OK THE THEORY OF RADIATIVE ELECTRON CAPTURE by GUY PAQUETTE . B . S c , U n i v e r s i t e de Montreal, 1951 •• M. A.,. U n i v e r s i t y o f B r i t i s h Columbia, 1953 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f Physi c s We accept this* t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1956 Faculty of Graduate Studies P R O G R A M M E O F T H E F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E O F . D O C T O R O F P H I L O S O P H Y 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 BUILDING C O M M I T T E E IN C H A R G E D E A N F . H . S O W A R D , Chairman W. O P E C H O W S K I C. A . M A C D O W E L L K . C. M A N N R. D. J A M E S G. M . V O L K O F F T. E. H U L L R. W . S T E W A R T J . G. A N D I S O N External Examiner: R. T. S H A R P McGill University O N T H E T H E O R Y O F R A D I A T I V E C A P T U R E O F ORBITAL ELECTRONS Abstract The continuous spectrum of gamma radiation which accompanies the cap-ture 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 differ-ent: the intensity of the gamma radiation is an order of magnitude lower accord-ing to Hess than according to Glauber and Martin. The purpose of the calcula-tions 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 incon-sistent, 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 Wu (1955) seems to agree quite well with Glauber and Martin's result. However, Lindqvist and Wu 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 Rayon-nement 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 Nuclear Physics K. C. Mann 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 Physics of the Solid State i [ R Brown " Low Temperature Physics J. M. Daniels Chemical Physics A. J. Dekker Other Studies: Differential and Integral Equations •. T. E. Hull Group Theory B. N. Moyls Theory of the Chemical Bond C. Reid Network Theory A. D. Moore Faculty of Graduate Studies P R O G R A M M E O F T H E F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y 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 I N C H A R G E D E A N F. H . SOWARD, Chairman W. O P E C H O W S K I C. A . M A C D O W E L L K . C. M A N N R. D. J A M E S G. M . V O L K O F F T. E. H U L L R. W . STEWART J. G. A N D I S O N External Examiner: R. T. SHARP McGil l University O N T H E T H E O R Y O F R A D I A T I V E C A P T U R E O'F ORBITXL ELECTRONS Abstract The continuous spectrum of gamma radiation which accompanies the cap-ture 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 differ-ent: the intensity of the gamma radiation is an order of magnitude lower accord-ing to Hess than according to Glauber and Martin. The purpose of the calcula-tions 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 " (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 incon-sistent, 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 Wu (1955) seems to agree quite well with Glauber and Martin's result. However, Lindqvist and Wu 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'Rayon-nement 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 Theory W . Opechowski Theoretical Nuclear Physics W . Opechowski !W . Opechowski G. M . Volkoff F. A. Kaempffer Nuclear Physics K . C. Mann 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 Physics of the Solid State....: { ] J . ™ % ™ O T E Low Temperature Physics ..]. M . Daniels Chemical Physics . . . . .A. J. Dekker Other Studies: Differential and Integral Equations T. E. Hull Group Theory B. N . Moyls Theory of the Chemical Bond C. Reid Network Theory A. D. Moore ACKNOWLEDGEMENTS I wish to express 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 suggesting t h i s problem and .f o r h i s continued i n t e r e s t and v a l u a b l e a d v i c e throughout the performance o f the r e s e a r c h . I wish a l s o to thank the N a t i o n a l Research C o u n c i l o f Canada f o r f i n a n c i a l help i n the form o f a Studentship and a F e l l o w s h i p . ABSTRACT The continuous spectrum of gamma radiation which accompanies the capture of orbital 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 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 i n this thesis has been to settle this disagreement and to explain i t s origin. To this effect the high energy part of the gamma spectrum, which i s almost entirely determined by the contribu-tions of the capture of the Is and 2s electrons, has been com-37 < puted for the case of A for which experimental data are avail-37 able. In view of the low nuclear charge of A- (z = 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: f i r s t , 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 0.81 of the intensity obtained by Glauber and Martin at 135 Kev, and to 0.24 at 675 KeV (the gamma spectrum limit being 6*16 KeV). 37 The gamma spectrum of A determined by Lindqvist and'.Wu (1955) seems to agree quite well with Glauber and Martin's r e s u l t . However, Lindqvist and Wu measured only r e l a t i v e i n t e n s i t i e s and had to apply many instrumental corrections so that i t i s not yet clear whether the measured spectrum would not agree as well with the spectrum computed i n t h i s t h e s i s . V Table of Contents. Page ACKNOWLEDGEMENTS i i ABSTRACT i i i Table of Contents v Introduction and Summary 1 Chapter I - General Formalism of the Theory of Electron Capture 8 A - Radiationless K-Capture % ' B - Radiative .K-Capture 1 4 C - The Matrix Element of Electromagnetic Interaction let Chapter II - The Expression f o r the Pr o b a b i l i t y of Electron Capture i n Terms of Approximate Wave Functions. 26 A - Wave Functions and the General Expression for AVftify^^ 27 B - Non-Relativistic Expression f o r f ^ ^ ^ j ^ C - The Approximation of Hess ( 5 5 ) 37 Chapter I I I - Radiative K-Capture i n a Non-Relativistic Approximation 39 A - Method of Glauber .and Martin ( 5 4 ) 4 0 B - Evaluation of ^ A V A * * C i n a Non-Relativistic Approximation for the Case of Is and 2s Electrons in the I n i t i a l State 0 4 4 1 ° Discrete States 4 4 (a) For the case of Is electrons 4 9 (b) For the case of 2s electrons 54 Table of Contents (Cont'd) Page 2° Continuous States (a) For the case of Is electrons (b) For the case of 2s electrons 3 0 Results and Conclusions a) The Case of A 3? b) The Case of C s 1 3 1 56 57 62 66 66 Tables I - Various Contributions to the Integral L , s (11.30 with n . 1) 70 II - Various Contributions to the Integral (11,30 with n = 2) 71 III - Values for the Probability Ratio 72 Appendices A B C On the neglecting of the second term of II.13a 73 On the expression II.15b 74 On the expression III.52 75 Figures 1 - Integrand r l S (III.68b) for a photon energy of 135 Kev 2 - Integrand Xis for a photon energy of 269 Kev 3 - Integrand I|s for a photon energy of 404 Kev 4 - Integrand lis for a photon energy of 673 Kev Facing Page 70 70 70 70 Table of Contents (Cont'd) v i i Facing Page 5 - The t h e o r e t i c a l gamma spectrum of t h i s thesis as compared with that of Glauber and Martin (54) . 72 Pafte References 77 Introduction and Summary 1 This thesis i s concerned with the theory of the "radiative capture" of an orbital electron by the nucleus. We ca l l the process of capture "radiative" when i t i s accompanied by the emission of a gamma radiation. When no gamma emission takes place, the process i s called "radiationless". 3 The radiative capture, although a factor 10 less probable than the radiationless capture, has been observed for several elements, and the corresponding continuous gamma spectrum has been determined (See, for instance, Lindqvist and Wu (55), where also references to earlier papers are given). The most important radiative capture process i s that accompanying the radiationless capture of a K-electron. The theory for this case has been developed by several authors (Morrison and Schiff (40), Glauber and Martin (-54) and Hess (55)). In order to describe briefly what these authors have done, and to indi-cate the contribution to the theory, presented in this thesis, we shall have f i r s t to sketch the common theoretical basis of a l l these calculations. The expression for the probability of 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 this formula, one factor arises from the electromagnetic interaction Hy a n ° - t n e o t ^ r from the Fermi interaction Hft The electromagnetic interaction induces a transition between an i n i t i a l state may correspond to an electron characterized by i n i t i a l state and an intermediate state. For. instance, the 2 the p r i n c i p a l and o r b i t a l quantum numbers n and respectively. The intermediate state then corresponds to an electron with the quantum numbers and JC" =0 (an s-electron), and to a photon. The Fermi interaction gives r i s e to the capture of the electron i n that intermediate state and to the creation of a neutrino. The assumption that JL*" - 0 (but n* arbitrary) i n the intermed-iate states, or, i n other words, that the captured, electron i s an s-electron, defines the radiative process corresponding to the "allowed" radiationless capture of an s-electron from the K - s h e l l . We s h a l l c a l l t h i s process the "radiative K-capture", although, i n view of the remark above, t h i s phrase should hot be interpreted i n too l i t e r a l a sense. We denote the p r o b a b i l i t y of radiative K-capture, i . e . the p r o b a b i l i t y per second that a photon of energy #ck in the range d(/ick) i s emitted during the K-capture process by /uT^ dLM. , and the p r o b a b i l i t y that a r a d i a -t i o n l e s s K-capture occurs per second by /ur c . It i s the ob-served r a t i o ^^rV^j^r » a s a function of k, that i s compared with theory. In p r i n c i p l e , a l l o r b i t a l electrons contribute to the experimental value f o r Ajj^ J l r l . This means that AAr^<Lr\ i s a sum of the p r o b a b i l i t i e s obtained f o r a l l values of n and A. that correspond to the occupied o r b i t s . However, i t turns out that only the o r b i t s characterized by n = 1, ^ : 0; n » 2, X> • 0 and JL s 1; and n • 3, X = 1, contribute an appreciable amount to the r a t i o ^ K ^ ^ / M J ^ • As i s well known, there are f i v e d i f f e r e n t interactions H p possible i n the Fermi theory of beta decay, giving r i s e to allowed and forbidden spectra. In t h i s t h e s is, as well as i n the publications quoted above, only allowed t r a n s i t i o n s have been considered. The f i r s t t h e o r e t i c a l evaluation of AAT^ dLK was made by Morrison and S c h i f f (40) who considered only the case of the K-shell electrons, i . e . the case i n which n s 1 and JL s 0. These authors made, among others, two important simplifying assumptions: f i r s t , they represented the f i n a l , intermediate and i n i t i a l states by plane waves, thereby neglecting the Coulomb int e r a c t i o n between the electron and the nucleus; second, they took into account only the vector i n t e r a c t i o n . They calculated . /UJ£ i n the same approximation and they f i n a l l y obtained the following simple formula: O f i ^ = £ K ( I - . - J f — where ft ~ J^rtr\/c}~ (dimenslonless) and aC i s the fin e structure constant. ... This formula more or les s agrees with experimental data f o r photons of large energy ( i . e . 