i  n  from  and I I . 9 :  I I . 8  G  II.1.0 was  used b o t h  by Hess a n d by G l a u b e r  T  11.10  and M a r t i n  in their  calculations. We now write  c o n s i d e r t h e e x p r e s s i o n f o r /Wj^JLK  t h e sum o v e r  a more d e t a i l e d element obtained  the intermediate  •  energy s t a t e s ,  form, u s i n g the e l e c t r o m a g n e t i c i n S e c t i o n C, C h a p t e r  t i o n s i n the intermediate  ^e  and i n i t i a l  I.  Since  E q . 1.12,  interaction t h e wave  have t h e same p a r i t y ,  e x p r e s s i o n 1.45 magnetic 1.12,  for(F|Hf]l)  i n t e r a c t i o n matrix  1.13,  1.15  and 1.45  will  in matrix  func-  s t a t e s a r e s - f u n c t i o n s and, mt  consequently,  first  the operator  -  >  -•»  T7  *~ ^ —  i n the  n o t c o n t r i b u t e t o the e l e c t r o -  element.  I t f o l l o w s t h e r e f o r e , from  that  11.11  31 where  Writing  £-6^^=  E-E  m s  +  -  ftch,  !  w  ^  e  P  u t  II.11  into the following form  i - i c ^ ) «fCR)A*$tf i f «>1&*>S*^9lVs f  +  where  1 1 , 1 3 8  e  M  *s  <  £  £  o  K  II..13b  We show i n Appendix A that the second l i n e of II.13a may be neglected.  r  We obtain, f i n a l l y  Ej.-^  *  1 1  •4 1  We now form the absolute square of the expression 11.14 and a p p l y the t h e operator operator  S.  on the t h e WJ wave f u n c t i o n product *tms4/»iS  Because of II.4 and II.6, we obtain  15a  32  where  I I . 15b  and where t h e prime on h ^ m e a n s t h a t we r e p l a c e A £ by A , £* 1  1  i n 11.13.  I t i s shown i n Appendix B t h a t  P -  1 -  £i  JL  11.16  Because o f t h e form o f t h e wave f u n c t i o n s  , as given  by I I . 2 , we may w r i t e E q . I I . 1 3 b as f o l l o w s :  We i n t e g r a t e o v e r t h e a n g l e s , u s i n g the r e l a t i o n  ( a n ^  = 4-T ^ L  1  ^  ii.18  and i n t r o d u c e t h e n o t a t i o n  . A»whA  j  <  ft  A -  KA.  ,  /^VSCA) A .  II.19a '**  E.  £ -  e  m  s  +• #cfc  33  = ^  UL.  ~g  )  Q  A .  K/L  *  I I . 19b  11.15 becomes  11.20  K The summation Eq.II.7,  J  over the f o u r n e u t r i n o s t a t e s g i v e s , as i n  the t r a c e o f the matrix  's o f Eq. 11.20.  contained between the two  There a l s o a r i s e s , because o f I I . 2 and 3,  a factor 8^  *^  .3  11.21*  i n f r o n t o f the whole e x p r e s s i o n .  In d e r i v i n g 11.21, we used  the equations o f energy c o n s e r v a t i o n 1.1 and I.11 and the d e f i n i t i o n 1.1 f o r W.  S $ o  v  Therefore,  — Q v° = BY. (( Ww- f Et c kai -a Ems" i E aE»w l 0  /V  ^ c ^ ( t f W A'  11.22  34 I t can be seen from t h e d e f i n i t i o n s 1.6 t h a t t h e t r a c e s o f e v e r y m a t r i x o f t h e form are z e r o . the t r a c e .  and A^/ffi^L  A'VOo^  Thus, t h e term Therefore,  U  = 1 , 2, 3.)  o f 11.22 w i l l n o t c o n t r i b u t e t o  one has o n l y t o e v a l u a t e  A l s t o n  the trace o f  CJX^K  11.23  i n t h e way e x p l a i n e d under E q . I I . d . A f t e r t h i s i s done, 11.22 becomes:  s,sJ * !*• - B Y i * t w - M ^ c - e ^  with T  1  and T  2  g i v e n by  T , ( Cs * and  x  Ta-  t ttf tCc?•> ^ ) | S ? | V S*l Sfr/ff  ^ S ^ I S l f  11.25.  '+ II.25b  such t h a t T, * T * = T  , d e f i n e d i n E q . I I . 9 - From 11.12, 11.24  and 1.14, we have f i n a l l y i.  "3  xi.[( L; i%\Cf)T l  s  t ciu:j - i £ r ) T j a  t  s  35 and,  11.10 and 11.26 •  from  N3  TT  AA/;  (W*)* . V  W  /. 11.27'  I  R  The above e x p r e s s i o n f o r ^ ^ / u r the  subsequent  discussions o f this  B -  be t h e b a s i s f o r  Chapter*  Non - R e l a t i v i s t i c for  will  Expression  /UTfcclk/^  When Z i s small,, t h e e x p r e s s i o n f o r t h e sum o v e r t h e i n t e r mediate . energy its  states,  non-relativistic 1°  function 2° ponding 3  ^ G  a s g i v e n b y 11.17,  form.  To t h i s  may be a p p r o x i m a t e d b y  effect, m a t r i x i n 11.17,  i s put equal t o a u n i t  s o t h a t the  d i s a p p e a r s completely from t h e e x p r e s s i o n ;  The f u n c t i o n solution  i s r e p l a c e d by  CJp  £  the corres-  t o Schroedinger*s equation;  The sum o v e r a l l e n e r g y  states  IE. becomes a sum o v e r EL  the d i s c r e t e  s t a t e s and an i n t e g r a l  over the positive  continuous  states. When t h e above a p p r o x i m a t i o n  i s made, t h e e x p r e s s i o n 11.23  36 becomes E.28  and  t h e t r a c e o f 11.28 g i v e s r i s e o n l y t o t h e f a c t o r T /  i n Eq. II.25a.  defined  I n t h i s a p p r o x i m a t i o n , we have, from 11.27 i a.  I f we make use o f t h e f a i r l y w e l l e s t a b l i s h e d f a c t (See,  experimental  f o r i n s t a n c e , K o n o p i n s k i and Langer (53)) t h a t t h e  mixed terms ( i . e . those p r o p o r t i o n a l t o C C &  n e g l i g i b l e , we may s e t  v  and C j .  a 0; and t h u s T J - T  a l s o r e p l a c e t h e wave f u n c t i o n  Rs<fO R  ) are  i n 11.29.  f o r t h e i n i t i a l e l e c t r o n by  the c o r r e s p o n d i n g s o l u t i o n t o S c h r o e d i n g e r * s e q u a t i o n ^ C R ) Hence, 11.29  becomes:  <VJi  where  U  =s  Expression  ^  '  "  -  .  11.30 i s t h e s t a r t i n g p o i n t o f o u r c a l c u l a t i o n s o f  Chapter I I I .  