UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Laser nonenhancement of beta decay Blevis, Ira 1984

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1984_A6_7 B59.pdf [ 3.58MB ]
Metadata
JSON: 831-1.0096047.json
JSON-LD: 831-1.0096047-ld.json
RDF/XML (Pretty): 831-1.0096047-rdf.xml
RDF/JSON: 831-1.0096047-rdf.json
Turtle: 831-1.0096047-turtle.txt
N-Triples: 831-1.0096047-rdf-ntriples.txt
Original Record: 831-1.0096047-source.json
Full Text
831-1.0096047-fulltext.txt
Citation
831-1.0096047.ris

Full Text

LASER NONENHANCEMENT OF BETA DECAY by IRA BLEVIS B . S c . , U n i v e r s i t y Of Toronto,1981 THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e a u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA June 1984 © IRA BLEVIS, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C olumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n permi s s i o n . Department of PHYSICS The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: June 1984 i i A b s t r a c t The p o s s i b i l i t y of i n f l u e n c i n g n u c l e a r beta decay w i t h h i g h i n t e n s i t y , low f r e q u e n c y e l e c t r o m a g n e t i c r a d i a t i o n (such as from a l a s e r ) i s examined. The motion of an e l e c t r o n i n an e.m. f i e l d i s found f i r s t a c c o r d i n g t o c l a s s i c a l mechanics and then a c c o r d i n g t o quantum mechanics. The l a t t e r d i s c u s s i o n y i e l d s the V o l k o v s o l u t i o n f o r the e l e c t r o n wave f u n c t i o n which i s then used i n a beta decay c a l c u l a t i o n p a t t e r n e d a f t e r Becker e_t a l . In t h i s c a l c u l a t i o n the V o l k o v s o l u t i o n i s s u b s t i t u t e d f o r the e l e c t r o n p l a n e wave f a c t o r i n the t r a n s i t i o n a m p l i t u d e i n the Fermi t h e o r y of beta decay. The r a t i o of t o t a l decay r a t e s R= r.^tr i s n u m e r i c a l l y e v a l u a t e d w i t h a computer f o r the case of 3H (and some o t h e r s ) where the emphasis i s on o n l y a l l o w e d n u c l e a r t r a n s i t i o n s and on phase space c o n s i d e r a t i o n s . No change i n r a t e t o 1% a c c u r a c y i s found, f o r the range of the d i m e n s i o n l e s s f i e l d s t r e n g t h i n t e n s i t y parameter V =^£- 6 [. 3 ,7 ], or the photon energy range u: e [2,8]eV. T h i s range of ]) c o r r e s p o n d s t o the range i n l a s e r i n t e n s i t y of I € [ 3 x 1 0 1 7 , 2 x 1 O 2 0 ] W / c m 2 . The e n d p o i n t s of the n e u t r i n o spectrum a r e found t o be u n m o d i f i e d by t h e l a s e r ( b o t h a n a l y t i c a l l y and n u m e r i c a l l y ) . Thus t h i s c a l c u l a t i o n s u g g e s t s t h a t the b a s i c b e t a decay p r o c e s s i s not a f f e c t e d by l a s e r i r r a d i a t i o n . i i i T a b l e of C o n t e n t s A b s t r a c t i i L i s t of T a b l e s i v L i s t of F i g u r e s v Acknowledgement v i Chapter I INTRODUCTION 1 1.1 H i s t o r y 1 1.2 F i r s t I m p r e s s i o n s 2 1 . 3 O u t l i n e 6 1.4 C o n v e n t i o n s 6 Chapter I I MOTION OF AN ELECTRON IN AN E.M. FIELD 8 2.1 C l a s s i c a l S o l u t i o n 8 2.2 V o l k o v S o l u t i o n 16 Chapter I I I INDUCED BETA DECAY 27 3.1 Be t a Decay H a m i l t o n i a n 27 3.2 Be t a Decay Kate 34 3.3 Induced Beta Decay Rate 38 Chapter IV NUMERICAL DECAY RATE 44 4 . 