1 - ' K ^ ^ ^ I ) . However, as was f i r s t shown by Saraf's experiments (54), the Morrison-Schiff formula f a i l s completely to explain the large r a t i o A*J\(^^/MJ" that one obtains experimentally for low energies of the emitted photon. In order to explain t h i s breakdown of the theory at low photon energies, one may extend Morrison and S c h i f f f s c a l c u l a t i o n s i n several d i r e c t i o n s , by taking into account: 1 s t t n e Coulomb inte r a c t i o n between the electron and the nucleus, 2nd an a r b i t r a r y mixture of the f i v e beta i n t e r a c t i o n s , 3rd the contribution to radiative capture of electrons with higher values of n and JL . Such an extension of the theory has been carried out by Glauber and Martin (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 himself, however, to taking into account the contribution of the Is electrons only. The cal c u l a t i o n s are approximate in both cases, but the method applied by Glauber and Martin i s quite d i f f e r e n t from that applied by. Hess. The re s u l t s do not agree: the i n t e n s i t y of gamma radiation i s an order of magnitude smaller according to Hess than i t i s according to Glauber and Martin. Glauber and Martin's (54) c a l c u l a t i o n i s n o n - r e l a t i v i s t i c as f a r as wave functions of the electrons are concerned; i n other words, they used the Schroedinger wave functions of an electron i n the Coulomb f i e l d of the nucleus. In that approxi-mation,, they could f i n d a r e l a t i v e l y simple closed expression for the sum over the intermediate energy states, using, i n an ingenious way, the properties of the Green's function. However, in order to evaluate the r e s u l t i n g expression f o r the pr o b a b i l i t y /or^ dlK » they s i m p l i f i e d the expression f o r the.matrix element of the electromagnetic i n t e r a c t i o n by neglecting the retardation f a c t o r . On t h i s simplifying assumption they came to the rather s t a r t l i n g conclusion that the (n s 1, JL s 0) - contribution to the photon spectrum i s exactly the same as that found by Morrison and S c h i f f (40), who, as w i l l be r e c a l l e d , completely neglected the Coulomb i n t e r a c t i o n . This conclusion seems, on the other hand, to be i n agreement with the high energy part of the measured spectrum, since t h i s part, as i t was mentioned before, i s well 5 represented by the Morrison-Schiff formula. Glauber and Martin . . also showed (using, the same simplifying assumption) that the observed sudden r i s e i n the gamma spectrum at low energies i s due to the contribution of p-electrons ( i . e . X = 1, n • 2, 3 ...) and that t h i s contribution becomes n e g l i g i b l e at high photon energies. Hess Ts calculations, on the other hand, sta r t from general . formulae i n which the Dirae r e l a t i v i s t i c wave functions are used. However, the actual computation of the expression f o r /\xr^<i-K so obtained, turned out to be impracticable even f o r the simplest case of the Is electron. Consequently, number of simplifying assumptions were made: the sum over the intermediate discrete states was assumed to be small compared with the sum over the continuous states, then a " s e m i - r e l a t i v i s t i c " approximation f o r the . r a d i a l part of the intermediate state wave function was i n t r o -duced, and the subsequent ca l c u l a t i o n was carried out numerically 131 f o r the case of Cs f o r which, at that time, the best experi-mental data were available (Saraf ( 5 4 ) ) . In view of the discrepancy between Hess's r e s u l t s and those of Glauber and Martin, and of the neglecting of the retardation ' e f f e c t s by the l a t t e r authors, i t was f e l t that an evaluation of AAr^d-k » i n which the n o n - r e l a t i v i s t i c Coulomb wave functions are used, but which i s otherwise rigourous , would greatly c l a r i f y the s i t u a t i o n . Such, an evaluation of the contribution of JL = 0 and n = 1 and 2 (the contribution i n case n '^3 i s - n e g l i g i b l e ) , i . e . the contribution that determines completely the photon energy spectrum not very f a r from i t s l i m i t , i s presented i n t h i s thesis 1) I t i s true that the f i n a l part of these c a l c u l a t i o n s has been done numerically. However, no a r b i t r a r y approximation has been introduced i n those numerical c a l c u l a t i o n s . On the other hand, the screening e f f e c t s are e n t i r e l y disregarded, which f o r a l i g h t atom l i k e 3^7 i s probably not too serious. 37 131 1) for the case of A and Cs 13 55 Our calculations settle the disagreement between Glauber and M a r t i n i results and those of Hess essentially in favour of the former authors. The discrepancy i s traced down to a rather well hidden i n -consistency in the argument which leads from the general r e l a t i -v i s t i c formulae to Hess's "semi-relativistic" expressions. This question i s discussed in detail in Chapter II, Section Cf of this thesis, as i t i s of some general interest. Although our results essentially confirm those of Glauber ..' and Martin, the fact that we did not neglect the-retardation < effects has some important quantitative consequences. It turns out that the taking into account of these effects leads to a decrease of the intensity of gamma radiation accompanying the 37 K-capture in A 9 the decrease increasing with the increasing photon energy, as can be seen in Fig. 5, in Chapter III (the corresponding numerical data are summarized in Table III, in Chapter III). 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 their earlier results, and also announce that they have applied 11 a more f u l l y 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 results of this treatment with 37 Lindqvist and Wu's experimental data for A i s ^ i v e n in a recent 1) Only one value of the energy spectrum for Cs i s evaluated, to make a comparison with Hess's result possible. On the other hand, the case of ^ 37, for which we have the most recent and most reliable experimental data, (Lindqvist and Wu (55)) i s considered i n detail. 2) A copy of the paper has kindly been made available by Professor Glauber to Professor Opechowski. n o t e by Wu and a l . ( - 5 6 ) . The h i g h energy p a r t o f the spectrum does not seem to be a f f e c t e d by t h e s e r e f i n e m e n t s i n t h e c a l c u -l a t i o n ; however, i t i s not c l e a r whether th e r e t a r d a t i o n e f f e c t s have been t a k e n i n t o a c c o u n t . The problem o f a d e t a i l e d comparison o f o u r t h e o r e t i c a l r e s u l t s w i t h the e x p e r i m e n t a l d a t a o f L i n d q v i s t and wu ( 5 5 ) i s b r i e f l y d i s c u s s e d i n Chapter I I I , S e c t i o n B. Chapter I General Formalism of the Theory of Electron Capture. In t h i s Chapter, we s h a l l be concerned with the general theory underlying the calculations of Chapter I I I . In Chapter I I , we consider, in more d e t a i l s , the wave functions to be used, and we discuss the passage from the general theory to the approxima-ti o n of Hess (55). In Chapter I I I we present our own calculations and conclusions, and compare them with those of Glauber and Martin (54). The notation that we use i n sketching the general theory of electron capture i s e s s e n t i a l l y that used by De Groot and Tolhoek (50). In Section A of th i s Chapter, we set up the general expres-sion f o r the pr o b a b i l i t y Mtf^ of radiationless K-capture. A more, complete treatment of the subject can be found i n the review a r t i c l e s of Rose (55), Konopinski (55), and Konopinski and Langer (53). In Section B, we consider the case of the rad i a t i v e K-capture pr o b a b i l i t y / U^olK . F i n a l l y , i n Section C, we derive an al t e r n a -t i v e expression f o r the electromagnetic int e r a c t i o n matrix element. This expression i s equivalent 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 - Radiationless K-capture The p r o b a b i l i t y per second of a r a d i a t i o n l e s s K-capture depends on a matrix element describing 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 tate. The i n i t i a l state (represented by the symbol 0) c o n s i s t s of a nucleus of charge Z i n an energy s t a t e VV^ o The f i n a l state (represented by the symbol. F) c o n s i s t s of a nucleus of charge Z - l i n an energy s t a t e \V^_j, of an emitted neutrino of energy ELy> and of a hole i n the K- s h e l l . Therefore, i f E j s i s the energy of the e l e c t r o n i n the K--shell, W-z. t- E . , s = W x _ , -+- E v 1.1 expresses the law of the conservation o f energy. W ^ j-V/ 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 , , On assuming the standard Fermi theory of beta processes, the matrix element which determines the p r o b a b i l i t y o f