We  37 C -  The A p p r o x i m a t i o n  o f Hess  I n s t e a d o f u s i n g t h e approximate e l e c t r o m a g n e t i c  interaction  m a t r i x element o b t a i n e d i n Chapter I , S e c t i o n C, which l e a d s t o Eq. 11.14, Hess used t h e g e n e r a l r e l a t i v i s t i c w i t h the o p e r a t o r  oc*^ °  e x p r e s s i o n I.16  T h i s means t h a t t h e m a t r i x P o f  I I . 15b was, i n h i s case,  or  P = since  5?f* «  IdS  H.31  *  - £c?  and  (o?.  I  On t h e o t h e r hand, we have seen i n S e c t i o n B t h a t , w i t h the approximate wave f u n c t i o n s o f t h e form I I . 2 , t h e sum M*\s the i n t e r m e d i a t e energy s t a t e s t a k e s t h e f o r m I I . 17.  o v e r i  Since  » 0, i t f o l l o w s from 11.17 and 11.31 t h a t t h e factor  ^/vvsPM^ o f I I - 1 5 a becomes 11.32  i.e.  t h a t t h e f u n c t i o n F.  d i s a p p e a r s a l t o g e t h e r from t h e sura  o v e r t h e i n t e r m e d i a t e energy s t a t e s .  Since the functions G f o r  the d i s c r e t e s t a t e s a r e s m a l l , Hess was l e d t o n e g l e c t t h e sum o v e r t h e i n t e r m e d i a t e d i s c r e t e s t a t e s as compared w i t h t h a t over the continuous  s t a t e s ; t h e r e s u l t o b t a i n e d was o f an o r d e r o f  magnitude s m a l l e r than t h a t o f M o r r i s o n and S c h i f f .  (See Chapter I I I )  3*  The source o f t h i s d i f f i c u l t y r e s i d e s i n the f a c t t h a t • making the approximation I I . 2 f o r t h e wave f u n c t i o n s and, a t the same time, keeping the o p e r a t o r o T * i n  i t s g e n e r a l form i n the  matrix element o f e l e c t r o m a g n e t i c i n t e r a c t i o n ,  i s inconsistent.  Chapter  Radiative  III  K-Capture  i n a Non  - Relativistic  Approximation  In S e c t i o n method employed derivation discuss  A of this  by G l a u b e r and M a r t i n  o f the e x p r e s s i o n  the s i m p l i f y i n g  11.30, and i n S e c t i o n that  C h a p t e r , we  expression  b r i e f l y describe (55)  (56),  the  i n their  11.30 f o r /WT^cU<Jtoj^  , and  we  a s s u m p t i o n s t h e y make i n e v a l u a t i n g  B, we  present  f o r the cases o f  o u r own A  13  37  evaluation of  131  and  Cs 55  40  A -  Let 1.12,  Method of Glauber and  Martin  us c o n s i d e r the g e n e r a l Jbrm o f the matrix element  u s i n g the e x p r e s s i o n s 1.15  a c t i o n matrix element and  and 1.16  f o r the beta  inter-  the electromagnetic i n t e r a c t i o n  matrix  element r e s p e c t i v e l y :  I  E  r  - E  \  D  «  • X  M  W H E R E  =  £  /  fa  r.  M„<  5t-?  III.la  h  Mrt^i^L  i n ,ib  As pointed out by Glauber and M a r t i n , case o f the Green's f u n c t i o n  G-  i s actually a special (V, A ! ' f o r t h e wave equa-  t i o n o f the e l e c t r o n i n a Coulomb f i e l d ,  oince we  n u c l e a r r a d i u s R equal to zero i n I I I . l a ,  we have  The  may  s e t the  reason f o r the success o f t h i s approach i s t h a t t h i s  one-argument Green's f u n c t i o n i n i t s n o n - r e l a t i v i s t i c form be obtained as a s o l u t i o n o f the Schroedinger e l e c t r o n i n a Coulomb f i e l d .  can  equation f o r an  However, Glauber and M a r t i n  start  41 out from the Green's function yfvj^r ^ - ^ ^  of the iterated Dirac  equation, and they arrive at a formula which i s equivalents to our equation 11.27. they replace $  Only i n a l a t e r stage of their calculations do by the n o n - r e l a t i v i s t i c Green's function, and i n  t h i s way obtain an equation which reduces to our Eq. XI.30 i n case of s-electrons. Expression 11.30 form.  cannot be evaluated i n a closed  In order to evaluate  a n a l y t i c a l way, factor  for  JL  *^^K/UJ^  /V,  0  /  in a relatively  simple  Glauber and Martin have set the retardation equal to one.  expression for L  t  This i s equivalent to using an  ^ V-fc V i s replaced by unity f o r  in which  V  /  a l l values of k, i . e . over the whole energy range of the emitted photon.  In that manner, and with the help of the a n a l y t i c a l  expression for the n o n - r e l a t i v i s t i c Green's function, they could evaluate A*J}^ ^^/f^  almost without numerical  calculations.  The argument they present i n order to justify' the above approximation  ( i . e . JL  ~ JL  ) over the whole energy range  i s rather lengthy and somewhat unconvincing; i t w i l l only be sketched i n t h i s t h e s i s . The argument i s of a d i f f e r e n t nature f o r each of the following three photon energy ranges:  JUk  (1)  (2)  ^V<^<..K  <  x&-«f  <  Z^/nvc*  (3) Z ^ < ^ c K In the range (1) of low photon energies, the photons have a wave length at least system.  (Z^O  In t h i s case, one may  * times' larger than the atomic therefore set ^^/AKA i n 11.30.  "1.3  42 C o n s e q u e n t l y , o n l y one t e r m i n t h e sum and i n t e g r a l  over  inter-  mediate s t a t e s i s d i f f e r e n t  from z e r o because o f t h e o r t h o -  normal p r o p e r t i e s o f the  * s , namely t h e t e r m f o r w h i c h  Then  becomes,  SUA  E q . II.3D becomes i d e n t i c a l  when n in  =  1 and #  =0.  with  containing In  while  Eq. III.4  (non-relativistic)  the intermediate  which c o n t a i n s a d e c r e a s i n g r e m a i n s much s m a l l e r t h a n  O,  ' cs f  .  i s d e r i v e d f r o m an  In t h e high into account.  