1 Summary 44 4.2 K i n e m a t i c Domain 45 4.3 A s y m p t o t i c Formulae For B e s s e l F u n c t i o n s 48 4.4 The Summation 50 4.5 D i f f e r e n t i a l Decay Rate 55 4.6 T o t a l Decay Rate 56 4.7 Program Notes 57 Chapter V CONCLUSIONS 71 BIBLIOGRAPHY 73 APPENDIX A - NOTATION AND CONVENTIONS 75 APPENDIX B - ESTIMATE FOR JGT INTEGRAL 79 APPENDIX C - NEUTRINO ENERGY 80 APPENDIX D - ENTIRE KINEMATIC DOMAIN 81 i v L i s t of T a b l e s I . LOCATION OF FEATURES 69 I I . NONENHANCEMENT 70 V L i s t of F i g u r e s 1 . VIRTUAL AND REAL /9 DECAY 4 2. LAB TIME AND PROPER TIME 13 3. KINEMATIC DOMAINS 59 4. SUMMAND OF IV. 10 60 5. OVERLAP OF ASYMPTOTIC FORMULAE 61 6. DIFFERENTIAL DECAY RATE 62 7. DIFFERENTIAL DECAY RATE 63 8. DIFFERENTIAL DECAY RATE 64 9. DIFFERENTIAL DECAY RATE 65 10. ENERGY SPECTRUM OF BECKER et a l 66 11. DIFFERENTIAL DECAY RATE 67 12. DIFFERENTIAL DECAY RATE 68 v i Acknowledgement Thanks a r e extended t o Dr. D. Beder of the Department of P h y s i c s of UBC f o r b o t h s u g g e s t i n g and s u p p o r t i n g t h i s r e s e a r c h . My g r a t i t u d e t o Dr Beder, Dr. N. Weiss, and t h e i r c o l l e a g u e s f o r the enrichment of my g r a d u a t e s c h o o l program i s a l s o due. 1 I . INTRODUCTION 1.1 Hi s t o r y The p o s s i b i l i t y of i n f l u e n c i n g n u c l e a r b e t a decay p r o c e s s e s by i n t e n s e l a s e r beams r e c e n t l y s t i m u l a t e d some i n t e r e s t f o l l o w i n g the p u b l i c a t i o n of Becker e_t a_l ( r e f . 2, 1981). The l a s e r m o d i f i e d d i f f e r e n t i a l decay r a t e was p r e s e n t e d a l o n g w i t h some d r a m a t i c n u m e r i c a l r e s u l t s . For low energy a l l o w e d d e c a y s , namely f o r 3H and 1 8 F w i t h e" e n e r g i e s of 18 KeV and 650 KeV, and f o r l a s e r i n t e n s i t i e s of about 1 0 1 8 W/cm2, they c l a i m e d enhancements i n the decay r a t e by f a c t o r s of up t o 10*. T h e i r paper o u t l i n e d the c a l c u l a t i o n showing key e x p r e s s i o n s and i m p o r t a n t parameters but o m i t t e d the d e t a i l s of a d i f f i c u l t i n t e g r a l . S u b s e q u e n t l y they p u b l i s h e d a note ( r e f . 4, 1983) r e f u t i n g t h e i r o r i g i n a l r e s u l t w i t h o u t commenting on i t d i r e c t l y . The f i r s t l e t t e r used a F o u r i e r B e s s e l s e r i e s e x p a n s i o n i n the V o l k o v s o l u t i o n , r e p r e s e n t i n g the e l e c t r o n , t o a r r i v e a t an e x p r e s s i o n f o r the d i f f e r e n t i a l decay r a t e . T h i s a p p a r e n t l y was i n t e g r a t e d a n a l y t i c a l l y , but d e t a i l s of the i n t e g r a t i o n were not i n c l u d e d . The subsequent note used the ap p r o x i m a t e i n t e g r a t i o n t e c h n i q u e of s t e e p e s t d e s c e n t t o a r r i v e a t a s i m p l e e x p r e s s i o n f o r the t o t a l decay r a t e from which the l a s e r parameters c a n c e l l e d . S i n c e the f i r s t l e t t e r o t h e r s have r e p e a t e d and augmented the work. John Hebron ( r e f . 14, 1983) i n c l u d e d an e x p l i c i t c a l c u l a t i o n of the d i f f e r e n t i a l decay r a t e e x p r e s s i o n i n h i s M.Sc. t h e s i s . He a l s o i n c l u d e d a 2 d i s c u s s i o n of a one d i m e n s i o n a l square w e l l n u c l e a r model t o argue a g a i n s t the f i r s t r e s u l t of Becker et a_l. In p r e p a r a t i o n of the p r e s e n t t h e s i s the d i f f e r e n t i a l decay r a t e c a l c u l a t i o n has been r e p e a t e d and the decay r a t e e x p r e s s i o n has a g a i n been v e r i f i e d . R i e s s ( r e f . 