e l e c t r o n cap-ture can be w r i t t e n i n the f o l l o w i n g form: ( F | H p | o ) = l j ( ^ t . s U C f g ^ T O A V where Hp i s . t h e i n t e r a c t i o n Hamiltonian i s the wave f u n c t i o n f o r the emitted neutrino ^ s i s the wave f u n c t i o n f o r the e l e c t r o n . i n the K - s h e l l "Vjr^ and T|7^ are the i n i t i a l and f i n a l n u c lear s t a t e s r e s p e c t i v e l y , t i s the symbol f o r the a d j o i n t , i . e . the complex conjugate and transpose, of a matrix i \ and -O,^ are the i n t e r a c t i o n operators o p e r a t i n g r e s p e c t i v e l y on the l e p t o n and 10 the nucleon wave functions } means that, the function i n the brackets i s evaluated at the po s i t i o n of the n^*1 nucleon ^dL 1^ means that the. integration i s carr i e d over the volume containing the n nucleon, i . e . the whole nuclear 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 interaction matrix elements to be r e l a -t i v i s t i c a l l y invariant, one can show that there are only f i v e possible choices f o r S\ , and t h i s , on very general assumptions. XL. , i n general, consists of two components, one of which i s "large" or n o n - r e l a t i v i s t i c and the other i s "small" or r e l a t i v i s -t i c (of order v/c where v i s the nucleon v e l o c i t y ) . The "large" and "small" components give r i s e to d i f f e r e n t selection r u l e s . In t h i s thesis, we only consider allowed beta t r a n s i t i o n s . This means, as i s well known, 1 s t , that we neglect the "small" 1) components of the f i v e Fermi interactions, and 2nd, that we assume that the emitted neutrino carries away zero o r b i t a l angular momentum. The large components of the f i v e i n t e r a c t i o n operators are the following: The Scalar i n t e r a c t i o n ^ I.3a The Vector int 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 Ax i a l Vector i n t e r a c t i o n <? I.3d 1) I f one neglects the small components, the r e s u l t i n g expression i s of 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 I . 3 e The Pseudoscalar i n t e r a c t i o n ^fifs " i , ^ i CT and y y are four by four matrices defined as follows: 1 » 1.4a where A , ^ and 1.4b are unit vectors and are matrices defined as follows: / © o o | \ o o I o o 1 o o y 1 o o oj o o © \ o o JL o JL 0 o o / O O j 0 o o o - l / I o o o o -1 o o, I . 4 d The f i v e i n t e r a c t i o n operators thus defined are Hermitian. The pseudoscalar i n t e r a c t i o n -<!.(&^5 i s r e l a t i v i s t i c of order v/c; there i s no "large" component i n t h i s case. Since the operator fl. of the general expression 1.2 may contain an a r b i t r a r y l i n e a r combination of these f i v e i n t e r a c t i o n s , we introduce f i v e corresponding constants 1.5 The subscripts r e f e r to the scalar, vector, tensor, a x i a l vector and pseudoscalar interactions respectively. However, the f i v e interactions 1.3 a c t u a l l y contain nine .A. d i f f e r e n t matrices /\ , and i t i s convenient to introduce the nine c o e f f i c i e n t s C„ defined as follows: A . r B l 2 3 V -5 6 7 .8 9 p 1 fo; h i P<r3 a; 03 i$y5 C s C v C T c T c c A C A C P With the notation 1.6, the i n t e r a c t i o n matrix element 1.3 takes the form: (F) H p I o) = GjL c A £ ( ( ^ V T f , s L A ^ A T * i . 7 Here we have supposed that the in t e r a c t i o n constants are normalized according to C c - r C + C + C + C „ = i S V T rS P ' and we have denoted the i n t e n s i t y f a c t o r (the "Fermi constant") by G. The C'^.'s can always be chosen r e a l . (See B l a t t and Weisskopf (52)). The assumption that the neutrino c a r r i e s no o r b i t a l angular momentum means that the neutrino wave function CO i n the express-ion 1.7 i s characterised by a t o t a l angular momentum quantum number j a \ . One can show that the expression 1.7 i s much smaller when <J^ corresponds to values of j higher than | ("Forbidden" t r a n s i t i o n s ) . 13 F i n a l l y , we make the u s u a l assumption t h a t the l e p t o n p a r t (Cj> A f^jsj/Yt o f t h e F e r m i i n t e r a c t i o n matrix element 1.7 i s a slow l y v a r y i n g f u n c t i o n i n s i d e the nu c l e u s . Hence, t h i s l e p t o n matrix element can be w r i t t e n o u t s i d e the i n t e g r a l s i g n and, a l s o , o u t s i d e the s i g n o f summation over a l l the nucleons i n the ex-p r e s s i o n 1.7. The l e p t o n matrix element i s then u s u a l l y e v a l u a t e d a t a d i s t a n c e R from the o r i g i n , R being the n u c l e a r radius.. One o b t a i n s ( F | H p | 0 ) = &i cWAV ) R ( lA*) where i s a n u c l e a r matrix element t h a t i s assumed d i f f e r e n t from zero only when a proton changes i n t o a neutron. E x p r e s s i o n 1.8 i s t h e r e f o r e the matrix element f o r an allowed 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 such an event i s then given by the f o l l o w i n g formula: r svSctnvso|(F|Hp|o) 1.9 where S v means t h a t we sum over the sp i n s o f the emitt e d n e u t r i n o , ^JXL, t h a t we i n t e g r a t e over the angles o f emi s s i o n of the n e u t r i n o , and S 0 > t n a t w e s u m over the s p i n s o f the i n i t i a l K-e l e c t r o n . 14 The lepton wave functions that we use i n t h i s thesis are normal-ised per unit energy i n t e r v a l ; hence, the neutrino wave function i m p l i c i t e l y contains the density of f i n a l states that otherwise would appear as a factor i n the expression 1.9. • In t h i s thesis, we treat the nucleons and t h e i r wave functions as n o n - r e l a t i v i s t i c . I t follows that f o r the Scalar and Tensor interactions I.3a and I.3b i n the nuclear matrix elements I.8b, one can write • |P = J l and 'a J ? 1.10 One can thus replace >^ by the unit matrix, since, i n our formalism, the +1 components of ^ w i l l connect the "large" non-r e l a t i v i s t i c components of the nucleon wave functions whereas the -1 components of ^ w i l l connect the "small" components, which we are neglecting. B - Radiative K - capture In the case i n which the electron capture i s accompanied by the emission of a gamma photon, one has to consider a l l . t h e intermediate states that the system may occupy, i n order to evaluate the p r o b a b i l i t y AAJ^ OLK . In accordance with the hole theory, two types of processes are possible i n the t r a n s i t i o n from the i n i t i a l to the f i n a l state. The i n i t i a l state of the 15 . system consists of the nucleus Nj^ i n the energy state Wz. ; a l l the negative energy states are occupied. The f i n a l state consists of a nucleus Nz_| i n the energy state W^-j , a hole i n place of the electron characterized by the quantum numbers n and JL , a photon of energy tfck, and a neutrino of energy E. v > a l l the negative energy states are again occupied. The pro-cesses that may take place are of the two following types: I. - The atomic electron (characterized by n and JL ) makes a t r a n s i t i o n to eithe r a) an unoccupied discrete state or b) an unoccupied positive continuum state with the emission of a photon. The electron i s then captured by the- nucleus with the emission of a. neutrino. I I . - An electron i n either a) an occupied discrete state or b) an occupied negative continuum state is. captured • by the nucleus with the emission of a neutrino/.. Another electron jumps into the remaining hole with the emission of a photon The conservation of energy (between the i n i t i a l and the f i n a l , states) i s expressed by the r e l a t i o n \A/ Z 1- E^j^ = W^_f -r- £ v *r *<,k . 1 . 1 1 where E. j_ i s the energy of the electron i n the s h e l l characterized by n. Therefore, W x t- - W ^ , - E y + ^ k i s the energy available to the t r a n s i t i o n . The matrix element entering the expression f o r the probability, AA/t d_K should involve a sum over a l l the intermediate states 16 of type I plus a sum over the intermediate s t a t e s of type I I . However, i t can be shown (see, f o r example, a s i m i l a r proof i n H e i t l e r (50), p. 147) that the two sums can be evaluated together as one sum taken over a l l the intermediate s t a t e s , p o s i t i v e and negative, occupied and not occupied. From the well-known formula o f time-dependent p e r t u r b a t i o n theory, the matrix element i s of the form ^ C F | r y U ) U j H x L g l £ E x - E o 1.12 where F, I and 0 are symbols f o r the f i n a l , intermediate and i n i t i a l s t a t e s r e s p e c t i v e l y . ( I | Hy ] 0) i s the electromagnetic i n t e r a c t i o n matrix element and (F ( J ~ | ^ j l ) i s the same beta i n t e r -a c t i o n matrix element as the one used i n the r a d i a t i o n l e s s case E-<^ . I . \0 T except that the l s - e l e c t r o n wave f u n c t i o n i s replaced by the intermediate 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 al s o an s - f u n c t i o n because we suppose the t r a n s i t i o n to be allowed) E l j and E- 0 are the energy values corresponding to the i n t e r -mediate and the i n i t i a l s t a t e s . In 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 of E, the energy of the e l e c t r o n i n the intermediate s t a t e , we have E l x - E 0 = E ~ E„JL 1- Ack i . i 3 The expression f o r w kdk, s i m i l a r to th a t f o r wc, Eq. I.10, i s as f o l l o w s : **** f ••5& s rf A r V S 'K , ->. 