free particle  the  initial  exponential,  o f the form  shows t h a t  «( i ) > 0  i t s range  t h e p h o t o n wave l e n g t h so t h a t t h e t o be u n i m p o r t a n t ,  i . e . again  r a n g e a l s o , one o b t a i n s  and S c h i f f when n = 1 and L  energy range  (3)  »  t h e r e t a r d a t i o n must-be  0  o  taken used  f o r m o f t h e G r e e n ' s f u n c t i o n and a p p r o x i m a t e d  wave f u n c t i o n b y a c o n s t a n t .  a r e t h e same a s t h e one u s e d b y M o r r i s o n result:  priori  However, i n t h a t r a n g e , G l a u b e r a n d M a r t i n  the  their  expression  f o r t h e G r e e n ' s f u n c t i o n )j  seem a g a i n  o f Morrison  neglected  (2) one may n o t a  However, a s t u d y  I t follows that i n this  the e x p r e s s i o n  ratio  consequence,  and S c h i f f c o m p l e t e l y  energy range  the a n a l y t i c a l expression  retardation effects  and S c h i f f  Coulomb wave f u n c t i o n s ) .  neglect the retardation e f f e c t s . of  the Morrison  (This i s a rather s t a r t l i n g  v i e w o f the f a c t t h a t M o r r i s o n  Coulomb e f f e c t s ,  S  simply  ** and  E-£^.  These  approximations  and S c h i f f  in deriving  namely, t h e n e g l e c t i o n o f a l l Coulomb e f f e c t s i n  the  intermediate  the  initial  s t a t e s wave f u n c t i o n s a n d t h e a s s u m p t i o n  e l e c t r o n may be c o n s i d e r e d  e n e r g y r e g i o n , G l a u b e r and M a r t i n  at rest.  that  F o r the high  u s e d t h e r e f o r e t h e same  / L  43 expression as that used by Morrison and S c h i f f . The neglecting of a l l retardation e f f e c t s allowed Glauber and Martin to evaluate the contribution to .Atfj^oOl  made by the  p-electrons of the L and M s h e l l s (n = 1, 2; L = 1) in-a r e l a t i v e l y simple way.  This contribution was  shown to explain the  sudden rise of the photon spectrum at low energies Martin  (Glauber and  (55)).  In an unpublished paper (56), Glauber and Martin  introduced  a r e l a t i v i s t i c correction to the Is and 2s state spectra by mean of a canonical transformation  applied to the Green's function.  This correction i s seen to apply only to the low and  intermediate  energy ranges of the photon spectra, as defined i n III.3 and. i t i s evaluated  again on the assumption that the retardation factor  may be put equal to one.  These corrected results were compared •  3  7  .  with the experimental data, f o r the case of A and wu  •  , by Lindqvist  (55) and there appears to be an essential agreement be-  tween theory and experiment. As stated i n the Introduction, a "more f u l l y  relativistic"  calculation carried out by Glauber and Martin has not yet been published i n d e t a i l s , but the results have been compared with  37 the experimental data f o r A  (See Wu and a l . (56)).  From t h i s  comparison, i t would appear that the correction r e s u l t i n g from these l a t e s t calculations does not a f f e c t the high energy part . of the t h e o r e t i c a l spectrum. Although there can be l i t t l e doubt that Glauber and r-iartin's results are e s s e n t i a l l y correct, we think that a direct, nonr e l a t i v i s t i c c a l c u l a t i o n of  /vAJ^^H^j^  free of additional-  simplifying assumptions i s s t i l l of some value.  Such a c a l c u l a -  t i o n , which we undertake i n Section B of this Chapter f o r the  44 c a s e o f I s and to  2s e l e c t r o n s ,  see more c l e a r l y  B -  the  Evaluation of  role  makes i t , i n p a r t i c u l a r , p l a y e d by t h e  '^^/AJS^  in  retardation  evaluating  separately states.  directly,  the d i s c r e t e  The  and  the  procedure adopted  g r a t i o n o v e r the space v a r i a b l e first  f o u r t e r m s o f t h e sum  s t a t e s a r e o b t a i n e d and the  remainder  tegral.  The  evaluated 2 1.5  mc  the  integral  r.  carried  The  over the  The  consider  c a r r y out the  inte-  exact formulae f o r the  intermediate discrete  integrand  e q u a l t o mc  and  as w i l l  The  e x p r e s s i o n and  error  involved  be a p p a r e n t  from  The  in  the  o Discrete  about  f o r h i g h e r e n e r g i e s can  simple a n a l y t i c  small,  states i s  results.  1  energy  expression i s derived-for  limits  out a n a l y t i c a l l y .  i s quite  must  w h i c h c a n be t r a n s f o r m e d i n t o , an i n -  by a r e l a t i v e l y  procedure  numerical  i s to f i r s t  over the continuous energy  respectively.  integration  Electrons.  ( E q . 11..30) one  an a p p r o x i m a t e  sum,  2s  continuous i n t e r m e d i a t e energy  n u m e r i c a l l y between t h e  approximated  that  of  effects.  Non-Relativistic  a  A p p r o x i m a t i o n f o r t h e Case o f I s and  In  possible  States  s p a c e d e p e n d e n c e o f the g e n e r a l n o n - r e l a t i v i s t i c  be the  45  >  "s-function" describing a p a r t i c l e i n a d i s c r e t e energy l e v e l  E^—E^is  where  and  given by (see,  a.  -  /  where  P.311)  goSi™*-  *  £$(|_,yv.« ^ . & M A \  f u n c t i o n as d e f i n e d , The  f o r instance, Kramers (3^),  i s the confluent  for instance,  hyper-geometric  i n MacRobert ( 5 4 ) ,  p.346.  f u n c t i o n I I I . 5 i s normalized such t h a t  III. 6  When r  s  R, the n u c l e a r  <?^(R)=  r a d i u s , Eq. I I I . 5 becomes  *(ft) **  •,  V  -'3  - 3  since  ^ r  and  since  =  Z^rrr^.  $e IO  may w r i t e  for R  |0 «wv  III.8  * ( * )' i s a f i n i t e polynomial o f the form  i +• f ( l ? ) i i one  III.7  ^ ^ f ^ i^ ir a  +  .  in.9  approximately  III.10  46 We c a l l  the part o f I—  I  v;hich c o n t a i n s the sum  over the intermediate d i s c r e t e s t a t e s . and  2s t h a t we s h a l l c o n s i d e r are represented by the wave f u n c -  tions  (see I I I . 5 ,  HI-9):  <PlS =  °?e,  = "V «•  III.Ha  and cp^£ i n II.30 by t h e i r v a l u e s I I I . 5 , I I I . 10  Replacing and  The i n i t i a l s t a t e s I s .  I I I . 1 1 . one o b t a i n s f o r t h e two  L  , n=l,2:  n  9 0  I L j  0  =  l e  |  =  where The  00 q 72, < * ^ <.  -i  _C__S  -  <JJ  energy  tV\ c  "  w  " >~  >  n» / / \  OA-  ,\  r  ___  III.12a  .—  III.12b  i n 12b i s the same hypergebinetric f u n c t i o n as i n 12a. i s given by the e x p r e s s i o n  I f one approximates  by the e x p r e s s i o n  III.14  47 one o b t a i n s  and  As we have a l r e a d y mentioned,  we s t a r t by i n t e g r a t i n g over  the space v a r i a b l e r i n Eqs. III.12.  To t h i s e f f e c t we expand  the hypergeometric f u n c t i o n s i n t o t h e i r  ( f i n i t e ) polynomial form  o f equation I I I . 9 and i n t e g r a t e term by term.  The space  integral  o f III.12a, f o r i n s t a n c e , y i e l d s  III.16  where  /\ ^ ^  t  ^  - Zf%  This i s r e a d i l y integrated to give  a*.k A* j '  ~*  L  ^A/  UA/ *!  "1.17  The e x p r e s s i o n i n the square bracket o f III.17 i s the o r d i n a r y hypergeometric s e r i e s of argument h e t t i n g e r , p.7).  \  \ r  ?^  ' (See Magnus and Ober-  Hence the i n t e g r a l III.16 can be w r i t t e n :  ^  i n which z stands f o r  48  ATVA  where  ^  t  -  -i— —  -  III.19  - *  ^—  III.20  We next use the r e l a t i o n : (See Magnus and O b e r h e t t i n g e r  J  Therefore  jS ( | - 3 L * >  r  ~ O-^^""  3LyZ.)'  With the h e l p o f I I I . 2 2 , I I I . 1 8 , I I I . 2 0 and I I I . 12a  1  m*i  In case o f  *>ts  III.15a, the e x p r e s s i o n  Lip  ?  */c  t h e second  +*}  t h i s l a s t formula  function  i s e a s i l y obtained u s i n g again I I I . 2 1 .  c a l c u l a t i o n s i m i l a r to t h a t above, y i e l d s then f o r L i p  £  *»  A-  MS  <  In III.23b, however, z i s  HI.23a  term i n the i n t e g r a n d i n E q .  III.12b l e a d s t o the hypergeometric  s  111.22  becomes  ^Dls  S  p.8)  ^*Li*(Vi),**l  A  (Eq.III.12b)  m.23b  In e v a l u a t i n g  L ^  and  D  we c a l c u l a t e the f i r s t  f o u r terms  o f the sum over n s e p a r a t e l y , and we d e r i v e an approximate exp r e s s i o n f o r the summand when t h a t we o b t a i n l a t e r w i l l  /v\ ^ 5"  •.  The numerical, v a l u e s  show t h a t t h i s approximation  duces an e r r o r o f l e s s than 1$ i n the expressions f o r L-O^s  of  a)  •  We f i r s t  c o n s i d e r the case o f LQ,  introL-p  J S  and  and then the case  s  L'Da.s  Case o f  ^Oic  Let us c o n s i d e r , i n the.summand o f Eq. III.23a the f a c t o r  u  =  [ *v*r~-'-<-] l  c  t  iii,24  and w r i t e z i n the form  Z. =  2&  = f**^ " 1  - Z. = where  md  ^  =  cD t  '  III.25a  III.25b [( |  t  ±f  to^T  1  +  III.25c  —^~ III.25d  50 I f I I I . 2 5 a and b are i n s e r t e d i n I I I . 2 4 , one o b t a i n s  8«  MlrJVf- ^  =  Therefore, from  - <*-»<?-] .. . . m  26  III.13a  four terms o f the sum I I I . 2 7 are given.below  The f i r s t  CO 1)  L.  ^  =  —  by u s i n g the d e f i n i t i o n s o f  r  -  ^  (p  ~  t  ,  .  and  '  III  (J^  .23  , JEqs. III.25c and d.  When the r e t a r d a t i o n f a c t o r i s put equal to one i n the e x p r e s s i o n III.12a, o n l y the term n = 1 i n the sum over n c o n t r i b u t e s to the value o f  L»  n  (because o f the o r t h o n o r m a l i t y o f t h e wave  f u n c t i o n s used i n the i n t e g r a n d ) .  » *  n  t h a t case, i s given  by III.28 i n which the x i n the f a c t o r rk + ^t) zero.  i s put equal, to  Therefore:  U  °»S  ^  JfaA  I I I . 28a  and one o b t a i n s the Morrison and S c h i f f e x p r e s s i o n f o r 1) CO4.  A/jr  k^/fijj'  There are an i n f i n i t e number o f c h o i c e s f o r t h e v a l u e s o f , Eq. I I I . 2 5 d .  m - ... - 2 , - 1 , the expression  0 , 1, III.  \  However, i f one w r i t e s <^ c£++/«gf| where ,  t=  2,  ... and  O £><f±^i &  , one sees t h a t  remains the same f o r any choice o f m.  51 Similarly  2)  III.29  (?)  L.  3)  III.30  5  The t e r m s f o r way. it and  may  be a p p r o x i m a t e d  L e t us c o n s i d e r t h e e x p r e s s i o n  i n powers o f s m a l l e r than  3  i n the expansion,  argument o f t h e s i n e f u n c t i o n i n I I I . 2 6 ;  *  From r i l . 2 5 d ,  and f r o m  , E q . I I I . 2 6 and expand  "tie n e g l e c t a l l t e r m s o f powers* e q u a l t o  . /nT "  i n the f o l l o w i n g  tie we  consider f i r s t t h e  have  III.32  }  the r u l e f o r the tangents  d i f f e r e n c e s o f a n g l e s , we  obtain  o f sums and  52  ^Vi-U 1  I +•  and  + <f- =  '  W  [ • _  J  I f one n e g l e c t s the terms i n  _^2L_  ^  III.33a  ~ '«F  ^  1  _  _  -  ]  III.33b  i n III.33, one o b t a i n s  X W'oC  III.34  In d e r i v i n g III.34, we used the s e r i e s expansion f o r the f u n c t i o n tan  with  (  J ^CL^^j  equal to o r s m a l l e r than We approximate  ^  +  <^  /rv and  |  )  and n e g l e c t e d the powers  . i n t h e same manner; from  III.25d,  III.35  53  T h e r e f o r e , from  III.35  {V?