19, 1983) has performed s i m i l a r work; but h i s emphasis was on f o r b i d d e n decays and as w i t h Hebron he d i d not g i v e n u m e r i c a l r e s u l t s . Ternov e_t a l ( r e f . 20, 1983) have now p u b l i s h e d n u m e r i c a l r e s u l t s t h a t show t h e r e i s no s i g n i f i c a n t enhancement. The p r e s e n t work has y i e l d e d n u m e r i c a l r e s u l t s a l s o showing t h a t t h e r e i s no enhancement i n n u c l e a r b e t a decay. The summation of l a r g e o r d e r B e s s e l f u n c t i o n s t h a t appears i n t h e d i f f e r e n t i a l decay r a t e , and the i n t e g r a t i o n t o f i n d the t o t a l decay r a t e were performed n u m e r i c a l l y . A l t h o u g h the d i f f e r e n t i a l r a t e was dependent on the i n t e n s i t y of the l a s e r , the t o t a l decay r a t e was i n s e n s i t i v e t o t h i s and was e q u a l t o the u n m o d i f i e d r a t e . As w e l l the n e u t r i n o s p e c t r a was found t o be u n s h i f t e d by the l a s e r . These r e s u l t s are t o be s u b m i t t e d f o r p u b l i c a t i o n s h o r t l y i n c o l l a b o r a t i o n w i t h D.S. Beder. 1.2 F i r s t I m p r e s s i o n s Becker e_t a_l proposed t h a t the quantum s t a t e s of the c h a r g e d p a r t i c l e s produced i n /3 decay were m o d i f i e d by the l a s e r f l u x , i n c r e a s i n g the number of a c c e s s i b l e s t a t e s f o r a decay and c o n s e q u e n t l y , u s i n g the Fermi Golden r u l e ( r e f . 5) i n c r e a s i n g the decay r a t e . To e l u c i d a t e t h i s s u g g e s t i o n , 3 f i r s t b e t a decay and the number of a c c e s i b l e s t a t e s i s c o n s i d e r e d , then g e n e r a l arguments c o n c e r n i n g the p o s s i b l e e f f e c t s of a l a s e r a r e p r e s e n t e d , i n t h i s c h a p t e r . A d e t a i l e d c a l c u l a t i o n and f i n a l r e s u l t s c o n t a i n e d i n the s u c c e d i n g c h a p t e r s complete th e d i s c u s s i o n . In decays c o n s i d e r e d h e r e i n , the energy and momentum ar e s h a r e d amongst t h r e e p a r t i c l e s i n the f i n a l s t a t e . I n i t i a l c o n s i d e r a t i o n s have t h e p a r t i c l e s c o n t a i n e d i n a l a r g e but f i n i t e volume i n c o o r d i n a t e space (a box) and t h u s t o have d i s c r e t e energy s p e c t r a . One p a r t i c l e , the n u c l e u s , i s much more massive than t h e o t h e r s and a c q u i r e s p r o p o r t i o n a t e l y l i t t l e of the decay energy as d i c t a t e d by the law of c o n s e r v a t i o n of momentum. Most of the energy i s d i s t r i b u t e d t o the two l e p t o n s . The c o n f i g u r a t i o n s a r e not c o n s t r a i n e d except by energy-momentum c o n s e r v a t i o n and a l l p o s s i b i l i t i e s would be r e a l i z e d i n a s t a t i s t i c a l ensemble of d e c a y s . I f the energy of t h e decay were h i g h e r , then the number of p o s s i b l e c o n f i g u r a t i o n s would a l s o be h i g h e r as would the t o t a l p r o b a b i l i t y of decay f o r a g i v e n time i n t e r v a l . The c o u n t i n g argument of the l a s t p a r a g r a p h i s s u g g e s t i v e r e g a r d i n g the c a s e of an i n f i n i t e box and c o n t i n u o u s s p e c t r a . The phase space domains a l l o w e d by