1.14 where Sy- i n d i c a t e s summation over the two d i r e c t i o n s of p o l a r i z a t i o n of the photon 17 S v and S Q i n d i c a t e s summation over the spins o f the neutrino and those of the i n i t i a l e l e c t r o n r e s p e c t i v e l y $o\fy and . ^ d L f X v mean, t h a t we i n t e g r a t e over the angles of emission of the photon and of the neutrino r e s p e c t i v e l y i s the photon c o n t r i b u t i o n to t h e dens i t y of f i n a l s t a t e s . The wave fun c t i o n s are supposed to be normalized per. u n i t energy i n t e r v a l . The matrix elements i n 1.14 are given by: (F I H^li) = GlLc^A^iM) 1.15 (the symbols have the same meaning as i n Sec t i o n A) and by (See H e i t l e r (50), p.95) (n.Hr|o) = e Jr^ -?K ' I > 1 6 where ^ = e. ( JIT**. \ — JL , the r e t a r d a t i o n f a c t o r Q —^ ji^ i s the p o l a r i z a t i o n v e c t o r of the photon whose wave v e c t o r i s K } |rt] — K oT i s the Dirac matrix defined i n 1.4 JL i s the charge of the e l e c t r o n YB and are the wave' f u n c t i o n s f o r the e l e c t r o n i n the intermediate and the 18 i n i t i a l s t a t e s r e s p e c t i v e l y , v"^" means the Hermitian 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 another expression f o r • • • » the electromagnetic i n t e r a c t i o n matrix element 1.16, the u s e f u l -ness of which w i l l be apparent i n the next Chapter. C - The M a t r i x Element of Electromagnetic I n t e r a c t i o n In t h i s S e c t i o n , we consider the general case of an e l e c t r o -magnetic t r a n s i t i o n between a s t a t e of energy E described by the e l e c t r o n wave f u n c t i o n vL»^ and a s t a t e of energy E^ " described by ^E 1 • and are Dirac 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 of an electromagnetic 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 matrix element of the electromagnetic i n t e r a c t i o n i n t r o -duced i n Section B, Eq. 1.16 i s then a s p e c i a l case of the f o l l o w -i n g matrix element: M = C E | « r / e * ) = C •Jf/flf-^hQvfE'*3? 1.17 We are now going to put t h i s matrix 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 introduced by Gordon (28) to decompose the Dirac p r o b a b i l i t y current 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 to use a covariant tensor notation (in t h i s , we more or less follow Pauli (33) and Somraerfeld in t h e i r presentation of Gordon's method). We f i r s t introduce the contravariant four-vector 0 , 1 , 2 , 3 i . i d of which the'components are chosen to be -1. a . . .2. y - 0 , y '•= P * i , y =• £ < * a , y J = (3<*3 1.19 ^ } OC.^ ^ 0 ^ a " d 0C.3 a r e defined i n Eq. 1.4. The components y ^ y ^ a n d y 3 are thus anti-Hermitian: - y" K , K~ i , 2 , 3 . The y/*" ' s thus chosen obey the following commutation r u l e : 1.20 where 0/ i s the well-known contravariant metric tensor: 1.21 and where I i s the unit matrix. The covariant i s defined such that where i s the Kronecker tensor. Therefore, 1.22 /1 o o o 0-1 0 0 Q O -( O \ 0 o .0 -|> 1.23 20 The p o s i t i o n f o u r - v e c t o r i s defined as OC^ - 1.24 such that the energy - momentum f o u r - v e c t o r i s r e l a t e d to the operator It a. - ^ V « . = Therefore, from 1.23 1.26 by the r u l e 3 V ^ = Z j r f> 1.27 1.28 The electromagnetic f i e l d can a l s o be expressed as a f o u r - v e c t o r A^= ( q > X ) J - 2 ' The covariant f o u r - v e c t o r A^c i s given by In the n o t a t i o n introduced above, the equation s a t i s f i e d -by and vp£, i s the f o l l o w i n g I n t r o d u c i n g the bar f u n c t i o n = T » w e s e e t h a t yV£ and ^ j , ! s a t i s f y the a d j o i n t equation %0{h.^t 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 and 1.31b for a n c i y E respectively, one obtains 1.32a 1.32b If we write for simplicity = JL and introduce the component JLQ ~ 0 , the matrix element M, Eq. 1.17, can be written in the form We now replace Vj^t and ^ i n I«33 each in turn by the expression I.32a and I.32b respectively, which yields: M - " ztifc^T^V&V)*-* 1.3*. and Writing again = ^ " " v f ^ » adding the two expressions 1.34 and dividing the result by 2, we obtain for M the following ex-pression: 22 where we have separated the f i e l d - dependent terms from the others. Using the fa c t that "f^Y* t Y^= 2 ^ w e separate the terms i n which y »yU. from those in which V ^ y u . to obtain 1.36 We now use the r e l a t i o n s JHe - O f — a n d Ay^ . = ""A for f<+ O . (See Egs. 1.21 and 1.30). Also we note that when V f ^ - , ^ y ^ / v * - ^ c A / V : , r r ^ v , where CT^ V i s the Dirac spin operator. We also separate the ^ s Q term from the terms i n which and v> are d i f f e r e n t from zero i n the l a s t l i n e of 1.36. In t h i s way, 1.36 becomes the sum of four terms: where 1.37 1.37a M J L - $ X Q. Y E Y E > I.37b 23 3 3 Je,m--i 1.37c We now make the assumption that one of the energy l e v e l s E, E"1" ( l e t us say, E ) i s a discrete l e v e l . Then the function vpe* vanishes at i n f i n i t y and the f i r s t term of M , Eq. 1.37a. can be integrated by parts to y i e l d : ~ - ( Ve. Q * V Y E ! since w = _^ . The l a s t term of I .3S i s zero since the vectors JL and are perpendicular to each other. Therefore, becomes M, * eA (rE 3 v «** " 1.39 The integrand of Eq. I.37c can be expressed as a c u r l : writing B * ^ for the quantity ^ V ^ g * , ( fc"** ^ - B*^, M3 becomes 1.40 where P = Y E °" Ye' » a n d °" n a s t h e components cr, - fl" , (Ti. = c r 3 1 , c r 3 = c r , a - . 1.41 Rearranging and integrating 1.40 by parts, one obtains M a = 1 = ^ ( ^ x B ) • 1.42 24 since and consequently B become zero on the i n f i n i t e bounding surface. Since *?C^ = - A R Q 1.42 becomes 1.43 In 1.43 we have denoted the vector by AZ . The expression i E<1« I.37d, can be written as follows since' the time-dependence of y = and M ' I i s ^iven by the factors X and A * respectively. From Eqs. 1.37, 1.39, 1*43 and 1.44, the expression f o r M equivalent to Eq. 1.17 i s the following: -r 1.45 -t —* K where >u. = -2-^ X For the case of an electron i n Coulomb f i e l d (A" ^ ), 1.45 becomes 25 I f we compare the 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 matrix element of S e c t i o n B, Eq. 1 . 1 6 w i t h the e x p r e s s i o n j u s t g i v e n , we can conclude t h a t the o p e r a t o r Z.^ i n the matrix element may be r e p l a c e d by the o p e r a t o r 1 . 4 7 i n t h i s sense, these two o p e r a t o r s are e q u i v a l e n t . In our general e x p r e s s i o n o f S e c t i o n B, Eq. 1 . 4 2 , we thus r e p l a c e 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 matrix element o f Eq. 1 . 1 6 by the e x p r e s s i o n 1 . 4 6 w i t h \^^\ r e p l a c e d by and r e p l a c e d 26 Chapter II The Expression f o r the Pr o b a b i l i t y of Electron Capture in Terms of Approximate Wave Functions In Section A of this Chapter, we introduce the wave functions used i n our calculations and we give the general r e s u l t i n g ex-pression f o r f*rk^//u*^ t when these wave functions are s u b s t i -tuted i n the expressions 1.9 f o r fuJ^ and 1.14 f o r AAJ^dLK • In Section B, t h i s general expression i s s i m p l i f i e d by introducing the n o n - r e l a t i v i s t i c approximation f o r the electron wave functions), and put into the form which w i l l serve as a s t a r t i n g point i n our calculations of Chapter I I I . In Section C, we discuss the passage from the general expression of Section A to that used by Hess. 27 A - Wave Functions and the General E x p r e s s i o n f o r A x r ^ d K / ^ We s t a r t by c o n s i d e r i n g the s p h e r i c a l wave s o l u t i o n t o the D i r a c equation f o r a p a r t i c l e i n a Coulomb f i e l d . These wave f u n c t i o n s are giv e n , f o r i n s t a n c e , i n Rose (37) o r i n Hess (55). They are normalized per u n i t energy i n t e r v a l and are c h a r a c t e r -i z e d by the f o l l o w i n g quantum numbers: j , the t o t a l a n g u l a r momentum quantum number, which takes h a l f - i n t e g r a l v a l u e s ; jx , which takes a l l h a l f - i n t e g r a l v a l u e s from - j t o +j; and K, which takes the va l u e s i ( j +• J ) . Since i n t h i s t h e s i s we o n l y need the wave f u n c t i o n s f o r which j = * (allowed case: no o r b i t a l a n g u l a r momentum i s c a r r i e d away by the p a r t i c l e s ) , we s h a l l not w r i t e the wave f u n c t i o n s i n t h e i r g e n e r a l form. For j : J, the wave f u n c t i o n s a r e : '« A. o v3 li "AT / 03^ IIo The f u n c t i o n s F G and G~ are the r a d i a l s o l u t i o n s to the Dir a c wave equation: we s h a l l not w r i t e them e x p l i c i t l y i n t h i s t h e s i s . The \i 's are the s p h e r i c a l harmonics d e f i n e d f o r i n s t a n c e , i n B l a t t and Weisskopf (52), p. 7#3. Since we c o n s i d e r o n l y allowed t r a n s i t i o n s , we s e t equal t o zero the components o f the wave f u n c t i o n I I.1 which c o n t a i n a s p h e r i c a l harmonics d i f f e r e n t from J 0 . We have t h e r e f o r e , i n t h a t approximation, 26* r - i K---1 1 o \ / • .\ / o \ 0 0 0 vf ' ° A. o w 1 0' j V° * / II. 2 If one sets Z, the nuclear charge, and m, the electron mass, equal to zero in II.2, one obtains the neutrino wave functions (S) which w i l l be used in the expressions 1.8a and 1.45. I f and the radial parts of the non zero corn-one denotes by ^ * ponents of these wave functions, evaluated at the nuclear sur-face, one easily shows that JL II.3 where Ey is the energy of the emitted neutrino. In the case of the Is electron, the two functions with K=*[in II.2 are identically zero. In the case of s electrons in higher discrete levels, these two functions no longer vanish but are small and w i l l be neglected in our calculations. The two remaining wave functions are H<'- */* I Y ° 5SA \ o 0 \ 0 II.4 With the help of II.4 (with n s 1), and of II.3, one readily evaluates the expression 1.9 for AJLT^ . The absolute square of 29 the matrix element i n 1.9 i s ( F| o) ' G z S% ^ ( ) ( J / f f ( V tis t i l A"' ^ ) R I 2 . 5 (see Eq. I.£). Operating with S G on the product ^ ^ i l y i e l d s , because of II.4, I±j3 a. I I . 6 where ^ i s defined i n Io4. I t i s also e a s i l y seen that the sum Sy over the four neutrino states y i e l d s where the operator Tr means that we take the trace of the matrix on the right of i t . In deriving II.7, we used II.2, II.3 and the fact that E v = W, the available energy (See E q . I . l ) . Thus, from II.5, 6 and 7, « S , S o | ( F l H . | 0 ) | . = Yo ^ 3 7 3 J G * x c ^ , ( t f X S / O V * A A ' N - * The matrices of which we have to evaluate the traces, i n Eq.II.o*, are of the form A^A* and A ^ A * , where the A ^ . a s defined In . 