-  -  0 + **)  ..  III.36  and  III.37  To d e r i v e  III.37,  we made use o f the r e l a t i o n k  ^  to./  With the help o f I I I . 3 5 ,  III.36,. I l l . 3 4  as an approximate e x p r e s s i o n f o r  It follows  therefore  We s h a l l evaluate  L  ^  and I I I . 2 7 ,  one may  write  when n i s l a r g e :  that  , Eq. I I I . 2 7 , as f o l l o w s  w i t h the terms i n the sum g i v e n r e s p e c t i v e l y by I I I . 2 3 a , 29, 3 0 , 31  and 4 0 .  54  b)  Case of  U  The case of  Lig^is  treated l i k e the one of  consider the expression III.23b, f o r k^c,  LIDIS.  .  We  and write  42.  where z i s given by III.23c.  One can show that  in analogy with Eq. III.26, with  III.43b  and  <?t =  P^' [ l  I t follows from III.43a and III.23b that  The f i r s t four terms of the sum III.44 are the following:  111.43c  55  and  for  ^  5"  In III.45e we have neglected  the powers equal to o r smaller than  when compared t o u n i t y .  From III.45e, i t f o l l o w s t h a t  oo  LIQ^^  w i l l be evaluated  as f o l l o w s  where the terms are given by Eqs. III.45 and III.46.  56 2°  Continuous  States  To e v a l u a t e t h e c o n t r i b u t i o n  L^  to  s  and  Li^  11.30,  , Eq.  i n v o l v i n g t h e i n t e g r a t i o n o v e r t h e c o n t i n u o u s e n e r g y s t a t e s , one must c o n s i d e r  the following  integral  E-  E +ft<* m  The n o n - r e l a t i v i s t i c wave f u n c t i o n s i n t h e continuum  i s the f o l l o w i n g  describing  (See Sommerfeld  an e l e c t r o n p.115 & f f . )  IT 1.49  This  wave f u n c t i o n ,  interval:  t h i s means  which  i sreal,  i s normalized per unit  energy  that  ) ^ ^ i C A ) tf C/v) = 1  . in.50  E  1 where is  i s any energy i n t e r v a l c o n t a i n i n g  r e l a t e d t o t h e momentum p o f I I I . 4 9  E = [cytfr*?)*]**  E .  The e n e r g y E  through the r e l a t i o n  or***  £  '  -  I  I  I  w  5  1  57 When r = R, one cannot s e t equal t o one the f a c t o r  IIIo52  as r e a d i l y as i n the case o f beeause,  in  III.48,  integration.  LQ^  (See Eqs.  III.7  t o III.10)  p becomes i n f i n i t e a t the upper l i m i t o f  T h i s f a c t o r i s s h o r t l y c o n s i d e r e d i n Appendix C  where we conclude t h a t s e t t i n g i t equal t o u n i t y does not a f f e c t the value o f  III.48.  The wave f u n c t i o n s initial  C^  s  T h e r e f o r e , we may w r i t e , when r «  and ^  s t a t e a r e g i v e n by Eqs.  We c o n s i d e r f i r s t the case o f L  With the use o f 111.49,  U  where  >  f o r the e l e c t r o n i n t h e  III.11a  and  III.lib  o  and then that o f  C | s  III.53 and"III.11a, III.43  i s given by Eq. nrrve  To e v a l u a t e III.54-*  w  X  e  f i  I -  ~  r  s  III.13- w i t h  t  becomes  III.54  - E L - E , -rfco*  =  respectively.  Lie..  a) E v a l u a t i o n o f  °  s  n • 1. j  express the c o n f l u e n t  III  hypergeometric  f u n c t i o n i n an i n t e g r a l form (See, f o r i n s t a n c e MacRobert (54) p. 3 4 6 ) .  R  o 5.5  53  $(WJLr&\'U&)  Then we in  —  -  r  (^^^l-tP^t  i n t e r c h a n g e the o r d e r o f the i n t e g r a t i o n s o v e r r and t  III.54; t h i s  i s allowed because o f the presence  creasing exponential factor  JL  over r f i r s t ,  t , one o b t a i n s  U  111.56  =  and then over  t r f r * *  (~  o f the de-  i n the i n t e g r a n d .  dLzrfrir'i+itit  Integrating  • . •  v  m  . j  7  111.56 is  the hypergeometric Z±  f u n c t i o n and III.59  -  One a r r i v e s a t S q s . I I I . 5 7 , 5& and 59 b y making u s e o f t h e integral  MacRobert  r e p r e s e n t a t i o n o f the h y p e r g e o m e t r i c  f u n c t i o n (See  (54), p.297):  III.60  From I I I . 2 1 ,  i t follows that  For computational  purposes,  following dimensionless  i t i s convenient  quantities:  to introduce the  59  With these n o t a t i o n s ,  III.59 becomes  -  With the h e l p o f  -  -tfr 1 ^- ^ C  .'111.63 "  III.61, one shows  that.  III.56  yields  III. —  A.  I  -tr :-III.65a  where  %  0  - i ^ x l  6+-6-  [| r (fr and  i n 65b  v o +  111.65c  5A  ^  D  III ,66a  ^  III.66b  R  connected with the m u l t i - v a l u e d  obtain  nj 65d  0 ^ 6 < TT/z  These i n e q u a l i t i e s a r e e s s e n t i a l f o r a v o i d i n g  Using  =t^V*),  the r e l a t i o n  difficulties  c h a r a c t e r o f the f u n c t i o n  P(z.)P(|-Z.)  -  "JJ/^TTz. "'  III.64. we  60 Iff  JL  '  '  III.67  -  I t follows from III.55, 6 2 , 64 and 67 that  where  L  - ^  J ,  =  r  f  -s  *c  III.68a  III.68b  1  I ~ j has been plotted as a function of q,  The integrand  s  and the i n t e g r a l III.68a evaluated numerically f o r the lowest values of q ( i . e . f o r q between 0 and q to an energy E of about 1.5 mc i n the case of Cs  ).  0  i n the case of A^  2  2mc  where <J  Q  7  corresponds and of  For the higher energies  about  > fyoj) c  an approximate integrand (which i s derived below) i s used i n place o f ' I j tically.  and the resulting integral i s performed analy-  s  Let us write Ltc  where  .0) ^c/s, i  the part of U  3  i s  - u  Cls  the part of  a s  r  , LJ  u  C / s  t n e  c  Cls  . . <i) evaluated numerically and v-»c  JS  If  £  0  i s the energy  , we have  =  L,  °^ two terms:  s u m  evaluated a n a l y t i c a l l y .  