1 . 6 , are products of the Dirac matrices p**0^ and 0^3 . " Such traces of products of Dirac matrices are given, for instance, i n H e i t l e r (50), p.87. When they are evaluated i n II.6*, the 30 double sum over r and r * becomes S S c e o ( V f l C S t f T T\ A* A * V = a . T Remembering t h a t the o p e r a t o r m u l t i p l i e s the e x p r e s s i o n 1.9 by a f a c t o r 4 TT and t h a t Yo s V j ^ T T » w e o b t a i n from 1.9, I I . 8 and I I . 9 : G T 11.10 II.1.0 was used both by Hess and by Glauber and M a r t i n i n t h e i r c a l c u l a t i o n s . We now c o n s i d e r the e x p r e s s i o n f o r /Wj^JLK • ^e f i r s t w r i t e the sum over the in t e r m e d i a t e energy s t a t e s , Eq. 1.12, i n a more d e t a i l e d form, u s i n g 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 matrix element obtained i n S e c t i o n C, Chapter I . S i n c e the wave f u n c -t i o n s i n the in t e r m e d i a t e and i n i t i a l s t a t e s are s - f u n c t i o n s and, mt > -•» T7 consequently, have the same p a r i t y , the o p e r a t o r - *~ ^ — i n the e x p r e s s i o n 1.45 f o r ( F | H f ] l ) w i l l not c o n t r i b u t e to the e l e c t r o -magnetic i n t e r a c t i o n matrix element. I t f o l l o w s t h e r e f o r e , from 1.12, 1.13, 1.15 and 1.45 t h a t 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 ^ ) «ffCR)A*$tf i fe«>1&*>S*^9lVs 1 1 , 1 3 8 where M * s < £ £ o K II..13b We show in Appendix A that the second line of II.13a may be neglected. We obtain, f i n a l l y r Ej.-^ * 1 1 •14 We now form the absolute square of the expression 11.14 and the operator S. on the WJ Because of II.4 and II.6, we obtain apply he e wave f u n c t i o n product *tms4/»iS 15a where 32 I I . 15b and where the prime on h^means that we replace A 1 £ by A1, £* i n 11.13. I t i s shown i n Appendix B that P - 1 - £i JL 11.16 Because of the form of the wave fu n c t i o n s , as given by I I . 2 , we may w r i t e Eq. II.13b as f o l l o w s : We i n t e g r a t e over the angles, u s i n g the r e l a t i o n ( a n ^ = 4 -T ^ L 1 ^ i i.18 and introduce the n o t a t i o n . A » w h A , / ^ V S C A ) j < ft A - K A . A . ' * * E . £ - e m s +• #cfc II.19a 33 UL. = ^ ~ g )Q A . K / L * I I . 19b 11.15 becomes K J 11.20 The summation over the four neutrino states gives, as i n Eq.II.7, the trace of the matrix contained between the two 's of Eq. 11.20. There also ar i s e s , because of II.2 and 3, a factor 8^ *^ .3 11.21* in front of the whole expression. In deriving 11.21, we used the equations of energy conservation 1.1 and I.11 and the def-i n i t i o n 1.1 f o r W. Therefore, S o $ v — Q v ° ( W - f t c k i - Ems" E»w = BY. 0 ( w E a a i E a l ^ c ^ ( t f W /V A' 11.22 34 I t can be seen from the d e f i n i t i o n s 1.6 t h a t the t r a c e s o f every matrix of the form A'VOo^ and A^/ffi^L U = 1 , 2, 3.) are zero. Thus, the term of 11.22 w i l l not c o n t r i b u t e to the t r a c e . Therefore, one has only to evaluate the t r a c e of A l s t o n C J X ^ K 11.23 i n the way explained under Eq. 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 ^ x w i t h T1 and T 2 given by T , ( Cs * t ttf tCc?•> ^ ) | S?|V S*l Sfr/ff 11.25. and T a - ^ S ^ I S l f '+ II.25b such that T, *T* = T , defined i n Eq. I I . 9 - From 11.12, 11.24 and 1.14, we have f i n a l l y i . "3 x i . [ ( l L ; s i % \ C f ) T t t ciu: j a- i £ s r ) T j 35 and, from 11.10 and 11.26 • N3 A A / ; TT (W*)* . V W /. 11.27' I R The above e x p r e s s i o n f o r ^ ^ / u r w i l l be the b a s i s f o r the subsequent d i s c u s s i o n s o f t h i s Chapter* B - Non - R e l a t i v i s t i c E x p r e s s i o n f o r / U T f c c l k / ^ When Z i s small,, the e x p r e s s i o n f o r the sum over the i n t e r -mediate . energy s t a t e s , as given by 11.17, may be approximated by i t s n o n - r e l a t i v i s t i c form. To t h i s e f f e c t , 1 ° ^ i s put equal t o a u n i t matrix i n 11.17, so t h a t the f u n c t i o n G dis a p p e a r s completely from the e x p r e s s i o n ; 2° The f u n c t i o n i s r e p l a c e d by CJp£ the c o r r e s -ponding s o l u t i o n t o Schroedinger*s equation; 3 The sum over a l l energy s t a t e s IE. becomes a sum over 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 p o s i t i v e continuous s t a t e s . When the above approximation i s made, the e x p r e s s i o n 11.23 36 becomes E . 2 8 and the t r a c e of 11.28 gives r i s e only t o the f a c t o r T/ defined i n Eq. II.25a. In t h i s approximation, we have, from 11.27 i a. I f we make use of the f a i r l y w e l l e s t a b l i s h e d experimental f a c t (See, f o r i n s t a n c e , Konopinski and Langer (53)) t h a t the mixed terms ( i . e . those p r o p o r t i o n a l to C & C v and Cj. ) are n e g l i g i b l e , we may set a 0; and thus TJ - T i n 11.29. We al s o replace the wave f u n c t i o n f o r the i n i t i a l e l e c t r o n by Rs<fO R the corresponding s o l u t i o n to Schroedinger*s equation ^ C R ) Hence, 11.29 becomes: <VJi where U =s ^ ' " - . Expression 11.30 i s the s t a r t i n g point of our c a l c u l a t i o n s o f Chapter I I I . C - The Approximation of Hess 37 Instead of using the approximate electromagnetic i n t e r a c t i o n matrix element obtained i n Chapter I , Section C, which leads to Eq. 11.14, Hess used the general r e l a t i v i s t i c expression I.16 wi t h the operator o c * ^ ° This means that the matrix P of I I . 15b was, i n h i s case, or P = IdS * H . 3 1 since 5?f* « - £c? and (o?. I On the other 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 of the form I I . 2 , the sum M*\s o v e r i the intermediate energy s t a t e s takes the form I I . 17. Since » 0, i t f o l l o w s from 11.17 and 11.31 that the f a c t o r ^ /vvsPM^ of II-15a becomes 11.32 i . e . t h a t the f u n c t i o n F. disappears a l t o g e t h e r from the sura over the intermediate energy s t a t e s . Since the f u n c t i o n s G f o r the d i s c r e t e s t a t e s are small, Hess was l e d t o neglect the sum over the intermediate d i s c r e t e s t a t e s as compared w i t h that over the continuous s t a t e s ; the r e s u l t obtained was of an order o f magnitude smaller than that of Morrison and S c h i f f . (See Chapter I I I ) 3* The source of t h i s d i f f i c u l t y resides i n the fact that • making the approximation II.2 f o r the wave functions and, at the same time, keeping the operator o T * i n i t s general form i n the matrix element of electromagnetic in t e r a c t i o n , i s inconsistent. Chapter I I I R a d i a t i v e K-Capture i n a Non - R e l a t i v i s t i c Approximation In S e c t i o n A of t h i s Chapter, we b r i e f l y d e s c r i b e the method employed by Glauber and M a r t i n ( 5 5 ) ( 5 6 ) , i n t h e i r d e r i v a t i o n o f the e x p r e s s i o n 11.30 f o r /WT^cU<Jtoj^ , and we d i s c u s s the s i m p l i f y i n g assumptions they make i n e v a l u a t i n g 11.30, and i n S e c t i o n B, we present our own e v a l u a t i o n o f 37 131 t h a t e x p r e s s i o n f o r the cases o f A and Cs 1 3 55 40 A - Method of Glauber and Martin Let us consider the general Jbrm of the matrix element 1.12, using the expressions 1.15 and 1.16 f o r the beta i n t e r -action matrix element and the electromagnetic i n t e r a c t i o n matrix element respectively: I E r - E D \ « / r. • X fa M „ < 5 t - ? h III. la W H E R E M = £ M r t ^ i ^ L i n ,ib As pointed out by Glauber and Martin, i s a c t u a l l y a s p e c i a l case of the Green's function G- (V, A ! ' for the wave equa-tion of the electron i n a Coulomb f i e l d , oince we may set the nuclear radius R equal to zero i n I I I . l a , we have The reason f o r the success of t h i s approach i s that t h i s one-argument Green's function in i t s n o n - r e l a t i v i s t i c form can be obtained as a solution of the Schroedinger equation f o r an electron i n a Coulomb f i e l d . However, Glauber and Martin 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. Only in a later stage of their calculations do they replace $ by the non-relativistic Green's function, and in this way obtain an equation which reduces to our Eq. XI.30 in case of s-electrons. Expression 11.30 for cannot be evaluated in a closed form. In order to evaluate /V,*^ 0 K^//UJ^ in a relatively simple analytical way, Glauber and Martin have set the retardation factor JL equal to one. This i s equivalent to using an expression for L t in which ^VV-fc/V i s replaced by unity for 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 analytical expression for the non-relativistic Green's function, they could evaluate A*J}^ ^ ^/f^ almost without numerical calculations. The argument they present in order to justify' the above approximation (i.e. JL ~ JL ) over the whole energy range is rather lengthy and somewhat unconvincing; i t w i l l only be sketched in this thesis. The argument is of a different nature for each of the following three photon energy ranges: (1) JUk < x&-«f (2) ^V<^<..K < Z ^ / n v c * " 1 . 3 (3) Z ^ < ^ c K In the range (1) of low photon energies, the photons have a wave length at least (Z^O * times' larger than the atomic system. In this case, one may therefore set ^^ /AKA in 11.30. 