corresponding to q  and  ^Cjc,  _  *z  w ) ,  J  T  _ (  ' 5  i  £  - j - ^ e  •  I  n  -  7  0  a  III.70b  61 We approximate T^  by expanding i t i n powers of '/^ and by  s  taking the f i r s t two terras. Eq. I I I . 6 4 i n powers of ^  The upper l i m i t  t  Q  The expansion of the binomials of i s allowed when  of the numerical integration III.70a i s  chosen such that the corresponding q  s a t i s f i e s III.71.  Q  We give  below, as an example, the f i r s t few terms of the expansion f o r one of the binomials i n I I I . 6 4 :  + (-It^Xi-^JhrJ"**^  / 1 r y \/  *\  .  X.I  111  U  1  .  7  2  i  t -5 When the expansions are carried out up to the power (iq) , i t i s se'en that A  is  3<j.  ^  v  fc  ? '  in.73  In the same approximation, one has, f o r the remaining factors of the integrand X  , ( E q . . III.68b)  ) s  H  I I I . 74a  T T /  **V  Z  die.  c<  i n . 74b  Z-^d-o.  III.74c  62 III.74d T h e r e f o r e , from III.74, 73,  62,  63b  and  70b:  3  LT  and  =  !iL*  We have f i n a l l y , f o r  with  L  /, . X  Evaluation of  Li  The e v a l u a t i o n o f  i s c a r r i e d out along v e r y  With the a i d o f I I I . l i b , 49,  with E ^  given by Eq. III.13 with n  x  =.  AT,C3X'  given by III.69  c  lines.  E  and 53,  =  and then over t i to o b t a i n :  similar  III.43 becomes  2:  *  With the help o f III.56 and I I I . 6 0 , r first,  1 I T  IS  given by III.41 and  b)  X-ZisQ  IH. g 7  one i n t e g r a t e s III.77  63  III.79  where  - -L ( £zf  ^  [  * LX, l * f i * i £ ) i3  >  a^-  =  -  H I . 8 1  In the notation of III.62, we have  and,  from  = £  t  III.21,  [£°i^Jbi  +  x-'-'Ci-^-'r*^]  1 1 1 , 8 3  where we introduced a notation s i m i l a r to the one i n III.65a  w Ok  .  64  $i - W ^Cfr**) 1  6^  and  have t h e same r a n g e o f v a l u e s  as  Q+.  and  6_  ( E q . III.66) Using  t h e above n o t a t i o n a n d E q . I I I . 6 ? , we  I  -  ^  ( ° °  &  have  A  R  III.85a  with  Ijic  —  ~  \  i s defined  U^,^  ~  —  :  ;  II 1.85b  —  e - A r z*<x  3  i n III.78.  was e v a l u a t e d  i n t h e same manner a s  , i . e . a s t h e sum  o f two t e r m s 0)  ,  Ljc ,' - L»c a  with  «  ^c-xs  X!l  U,  -  Hi^.Ji£-  \  l £  =  ^  (  • -i§-  III.86a  III.86b  I"  cL  U)  ,0) L>  111.86c  L  i s the part o f u  C3kS  part X»vS  evaluated a  t  high  ponding q  c  C  a  evaluated  s  analytically,  energies.  ( E q . III.62)  £  a  numerically,  being i s chosen  satisfies  a n d >-JCa^ t h e  an a p p r o x i m a t i o n f o r such t h a t the c o r r e s -  the r e l a t i o n :  65  4r 2^  ±  III.37 rv  The approximate i n t e g r a n d  Xj^  i s d e r i v e d by d e v e l o p i n g the  e x p r e s s i o n III.35b i n powers o f l / q .  The f i r s t two terms o f  the expansion are - 7 * ft  TT«[f  V  3  /  */\7+<f  I t f o l l o w s from 36c and from the r e l a t i o n  III.S3  cLc^^d^  that  III.39  We have f i n a l l y ,  *  with ^-»o  3kS  for  L  Px  L* c, a  + s  given by Eqs. I l l . 4 5 and III.47 and  Eqs. III.36a, 86b and 39.  111.90  g i v e n by  66 o 3  We to  have a p p l i e d t h e f o r m u l a e  t h e case  reliable value  of A  data  and methods o f t h i s  Among  exist  In a d d i t i o n  t h e I s - spectrum  this  i s t h e one w i t h approximation  the smallest i snot  t o t h a t , we have e v a l u a t e d one p o i n t  f o r the case  o f Cs^" " 1  (Z s 55) u s i n g o u r  with a view t o comparing t h e r e s u l t  t h e same  Section  t h e few e l e m e n t s f o r w h i c h  o f Z so t h a t t h e n o n - r e l a t i v i s t i c  formulae, for  (Z r 1 8 ) .  experimental  unjustified. of  R e s u l t s and C o n c l u s i o n s  with Hess's  result  case.  37 a)  The Case o f A  37 In  t h e case  ^^/us^  ^  o  f°  r  F o r $ck  u  of A  different  r  "  Jrfck » 4 0 4 Jrfck = 673 limit The and  integral the and  o f t h e spectrum results  f o rthis  Table  " x = 4 " x » 6 " x = 10  n  i s a t #ck = case  £16 K e v .  a r e summarized  i n Tables I , I I  the v a r i o u s c o n t r i b u t i o n s t o the  (Eq. 11.30, ( n  photon energy  ^  "  5.  I , we l i s t  L-*^  spectrum:  to x • 2  "  n  "  I I I and i n F i g . In  p o i n t s o f the photon  135 Kev c o r r e s p o n d i n g  =  jfick = 269  The  (Z = 18), we have computed t h e r a t i o  listed  =  above.  1)), f o r the f o u r v a l u e s o f  , <» ,W . L °is ' Dia  U  n  are given by the formulae  X»  ,w  , . L»« 1 L °^ w»s I I I . 23, 2 9 , 3 0 , 31 and n  7  n  67 40  respectively.  Eq. III.40,  To get an idea about the r e l i a b i l i t y of  l—p  has also been calculated fro,m the approximate  expression III.39.  The values obtained (in i t a l i c s ) when . to  compared with the exact values f o r L. ^ l e s s than 0.5 percent.  , show a deviation of  The error i n evaluating  i-  L - » D , C  o£  s  ,0)  the same order of magnitude.  The values for  are obtained  by numerical integration of Eq. III.70a f o r four d i f f e r e n t values of x.  Since the integrand i s too complicated to be studied analy-  t i c a l l y , we give i n F i g . 1, 2, 3 and 4 the curves representing the integrand as a function of the integration variable q f o r the four values of x.  The curves \ju'S"t helped us to choose the  appropriate lengths of intervals when carrying out the numerical integration. each case.  The upper l i m i t of integration q L.  The main error i n the expression f o r U,  LJ  c  comes from the numerical  ; the error made i n evaluating  are small compared to i t .  