42 Consequently, o n l y one term i n the sum and i n t e g r a l over i n t e r -mediate s t a t e s i s d i f f e r e n t from zero because of the o r t h o -normal p r o p e r t i e s o f the *s, namely the term f o r which E-£^ S. Then becomes, simply * * SUA and Eq. II.3D becomes i d e n t i c a l w i t h the Morrison and S c h i f f r a t i o when n = 1 and # =0. ( T h i s i s a r a t h e r s t a r t l i n g consequence, i n view of the f a c t t h a t Morrison and S c h i f f completely n e g l e c t e d Coulomb e f f e c t s , while Eq. III.4 i s d e r i v e d from an e x p r e s s i o n c o n t a i n i n g ( 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 ) . In the intermediate energy range (2) one may not a p r i o r i n e g l e c t the r e t a r d a t i o n e f f e c t s . However, a study o f the form of the a n a l y t i c a l e x p r e s s i o n f o r the Green's f u n c t i o n )j «( 0i / L) > which c o n t a i n s a d e c r e a s i n g e x p o n e n t i a l , shows t h a t i t s range remains much sm a l l e r than the photon wave l e n g t h so t h a t the r e t a r d a t i o n e f f e c t s seem again to be unimportant, i . e . a g a i n O, ' cs f . I t f o l l o w s t h a t i n t h i s range a l s o , one o b t a i n s the e x p r e s s i o n o f Morrison and S c h i f f when n = 1 and L » 0 o In the high energy range ( 3 ) the r e t a r d a t i o n must-be taken i n t o account. However, i n t h a t range, Glauber and M a r t i n used the f r e e p a r t i c l e form o f the Green's f u n c t i o n and approximated the i n i t i a l wave f u n c t i o n by a constant. These approximations are the same as the one used by Morrison and S c h i f f i n d e r i v i n g t h e i r r e s u l t : namely, the n e g l e c t i o n of a l l Coulomb e f f e c t s i n the i n t e r m e d i a t e s t a t e s wave f u n c t i o n s and the assumption t h a t the i n i t i a l e l e c t r o n may be considered a t r e s t . For the high energy r e g i o n , Glauber and M a r t i n used t h e r e f o r e the same 43 expression as that used by Morrison and Schiff. The neglecting of a l l retardation effects allowed Glauber and Martin to evaluate the contribution to .Atfj^ oOl made by the p-electrons of the L and M shells (n = 1, 2; L = 1) in-a rela-tively simple way. This contribution was shown to explain the sudden rise of the photon spectrum at low energies (Glauber and Martin (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 is seen to apply only to the low and intermediate energy ranges of the photon spectra, as defined in III.3 and. i t is 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, for the case of A , by Lindqvist and wu (55) and there appears to be an essential agreement be-tween theory and experiment. As stated in the Introduction, a "more f u l l y r e l a t i v i s t i c " calculation carried out by Glauber and Martin has not yet been published in details, but the results have been compared with 37 the experimental data for A (See Wu and a l . (56)). From this comparison, i t would appear that the correction resulting from these latest calculations does not affect the high energy part . of the theoretical spectrum. Although there can be l i t t l e doubt that Glauber and r-iartin's results are essentially correct, we think that a direct, non-r e l a t i v i s t i c calculation of /vAJ^^H^j^ free of additional-simplifying assumptions i s s t i l l of some value. Such a calcula-tion, which we undertake in Section B of this Chapter for the 44 case o f I s and 2s e l e c t r o n s , makes i t , i n p a r t i c u l a r , p o s s i b l e to see more c l e a r l y the r o l e played by the r e t a r d a t i o n e f f e c t s . B - E v a l u a t i o n o f '^^/AJS^ i n a N o n - R e l a t i v i s t i c Approximation f o r the Case o f I s and 2s E l e c t r o n s . In e v a l u a t i n g d i r e c t l y , (Eq. 11..30) one must c o n s i d e r s e p a r a t e l y the d i s c r e t e and the continuous i n t e r m e d i a t e energy s t a t e s . The procedure adopted i s to f i r s t c a r r y out the i n t e -g r a t i o n over the space v a r i a b l e r . The exact formulae f o r the f i r s t f o u r terms of the sum over the i n t e r m e d i a t e d i s c r e t e energy s t a t e s are o b t a i n e d and an approximate e x p r e s s i o n i s d e r i v e d - f o r the remainder o f the sum, which can be transformed into, an i n -t e g r a l . The i n t e g r a l over the continuous energy s t a t e s i s evaluated n u m e r i c a l l y between the l i m i t s equal to mc and about 2 1.5 mc r e s p e c t i v e l y . The i n t e g r a n d f o r h i g h e r e n e r g i e s can be approximated by a r e l a t i v e l y simple a n a l y t i c e x p r e s s i o n and the i n t e g r a t i o n c a r r i e d out a n a l y t i c a l l y . The e r r o r i n v o l v e d i n t h a t procedure i s q u i t e s m a l l , as w i l l be apparent from the numerical r e s u l t s . o 1 D i s c r e t e S t a t e s The space dependence of the g e n e r a l n o n - r e l a t i v i s t i c 45 > "s-function" describing a particle in a discrete energy level E ^ — E ^ i s given by (see, for instance, Kramers (3^), P.311) where a. - goSi™*-/ * and where £$(|_,yv.« ^ . & M A \ i s the confluent hyper-geometric function as defined, for instance, i n MacRobert ( 5 4 ) , p.3 4 6 . The function I I I . 5 i s normalized such that I I I . 6 When r s R, the nuclear radius, Eq. III.5 becomes < ? ^ ( R ) = * ( f t ) V ** •, I I I . 7 - 3 - '3 since ^ = Z^rrr^. $e IO for R |0 «wv III.8 r * ( * ) ' and since i s a f i n i t e polynomial of the form i +• f ( l ? ) i i ^ ^ f ^ i ^ a i r + . i n . 9 one may write approximately III.10 46 We c a l l I the part of I— v;hich contains the sum over the intermediate discrete states. The i n i t i a l states Is . and 2s that we s h a l l consider are represented by the wave func-tions (see III.5 , H I - 9 ) : <PlS = °?e, = "V «• III.Ha Replacing and cp^ £ i n II.30 by t h e i r values III.5 , I I I . 10 and III.11. one obtains f o r the two L n , n=l ,2: 9 0 00 I q 72, < tV\ " w " > ~ > n» / / \ OA-L j 0 l e = * ^ < . -i c r , \ III.12a | = _ C _ _ S - ___ . — III.12b where <JJ i n 12b i s the same hypergebinetric function as i n 12a. The energy i s given by the expression I f one approximates by the expression III.14 47 one obtains and As we have already mentioned, we start by integrating over the space variable r in Eqs. III.12. To t h i s e f f e c t we expand the hypergeometric functions into t h e i r ( f i n i t e ) polynomial form of equation III.9 and integrate term by term. The space i n t e g r a l of III.12a, f o r instance, 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 expression i n the square bracket of III.17 i s the ordinary hypergeometric series of argument ? ^ ' (See Magnus and Ober-hettinger, p.7). Hence the i n t e g r a l III.16 can be written: \ \ r ^ in which z stands for 48 ATVA t -i— — III.19 where ^ - - - * ^ — III.20 We next use the r e l a t i o n : (See Magnus and Oberhettinger p.8) J r Therefore jS ( | - 3 L * > 3LyZ.)' ~ O - ^ ^ " " 1 111.22 With the help of III.22, III.18, III.20 and III.15a, the expression I I I . 12a becomes ^Dls m*i *>ts * / c +*} HI.23a In case of L i p ?the second term i n the integrand i n Eq. III.12b leads to the hypergeometric function t h i s l a s t formula i s e a s i l y obtained using again III.21. A c a l c u l a t i o n s i m i l a r to that above, y i e l d s then f o r L i p (Eq.III.12b) S s £ *» A- M S < ^ * L i * ( V i),**l m . 2 3 b In III.23b, however, z i s In evaluating L D^ and we calculate the f i r s t four terms of the sum over n separately, and we derive an approximate ex-pression f o r the summand when /v\ ^ 5" •. The numerical, values that we obtain l a t e r w i l l show that t h i s approximation i n t r o -duces an error of les s than 1$ i n the expressions f o r L - p J S and L-O^s • We f i r s t consider the case of LQ,s and then the case of L'Da.s a ) Case of ^ O i c Let us consider, i n the.summand of Eq. III.23a the factor u = [ *v*r~l-'c-<t-] iii,24 and write z i n the form Z. = - Z. = 2& = f * * ^ 1 " ' III.25a III.25b where ^ = [( | t ±f + III.25c md cDt - t o ^ T 1 — ^ ~ III . 2 5 d 50 I f III.2 5 a and b are inserted i n I I I . 2 4 , one obtains 8« = MlrJVf- ^ - <*-»<?-] .. m . 2 6 . Therefore, from III.13a The f i r s t four terms of the sum III.2 7 are given.below CO 1) L. = ^ — r - ^ ~ , . ' III .23 by using the d e f i n i t i o n s of (p t and ( J ^ , JEqs. III.25c and d. When the retardation factor i s put equal to one i n the expression III.12a, only the term n = 1 i n the sum over n contributes to the value of L»n (because of the orthonormality of the wave functions used i n the integrand). » * n that case, i s given by III.28 i n which the x i n the factor rk + ^ t) i s put equal, to zero. Therefore: °»S U ^ JfaA I I I . 28a and one obtains the Morrison and S c h i f f expression for A/jrk^/fijj' 1) There are an i n f i n i t e number of choices for t h e values of \ CO4. , Eq. I I I . 2 5 d . However, i f one writes < t^=c£++/«gf|,where m - ... - 2 , - 1 , 0 , 1, 2 , ... and O £><f±^i & , one sees that the expression I I I . remains the same f o r any choice of m. S i m i l a r l y 51 2) III.2 9 3) L. (?) III.3 0 The terms f o r 5 may be approximated i n the f o l l o w i n g way. Let us c o n s i d e r the e x p r e s s i o n , Eq. III.26 and expand i t i n powers o f . "tie n e g l e c t a l l terms o f powers* equal t o and s m a l l e r than /nT 3 " i n the expansion, tie c o n s i d e r f i r s t t h e argument of the sine f u n c t i o n i n III.2 6 ; we have * } III.3 2 From r i l . 2 5 d , and from the r u l e f o r the tangents o f sums and d i f f e r e n c e s o f angles, we o b t a i n 52 I +• ^Vi-U III.33a 1 ~ '«F and + <f- = W ' [ • _ J ^ 1 _ _ - ] III.33b I f one neglects the terms in i n III.33, one obtains _^2L_ ^ X W ' o C III.34 In deriving III.34, we used the series expansion f o r the function tan with ( J ^CL^^j <^ | ) and neglected the powers equal to or smaller than /rv . We approximate ^ + and i n the same manner; from III .25d, III.35 Therefore, from III. 3 5 53 { V ? - - 0 + **) .. III . 3 6 and III . 3 7 To derive I I I . 3 7 , we made use of the r e l a t i o n k ^ t o . / ^ With the help of I I I . 3 5 , I I I . 3 6 , . I l l . 3 4 and I I I . 2 7 , one may write as an approximate expression f o r when n i s large: I t follows therefore that We s h a l l evaluate L , Eq. III.27, as follows with the terms i n the sum given respectively by III.23a, 29, 3 0 , 3 1 and 4 0 . 54 b) Case of U The case of Lig^is treated like the one of L I D I S . . We consider the expression III.23b, for k^c, and write 42. where z i s given by III.23c. One can show that in analogy with Eq. III.26, with and III.43b < ? t = P^'l[ 111.43c It follows from III.43a and III.23b that The f i r s t four terms of the sum III.44 are the following: 55 and f o r ^ 5" In III.45e we have neglected the powers equal to or smaller than when compared to unity. From III .45e, i t follows that L I Q ^ ^ w i l l be evaluated as follows where the terms are given by Eqs. III.45 and III.46. oo 56 2° Continuous S t a t e s To evaluate the c o n t r i b u t i o n to L^ s and L i ^ , Eq. 11.30, i n v o l v i n g the i n t e g r a t i o n over the continuous energy s t a t e s , one must c o n s i d e r the f o l l o w i n g i n t e g r a l E - E m +f t<* 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 d e s c r i b i n g an e l e c t r o n i n the continuum i s the f o l l o w i n g (See Sommerfeld p.115 & f f . ) IT 1.49 T h i s wave f u n c t i o n , which i s r e a l , i s normalized per u n i t energy i n t e r v a l : t h i s means t h a t ) ^ ^ i C A ) tf EC/v) = 1 . in.50 1 where i s any energy i n t e r v a l c o n t a i n i n g E . The energy E i s r e l a t e d t o the momentum p o f III.49 through the r e l a t i o n E = [cytfr*?)*]** o r * * * £ ' - I I I w 5 1 57 When r = R, one cannot set equal to one the f a c t o r IIIo52 as r e a d i l y as i n the case of LQ^ (See Eqs. III.7 to III.10) beeause, i n III.48, p becomes i n f i n i t e at the upper l i m i t of integration. This f a c t o r i s shortly considered i n Appendix C where we conclude that setting i t equal to unity does not a f f e c t the value of III.48. Therefore, we may write, when r « R The wave functions C^ s and ^ s for the electron i n the i n i t i a l state are given by Eqs. III.11a and III.lib respectively. We consider f i r s t the case of L C | s and then that of a) Evaluation of Lie.. With the use of 111.49, III.53 and"III.11a, III.43 becomes ° U > o - E L - E , - r f c o * III.54 where i s given by Eq. III.13- with n • 1. = nrrve X ~ I - j III o 5 .5 To evaluate III.54-* w e f i r s t express the confluent hypergeometric function i n an i n t e g r a l form (See, f o r instance MacRobert (54) p. 346). 53 $(WJLr&\'U&) - — r ( ^ ^ ^ l - t P ^ t 111.56 Then we interchange the o r d e r o f the i n t e g r a t i o n s over r and t i n III.54; t h i s i s allowed because o f the presence o f the de-c r e a s i n g e x p o n e n t i a l f a c t o r JL i n the i n t e g r a n d . I n t e g r a t i n g over r f i r s t , and then over t , one o b t a i n s U = t r f r * * (~ dLzrfrir'i+itit v • . • m . j 7 111.56 i s the hypergeometric f u n c t i o n and Z± - I I I . 5 9 One a r r i v e s a t Sqs. III.57, 5& and 59 by making use o f the i n t e g r a l r e p r e s e n t a t i o n o f the hypergeometric f u n c t i o n (See MacRobert (54), p.297): III.60 From III.21, i t f o l l o w s t h a t For computational purposes, i t i s convenient to i n t r o d u c e the f o l l o w i n g d i m e n s i o n l e s s q u a n t i t i e s : 59 With these notations, III.59 becomes - - -tfr .'111.63 1 -^ ^ C " With the help of III.61, one shows that. III.56 y i e l d s III. I — A. -tr where :-III.65a % v o + in 0 65b 6+-6- - i ^ x l 111.65c [| r (fr ^ 5 A = t ^ V * ) , n j D 65d and 0 ^ 6R < TT/z III ,66a ^ III.66b These i n e q u a l i t i e s are essential f o r avoiding d i f f i c u l t i e s connected with the multi-valued character of the function III.64. Using the re l a t i o n P(z.)P ( | -Z . ) - "JJ/^TTz. "' we obtain 60 JL Iff ' ' - I I I . 6 7 It follows from III.55, 6 2 , 64 and 67 that L - ^ f r * c III.68a where J , = -s 1 III.68b The integrand I ~ j s has been plotted as a function of q, and the integral III.68a evaluated numerically for the lowest values of q(i.e. for q between 0 and q Q where <J0 corresponds to an energy E of about 1.5 mc2 in the case of A^7 and of about 2mc in the case of Cs ). For the higher energies >cfyoj) an approximate integrand (which i s derived below) i s used in place o f ' I j s and the resulting integral i s performed analy-t i c a l l y . Let us write ^Cjc, a s t n e s u m °^ two terms: L t c i s - u C l s r u C l s .0) , . . <i) where ^c/s, i 3 the part of L J C / s evaluated numerically and v-»cJS the part of U evaluated analytically. If £ 0 i s the energy corresponding to q c , we have = * z w ) , T ' 5 i £ • I n - 7 0 a and L, _ J _ ( - j - ^ e III.70b 61 We approximate T s^ by expanding i t in powers of '/^ and by taking the f i r s t two terras. The expansion of the binomials of Eq. III.6 4 in powers of ^ i s allowed when The upper limit t Q of the numerical integration III.70a i s chosen such that the corresponding q Q satisfies III.71. We give below, as an example, the f i r s t few terms of the expansion for one of the binomials in III.6 4 : + ( - I t ^ X i - ^ J h r J " * * ^ U 1 . 7 2 / 1 r y \/ *\ . X.I 1 1 1 i t -5 When the expansions are carried out up to the power (iq) , i t is se'en that A i s ^ 3<j.fc v ? ' i n . 7 3 In the same approximation, one has, for the remaining factors of the integrand X ) s , ( E q . . III.68b) H T T / III. 74a Z**V i n . 74b die. c< Z - ^ d - o . III.7 4 c 62 III.74d Therefore, from III.74, 73, 62, 63b and 70b: 3 and LT = !iL* /, . X X - Z i s Q 1 I T We have f i n a l l y , f o r L I S with given by III.41 and given by III.69 b) Evaluation of L i c The evaluation of i s carried out along very s i m i l a r l i n e s . With the aid of I I I . l i b , 49, and 53, III.43 becomes with E ^ given by Eq. III.13 with n = 2: E x =. A T , C 3 X ' * I H . 7 g With the help of III.56 and I I I . 6 0 , one integrates III.77 r f i r s t , and then over t i to obtain: 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 III.21, = £ t [£°i^Jbi + x-'-'Ci-^-'r*^] 1 1 1 , 8 3 where we introduced a notation similar to the one in III.65a w Ok 64 $i - W 1 ^Cfr**) 6^ and have the same range o f v a l u e s as Q+. and 6_ (Eq. III.66) Using the above n o t a t i o n and Eq. I I I . 6 ? , we have I - ^ & ( ° ° R A I I I . 8 5 a with I j i c — ~ ~ — ; : — I I 1 . 8 5 b 3 e -A r z*<x \ i s d e f i n e d i n III.7 8 . U^,^ was evaluated i n the same manner as , i . e . as the sum o f two terms 0) , « Ljca,' - L»cX!l c^-xs I I I . 8 6 a with U , - H i ^ . J i £ - \ I I I . 8 6 b l £ = ^ • -i§- ( I" c L L 1 1 1 . 8 6 c ,0) U ) L> C 3 k S i s the pa r t o f u C a s evaluated n u m e r i c a l l y , and >-JCa^ the pa r t e v a l u a t e d a n a l y t i c a l l y , b e i n g an approximation f o r X»vS a t high e n e r g i e s . £ a i s chosen such t h a t the c o r r e s -ponding q c (Eq. III.62) s a t i s f i e s the r e l a t i o n : 65 4r ± 2^ III.37 rv The approximate integrand Xj^ i s derived by developing the expression III.35b in powers of l / q . The f i r s t two terms of the expansion are - 7 * ft TT«[f V 3 */\7+<f / III.S3 It follows from 36c and from the r e l a t i o n c L c ^ ^ d ^ that I I I . 3 9 We have f i n a l l y , f or L* a c , * L P x s + 1 1 1 . 9 0 with ^-»o3kS given by Eqs. I l l . 4 5 and III.47 and given by Eqs. III.36a, 86b and 39. 66 o 3 R e s u l t s and Co n c l u s i o n s We have a p p l i e d the formulae and methods o f t h i s S e c t i o n to the case o f A (Z r 18). Among the few elements f o r which r e l i a b l e experimental data e x i s t t h i s i s the one with the s m a l l e s t value o f Z so t h a t the n o n - r e l a t i v i s t i c approximation i s not u n j u s t i f i e d . In a d d i t i o n t o t h a t , we have e v a l u a t e d one p o i n t of the I s - spectrum f o r the case o f Cs^" 1" (Z s 55) u s i n g our formulae, with a view t o comparing the r e s u l t with Hess's r e s u l t f o r the same case. 37 a) The Case o f A 37 In the case o f A (Z = 18), we have computed the r a t i o ^^/us^ ^ o r f ° u r d i f f e r e n t p o i n t s o f the photon spectrum: For $ck = 135 Kev corresponding t o x • 2 jfick = 269 " " " x = 4 Jrfck » 404 n " " x » 6 Jrfck = 673 " n " x = 10 The l i m i t o f the spectrum i s a t #ck = £16 Kev. The r e s u l t s f o r t h i s case are summarized i n Tables I , I I and I I I and i n F i g . 5. In Table I , we l i s t the v a r i o u s c o n t r i b u t i o n s t o the i n t e g r a l L-*^ (Eq. 11.30, (n = 1)), f o r the f o u r v a l u e s o f , <» , W X» , w the photon energy l i s t e d above. U n . L n , . L»« L n ^ °is ' Dia 7 ° ^ 1 w»s and are given by the formulae I I I . 23, 2 9 , 3 0 , 31 and 67 40 respectively. To get an idea about the r e l i a b i l i t y of Eq. III.40, 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 for L. ^ , show a deviation of less than 0.5 percent. The error in evaluating L - » D , C i - s o£ ,0) the same order of magnitude. The values for are obtained by numerical integration of Eq. III.70a for four different values of x. Since the integrand is too complicated to be studied analy-t i c a l l y , we give in Fig. 1, 2, 3 and 4 the curves representing the integrand as a function of the integration variable q for 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. The upper limit of integration q 0 is different in each case. L . c i s evaluated analytically from Eq. III.75b. The main error in the expression for U, comes from the numerical integration of U c ; the error made in evaluating U p and L J C | S are small compared to i t . The total error on Ujj » however, i s probably less than one unit in the third significant figure. In Table II we l i s t the contributions to.the integral L a s , (Eq. 11.32 ( n s 2)) for the same four values of the Photon energy. L 0 ^ , U 0 ^ , L , L D j 5 , L 0 3 k & are evaluated from Eq. III.45, 46. | , calculated from the , CD approximate Eq. III.45e i s also included. L. c i s given by Eq. III.36b which was also integrated numerically. However, the curves for the integrand I ( E q . III.85b) have not been included. i s given by Eq. III.89- The error, in the values obtained for L *^ are of the same order of magni-tude as in the case of L i ^ . 68 In Table I I I , we g i v e the v a l u e s f o r the p r o b a b i l i t y r a t i o * /AAJ^ as e v a l u a t e d from Eq. 11.30 and from the v a l u e s of Table I and Table I I f o r L |s and L.