C|S  i s different i n  i s evaluated a n a l y t i c a l l y from Eq. III.75b.  c  integration of U  0  Up  and  The t o t a l error on Ujj  »  however, i s probably less than one unit i n the t h i r d s i g n i f i c a n t figure. In Table II we l i s t the contributions to.the integral L  a s  , (Eq. 11.32 ( n s 2)) f o r the same four values of the  Photon energy. L 0 ^ , U0 ^ evaluated from Eq. III.45, 46.  ,L  ,L  , L  are , calculated from the  |  D j 5  , CD  approximate Eq. III.45e i s also included.  L.  c  i s given by  Eq. III.36b which was also integrated numerically. the curves for the integrand been included.  I  (  E  q  L*^  tude as i n the case of  Li^  However,  . III.85b) have not  i s given by Eq. III.89-  the values obtained f o r  0 3 k &  The error, i n  are of the same order of magni.  68 In Table * /AAJ^ Table  I I I , we  as e v a l u a t e d f r o m  I and  Table  which appears  i n Table  11.30  / is  /W-  0  In  is  F i g . 5,  we  + 2.S  The  identical Schiffs We  The  curve  both  ^  of  quantity  as  V  WL  i n f l u e n c e of the  t h e two  retardation  i n our  case  ( *ij^)* A  n  t  n  e  c  a  w i t h t h a t w h i c h w o u l d be  s  and  °f  e  ratio  quantities i n the  (5^-), a s a f u n c t i o n o f t h e e n e r g y  Martin  photon.  .  the v a l u e s  ratio  c o n t r i b u t i o n s to the p r o b a b i l i t y  have p l o t t e d  ~~ ( ^ f c ^ ^ Y_ . , . OK \ / u j ^ /IS -t-iS  probability  from  L.^  and  III i s defined  LA  the  f o r the and  i s t h e r e f o r e a measure o f t h e  f a c t o r on  and  Eq. L |s  II f o r  _  P  ^  g i v e the v a l u e s  case  of  Glauber  o f the. e m i t t e d  Glauber  obtained  ^  from  and  Martin i s  Morrison  and  formula. can  say v e r y  little,  at the present  comparison o f our t h e o r e t i c a l  results  time,  about  for  the in  case  37 of A The  w i t h the procedure  experimental  used  theory consists, c o r r e c t i o n s to n o r m a l i z i n g the  by t h e s e  first,  r e s u l t s o f L i n d q v i s t and authors  measured s p e c t r u m intensities  c o r r e c t i o n s woul.i t h u s  have t o be  by L i n d q v i s t  the- d i f f e r e n c e B ( l s -+• 2s)  cannot  i n the  do  and  to the  are being  s o l e l y on  i u (55).  be  reduced  so  I t i s not  in  obtained The  theoretical  the b a s i s of  curves  next,  compared).  a p p l i e d to our  s l o p e o f the  i n F i g . 5 may  spectrum  and  experimental  gamma s p e c t r u m , and,  (so t h a t o n l y r e l a t i v e  published  compare e x p e r i m e n t s  i n applying a l l kinds of  the t h e o r e t i c a l  s p e c t r u m , w h i c h we  to  '.vu ( 5 5 ) .  the  data  impossible that  A ( I s + 2s)  considerably after  and the  n d  corrections have been applied.  b)  The Case of C s  1 3 1  •  ' ,, . %  55)  In the case of C s ^ ^  we have calculated  A  ^^&&£,  only f o r one value of the photon energy, namely f o r #cka-2^0^ Kev (x = 1), and we r e s t r i c t e d ourselves ,to the i s - electron c o n t r i bution.  . -  - '•  An application of the formulae of t h i s Section and a numeric cal integration carried out i n the way indicated i n Subsection> a) give, f o r G s  1 3 1  :  "  Therefore,  | .Tfi  JL  The r a t i o  i s found to be equal to  LWASJH'  0.79.  I f only the integration over the intermediate continuousstates i s taken into account, becomes only 0.03.  L, Q ^  0  > and the r a t i o  This small value f o r  accounts f o r the .  larger part of the discrepancy between the r e s u l t s of Hess (who neglected the sum over the discrete states)' and the. r e s u l t s of Morrison and S c h i f f .  1.0  0.8  Flq. 0-6  0.4-  0.2.  rr>t«.^\At\a  op  m  1 . 70  i  Poi\. % «  £  Patincj patje 70  P&ctrvj  pA«j£  (0  TO Table  is  Kck » 135  . (x - 2)  I  #ck = 269 (x - 4) '  #ck = 404 (x . 6)  tfck = 673 (x - 10)  li°  0.25000  0.02000  0.003333  0.0002959  •(»  0.03198  0.00266  0.000425  0.0000372  0.01225  0.00103  0.000168  0.0000147  Co  0.00401  0.00033  0.000053  0.0000047  ,w  0.OW0O7  0.00Q32Q  r  0.00513  0.00042  0.000069  0.00000601  0.3034  0.0244  0.0040  0.00036  0.5962  0.3677  0.2093  0.09953  0.0063  0.0005  0.0012  0.00029  0.6025  0.3682  0.2105  0.09982  0.906  0.393  0.215  0.100  Table I  showing the various contributions to  e.  u L  the i n t e g r a l  0. 0000SS  L ,  s  <?  (.in units of  for four values of the photon energy Jick ( i n Kev)  {^^j  Table  tfck = 135 (x = 2)  II  -ftck = 269 (x = 4)  tfck = 4 0 4 (x = 6)  0.08335  0.007306  0.001206  0.0001079  0.00594  0.000984  0.000156  0.0000134  0.00451  0.000308  0.000047  0.0000040  0.00147  0.000124  0.000020  0.0000017  0.001464  0.0001249  0.0000197  0.00000168  0.00137  0.000160  0.000025  0.0000022  0.0977  0.0039  0.00146  0.00013  0.2084  0.1312  0.07327  0.03062  0.0019  0.0007  0.00039  0.00015  0.2103  0.1319  0.07366  0.03077  0.303  0.141  0.0752'  O.O309  Table I I  showing the v a r i o u s c o n t r i b u t i o n s ,to the i n t e g r a l  L  -Kck = 673 (x = 10)  ' (in units of  f?&£M  f o r f o u r values o f the photon energy )ick ( i n Kev)  O  lOO  400  600  8OO  7a Table I I I  Jick . 135 (x * 2)  Jick « 269 (x . 4)  #ck . 404 (x - 6)  673 (x . 10)  jfek -  0.637  0.665  0.384  0.0470  0.080  0.087  0.047  0.0047  0.314  0.617  0.415  0 243  Table I I I  showing the contributions to  O  from the Is and 2s electrons ( i n photons per desintegration per unit Kev energy i n t e r v a l  / |0  )  f o r four values of the photon energy fo-K (in Kev). the ratio  The table also shows  of the sum of these con-  t r i b u t i o n s to the corresponding r e s u l t of Glauber and Martin (54).  73 Appendix  We  want t o s k e t c h t h e p r o o f o f t h e s t a t e m e n t  JL 71*. R  is  that  ^c OAi af( ) A * $ < * 2 £ r e ^ ^ m ^ i R ^ s . a  R  E-  1.  negligible  (where I i s . t h e u n i t spherical  -  . l  A  *  f o r s m a l l v a l u e s o f R.  M a k i n g U3e o f t h e c l o s u r e  of  A  property,  matrix), expanding  into  h a r m o n i c s and t h e B e s s e l f u n c t i o n s , and  i n t e g r a t i n g over  angles,  products finally  one o b t a i n s f o r A . l an e x p r e s s i o n  pro-  portional to  to f\  'r\  (we have h e r e  *  assumed, f o r s i m p l i c i t y ,  the n o n - r e l a t i v i s t i c  f o r m f o r the wave f u n c t i o n o f t h e K - e l e c t r o n ) . consequently,  A . l becomes  proportional to  3k.  w h i c h i s o f the O r d e r  (\  for  f{—> 0  Hence A.2 a n d ,  74Appendix  We  sketqh  the p r o o f o f t h e  B  identity:  B.l where  AZ - jj^ % K*  and  Making use o f t h e r e l a t i o n s  K  =  O  £<r =t <?3  , (32 a, -5/3  (J** 1  ,  -  t h e e x p r e s s i o n on t h e l e f t - h a n d s i d e o f B . l c a n be w r i t t e n a s  JL  l  •  3.2  or  B.3  Now,  2* = |  So  t h a t the second  just  the e x p r e s s i o n  line  o f B.3  vanishes,  on t h e r i g h t - h a n d  and  the f i r s t  side o f B . l  e  Dine  gives  75 Appendix  C  We want t o make i t p l a u s i b l e  in  that the f a c t o r  E q . I I I . 4 © may be s e t e q u a l t o u n i t y o v e r  of i n t e g r a t i o n , The  without  t o o much  t h e whole r a n g e  error.  g e n e r a l form o f Eq. 1 1 1 . 5 4 i s  ( S ^ - . H I -  6  8  )  1  C.l  where  is  t h e f a c t o r i n q u e s t i o n , and  i s given by I I I . 6 4 .  -3  yixr\ 0± | 0  Since gration  , one h a s , i n f a c t ,  r e g i o n where t h e r e m a i n i n g  large,  i . e . f o r 0<v^-<i5" »  Fig.  to  1  For without so  a  s  factor  ^  1  i n the i n t e - . ,  o f the integrand i s  c a n be s e e n f r o m t h e g r a p h s i n  4 .  qj- 7 j 5"  , the integrand  i n C . l c a n be r e p l a c e d ,  a p p r e c i a b l e e r r o r , by a s i m p l e r e x p r e s s i o n  (see Eq.  III.75a)  that  Consider large  now v e r y  large values  q's c a n be o b t a i n e d  o f q.  The b e h a v i o u s o f f f o r  from t h e asymptotic  expansion  o f the  76 confluent we  hypergeometric f u n c t i o n  l  find  f r  ( J a h n k e and  Snide,  p.275);  R  v. ' y ! .  and  f  —*» O  Erade, p.  10).  But  is  practically  is  harmless. For  the  remainder o f f  c£  for  <**  for large  zero,  so  that  intermediate the  = 1 i s o f no  integrand  great  i  , because q's,  the  putting  values  of  (Jahnke  integrand, a  ^  q,  |  and  excluding  (instead of  f remains f i n i t e  i s s t i l l very  consequence.  X  small,  so  that  f, f.<£l)  while  the  putting  77  References  (52).  B l a t t , J.K. and Weisskopf, V.F.  "Theoretical  P h y s i c s " , John Wiley and Sons,  Nuclear  1952.  De Oroot, S . R . and Tolhoek, H.A. ( 5 0 ) . Physica 1 6 , 456 (1950).  Fermi, E. (34)•  Z. Phys..88, 161 (1934).  Glauber, R.J. and M a r t i a , F . C . (55).Glauber,  R . J . and M a r t i n , ?.C. ( 5 6 ) . Ill,  page 43 o f t h i s  Q£, 572 (1954).  Phys. Rev.  (Unpublished;  see Chapter.  thesis).  Gordon, W. ( 2 8 ) . Z. Phys. £ 0 , 630 (1928). H e i t l e r , W. ( 5 0 ) .  "The Quantum Theory o f R a d i a t i o n " , 2 n d Ed*, 1950.  Oxford U n i v e r s i t y Press, Hess, G. (55). June  T h e s i s , The U n i v e r s i t y o f B r i t i s h Columbia, 1955.  Konopinski, E . J . and Langer,  L.M. (53).  Ann. Rev. o f N u c l .  Sc. 2, 261 (1953)• Konopinski, E . J . (55).  Chapter X o f K. S i e g b a h n s "Beta and f  Gamma Ray Spectroscopy",  North H o l l a n d P u b l i s h i n g Co. 1955•  Kramers, H.A., "Die Quantentheorie Strahlung".  Leipzig,  des E l e k t r o n s und d e r  1938.  L i n d q v i s t , I . and Wu, C S . (55), Phys. Rev. MacRobert, T . M . (54) MacMillan,  100. 145 (1955)-.  " F u n c t i o n s o f a Complex V a r i a b l e " , 4th Be*.., 1954-  Iladansky, L. and R a s e t t i , F. ( 5 4 ) .  Phys. Rev. 2kt 407  Magnus, W. and O b e r h e t t i n g e r , F. (49).  "Formulas'and Theorems  f o r the S p e c i a l F u n c t i o n s o f Mathematical Chelsea P u b l i s h i n g Co.,  1949.  (1954)-  Physics",  76  Morrison, Pauli,  L . I . (40).  P. a n d S c h i f f ,  W. (33).  0  R e v . 5|_, 24 (1%Q)  c  "Die allgemeinen P r i n z i p i e n . der  Hand, d e r Phys.,  Rose, xM.S.  Phys  (37).  3d. X X I V / i ,  Wellenmechanik",  (1933).  F h y s . Rev. 51, 4#4 (1937) »  Rose, M.E. (55).  Chapter  I>I o f S i e g b a h n ' s  " B e t a a n d Gamma R a y  S p e c t r o s c o p y " , N o r t h H o l l a n d , 1955.  S a r a f , B. (54).  P h y s . Rev. 94, 642 (1954).  S o m m e r f e l d , A. " W e l l e n m e c h a n i k " , New  Ungar P u b l i s h i n g  Company,  York..  Wu, C S . (55).  " 3 e t a and Gamma Ray S p e c t r o s c o p y " , . N o r t h  Holland,  1955.  Wu, C.3., L i n d q v i s t ,  ( p . 649).  T., G l a u b e r ,  R.G., a n d M a r t i n , P.O. (56).  P h y s . Rev. 101, 905 (1956).  •  

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