^ . The q u a n t i t y which appears i n Table I I I i s d e f i n e d as P _ L A /W-0 / is V WL ^ i s t h e r e f o r e a measure o f the i n f l u e n c e of the r e t a r d a t i o n f a c t o r on the i s + 2.S 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 r a t i o In F i g . 5, we have p l o t t e d the two q u a n t i t i e s ^ n d ~~ ( ^ f c ^ ^ Y_ . , . both i n our case and i n the case o f Glauber OK \ /uj^ /IS -t-iS and M a r t i n (5^-), as a f u n c t i o n of the energy o f the. emitted photon. The curve ^ ( A*ij^)* n t n e c a s e °f Glauber and M a r t i n i s i d e n t i c a l w i t h t h a t which would be obtained from M o r r i s o n and S c h i f f s formula. We can say very l i t t l e , a t the present time, about the comparison o f our t h e o r e t i c a l r e s u l t s f o r i n case 37 o f A w i t h the experimental r e s u l t s o f L i n d q v i s t and '.vu ( 5 5 ) . The procedure used by these authors to compare experiments and theory c o n s i s t s , f i r s t , i n a p p l y i n g a l l k i n d s of experimental c o r r e c t i o n s to the t h e o r e t i c a l gamma spectrum, and, next, i n n o r m a l i z i n g the measured spectrum t o the spectrum so o b t a i n e d (so t h a t o n l y r e l a t i v e i n t e n s i t i e s are being compared). The c o r r e c t i o n s woul.i thus have t o be a p p l i e d to our t h e o r e t i c a l spectrum, which we cannot do s o l e l y on the b a s i s of the data pub l i s h e d by L i n d q v i s t and i u (55). I t i s not i m p o s s i b l e t h a t the- d i f f e r e n c e i n the slope o f the curves A (Is + 2s) and B ( l s -+• 2s) i n F i g . 5 may be reduced c o n s i d e r a b l y a f t e r the corrections have been applied. b) The Case of C s 1 3 1 • ' ,, . % In the case of Cs^^ 55) we have calculated A ^ ^ & & £ , only for one value of the photon energy, namely for #cka-2^0^ Kev (x = 1), and we restricted ourselves ,to the i s - electron contri-bution. . - - '• An application of the formulae of this Section and a numeric cal integration carried out in the way indicated in Subsection> a) give, for Gs 1 3 1: " Therefore, | .Tfi JL The ratio LWASJH' i s found to be equal to 0.79. If only the integration over the intermediate continuous-states i s taken into account, L, Q ^ 0 > and the ratio becomes only 0.03. This small value for accounts for the . larger part of the discrepancy between the results of Hess (who neglected the sum over the discrete states)' and the. results of Morrison and Schiff. 1.0 0.8 0-6 0.4-0.2. F l q . 1 rr>t«.^\At\a op m . 70 i Poi\. % « £ Patincj patje 70 P&ctrvj pA«j£ (0 Table I T O i s Kck » 135 . (x - 2) #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 e. 0.01225 0.00103 0.000168 0.0000147 Co 0.00401 0.00033 0.000053 0.0000047 ,w 0.OW0O7 0.00Q32Q 0. 0000SS <? 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 u 0.6025 0.3682 0.2105 0.09982 L 0.906 0.393 0.215 0.100 Table I showing the various contributions to the integral L ,s (.in units of {^^j for four values of the photon energy Jick (in Kev) Table II tfck = 135 (x = 2) -ftck = 269 (x = 4) tfck = 404 (x = 6) -Kck = 673 (x = 10) 0.08335 0.00594 0.00451 0.00147 0.001464 0.00137 0.0977 0.2084 0.0019 0.2103 0.303 0 . 0 0 7 3 0 6 0.000984 0.000308 0 . 0 0 0 1 2 4 0.000160 0.0039 0.1312 0.0007 0.1319 0.141 0.001206 0.000156 0.000047 0.000020 0.0001249 0.0000197 0.000025 0.00146 0.07327 0.00039 0.07366 0.0752' 0 . 0 0 0 1 0 7 9 0.0000134 0 . 0 0 0 0 0 4 0 0.0000017 0.00000168 0 . 0 0 0 0 0 2 2 0.00013 0 . 0 3 0 6 2 0.00015 0 . 0 3 0 7 7 O .O309 Table II showing the various contributions ,to the i n t e g r a l L ' ( i n units of f ? & £ M fo r four values of the photon energy )ick (in Kev) O lOO 400 600 8OO Table III 7a Jick . 135 Jick « 269 #ck . 404 jfek - 673 (x * 2) (x . 4) (x - 6) (x . 10) 0.637 0.665 0.384 0.0470 0.080 0.087 0.047 0.0047 0.314 0.617 0.415 0 O 2 4 3 Table III showing the contributions to from the Is and 2s electrons (in photons per desintegration per unit Kev energy interval / |0 ) for four values of the photon energy fo-K (in Kev). The table also shows the ratio of the sum of these con-tributions to the corresponding result of Glauber and Martin (54). 73 Appendix A We want to sketch the proof o f the statement t h a t JL ^ c a O A i a f ( R ) A * $ < * 2 £ r e ^ ^ m ^ i R ^ s . - A . l 71*. R 1. E- * i s n e g l i g i b l e f o r small v a l u e s o f R. Making U3e o f the c l o s u r e p r o p e r t y , (where I i s . the u n i t m a t r i x ) , expanding i n t o products o f s p h e r i c a l harmonics and the B e s s e l f u n c t i o n s , and f i n a l l y i n t e g r a t i n g over angles, 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-p o r t i o n a l to 'r\ to f\ * (we have here 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 form f o r the wave f u n c t i o n o f the K - e l e c t r o n ) . Hence A.2 and, consequently, A . l becomes p r o p o r t i o n a l t o 3k. which i s of the Order (\ f o r f{—> 0 Appendix B 74-We sketqh the proof o f the i d e n t i t y : B . l where AZ - jj^ % K* and K = O Making use o f the r e l a t i o n s £<r =t <?3 , (32 a, -5/3 (J**-1 , the e x p r e s s i o n on the l e f t - h a n d s i d e o f B . l can be w r i t t e n as JL l • 3.2 or B .3 Now, 2* = | So t h a t the second l i n e of B . 3 v a n i s h e s , and the f i r s t Dine g i v e s j u s t the e x p r e s s i o n on the r i g h t - h a n d s i d e o f B . l e Appendix C 75 We want t o make i t p l a u s i b l e t h a t the f a c t o r i n Eq. III. 4 © may be set equal to u n i t y over the whole range of i n t e g r a t i o n , without too much e r r o r . The g e n e r a l form of Eq. 1 1 1 . 5 4 i s ( S ^ - . H I - 6 8 ) 1 C . l where i s the f a c t o r i n q u e s t i o n , and i s giv e n by I I I . 6 4 . - 3 Since yixr\ 0± | 0 , one has, i n f a c t , ^ 1 i n the i n t e - . , g r a t i o n r e g i o n where the remaining f a c t o r o f the i n t e g r a n d i s l a r g e , i . e . f o r 0<v^-<i5" » a s can be seen from the graphs i n F i g . 1 to 4 . For qj- 7 j 5" , the i n t e g r a n d i n C . l can be r e p l a c e d , without a p p r e c i a b l e e r r o r , by a si m p l e r e x p r e s s i o n (see Eq. III.75a) so t h a t Consider now very l a r g e v a l u e s o f q. The behavious of f f o r l a r g e q's can be obtained from the asymptotic expansion o f the 76 c o n f l u e n t hypergeometric f u n c t i o n (Jahnke and Snide, p.275); we f i n d l f r R v. ' y ! . and f —*» O f o r c£ <** , because i X (Jahnke and Erade, p. 1 0 ) . But f o r l a r g e q's, the i n t e g r a n d , e x c l u d i n g f , i s p r a c t i c a l l y zero, so t h a t p u t t i n g ^ a | ( i n s t e a d o f f.<£l) i s harmless. For the i n t e r m e d i a t e v a l u e s o f q, f remains f i n i t e while the remainder o f the i n t e g r a n d i s s t i l l v e r y s m a l l , so t h a t p u t t i n g f = 1 i s of no great consequence. 77 References B l a t t , J.K. and Weisskopf, V.F. (52). "Theoretical Nuclear Physics", John Wiley and Sons, 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 Martia, F . C . (55).- Phys. Rev. Q£, 572 (1954). Glauber, R.J. and Martin, ?.C. ( 5 6 ) . (Unpublished; see Chapter. I l l , page 43 of t h i s t h e s i s ) . Gordon, W. (28). Z. Phys. £0 , 630 (1928). H e i t l e r , W. ( 5 0 ) . "The Quantum Theory of Radiation", 2 n d Ed*, Oxford University Press, 1950. Hess, G. (55). Thesis, The University of B r i t i s h Columbia, June 1955. Konopinski, E.J. and Langer, L.M. (53). Ann. Rev. of Nucl. Sc. 2, 261 (1953)• Konopinski, E.J. (55). Chapter X of K. Siegbahn fs "Beta and Gamma Ray Spectroscopy", North Holland Publishing Co. 1955• Kramers, H.A., "Die Quantentheorie des Elektrons und der Strahlung". Leipzig, 1938. Lindqvist, I. and Wu, C S . (55), Phys. Rev. 100. 145 (1955)-. MacRobert, T .M. (54) "Functions of a Complex Variable", 4th Be*.., MacMillan, 1954-Iladansky, L. and Rasetti, F. ( 5 4 ) . Phys. Rev. 2kt 407 (1954)-Magnus, W. and Oberhettinger, F. (49). "Formulas'and Theorems fo r the Special Functions of Mathematical Physics", Chelsea Publishing Co., 1949. 7 6 M o r r i s o n , P. and S c h i f f , L . I . (40). Phys 0 Rev. 5|_, 24 (1%Q) c P a u l i , W. (33). "Die allgemeinen P r i n z i p i e n . der Wellenmechanik", Hand, der Phys., 3d. XXIV/i, (1933). Rose, xM.S. (37). Fhys. Rev. 51, 4#4 (1937) » Rose, M.E. (55). Chapter I>I o f Siegbahn's "Beta and Gamma Ray • Spectroscopy", North H o l l a n d , 1955. S a r a f , B. (54). Phys. Rev. 94, 642 (1954). Sommerfeld, A. "Wellenmechanik", Ungar P u b l i s h i n g Company, New York.. Wu, C S . (55). "3eta and Gamma Ray Spectroscopy",.North Holland, 1955. (p. 649). Wu, C.3., L i n d q v i s t , T., Glauber, R.G., and M a r t i n , P.O. (56). Phys. Rev. 101, 905 (1956).
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On the theory of radiative electron capture Paquette, Guy 1956
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Title | On the theory of radiative electron capture |
Creator |
Paquette, Guy |
Publisher | University of British Columbia |
Date Issued | 1956 |
Description | The continuous spectrum of gamma radiation which accompanies the capture of orbital electrons has been recently calculated independently 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 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 1s and 2s electrons, has been computed for the case of A³⁷ for which experimental data are available. In view of the low nuclear charge of A³⁷ (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 0.81 of the intensity obtained by Glauber and Martin at 135 Kev, and to 0.24 at 675 KeV (the gamma spectrum limit being 816 KeV). The gamma spectrum of A³⁷ determined by Lindqvist and Wu (1955) seems to agree quite well with Glauber and Martin's result. However, Lindqvist and Wu 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. |
Subject |
Radiation Electrons |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085942 |
URI | http://hdl.handle.net/2429/40400 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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