energy c o n s e r v a t i o n a r e Riemann i n t e g r a b l e , c o n t i n o u s , and c o n s t r a i n e d by the t o t a l energy a v a i l a b l e f o r the decay. L a r g e r decay e n e r g i e s e n t a i l l a r g e r domains i n phase space which e n t a i l h i g h e r decay p r o b a b i l i t i e s , as f o r the d i s c r e t e 4 c a s e . I t i s a s u c c e s s of /i decay t h e o r y t h a t the wide v a r i a t i o n s i n decay r a t e s found i n n a t u r e can be e x p l a i n e d by such phase space arguments. In a s e m i - c l a s s i c a l approach t o / j decay i n a l a s e r f i e l d , t he energy, Q>Kev, might be thought of as the energy n e c e s s a r y t o c o n v e r t a v i r t u a l decay i n t o a r e a l one: f i g 1 . F i g u r e 1 - V i r t u a l and R e a l / 3 Decay T h i s energy and the H e i s e n b e r g u n c e r t a i n t y p r i n c i p l e d e t ermine a time i n t e r v a l and thus an energy a b s o r p t i o n r a t e f o r the v i r t u a l e" Q *-t > * • p = 0_ - CL ~ lo' 5 " M*V/S At Jfr I f the v i r t u a l e~ responds t o the e l e c t r i c f o r c e (the L o r e n t z f o r c e i s weaker by 1/c) of a l a s e r v> = 1 , u> =2eV, f o r t h i s 6 t , then the absorbed power i s p cc ^m-fc u o V * Z ^ \Oi0 MeV/s C o n s e q u e n t l y we a r e not l e d t o an e x p e c t a t i o n of enhancement of /3 decay by t h i s r o u t e . In a quantum m e c h a n i c a l approach t o ^ decay i n a l a s e r f i e l d , t h e l a r g e d i f f e r e n c e i n s c a l e s i z e (wavelength) of 5 the /3 decay phenomena (Q>KeV) and t h e l a s e r f i e l d photons (co^eV) l e a d s i n i t i a l l y t o no e x p e c t a t i o n s of a s i g n i f i c a n t i n t e r a c t i o n i n a n a l o g y t o the resonance phenomenon. But the most i n t e n s e l a s e r s a v a i l a b l e today can p r o v i d e -WMrenSi t leS-of l i g h t t o atomic s i z e d r e g i o n s of about 1 0 Z O t i m e s -tha-fe- of the u s u a l l a b o r a t o r y environment f o r decay e x p e r i m e n t s (about lO^W/cm 2). Thus even i f t h e r e i s o n l y a s m a l l c o u p l i n g , a t t h i s i n t e n s i t y the m o d i f i c a t i o n of the r a t e c o u l d p o s s i b l y be s i g n i f i c a n t . 1 To c a l c u l a t e the magnitude of t h i s e f f e c t the e l e c t r i c a l l y c harged p a r t i c l e s a r e c o u p l e d t o t h e l a s e r photon f i e l d . Then the t r a n s i t i o n r a t e f o r the /3 decay p r o c e s s i s r e c a l c u l a t e d . As a f i r s t a p p r o x i m a t i o n the n u c l e a r response t o the l a s e r f l u x may be n e g l e c t e d r e l a t i v e t o the p a r t i c l e ' s r e s p onse. T h i s c a l c u l a t i o n c o m p r i s e s the body of the t h e s i s and i s o u t l i n e d i n the next s e c t i o n . In l i g h t of the above d i s c u s s i o n of ^ decay r a t e s a t t e n t i o n w i l l be g i v e n t o whether t h e l a s e r f l u x i n c r e a s e s the number of a c c e s s i b l e s t a t e s . I t w i l l be seen t h a t t h e t o t a l decay r a t e does not change (sec 4.6) and t h a t the n e u t r i n o spectrum r e t a i n s i t s o r i g i n a l bounds (sec 4.2); the l a t t e r r e s u l t i s c o n s i s t e n t w i t h no i n c r e a s e i n r a t e , and no m o d i f i c a t i o n of the /i decay p r o c e s s . 1 The assumption t h a t the n u c l e u s of an atom or i o n may be exposed t o the l a s e r f l u x has been debated i n t h e l i t e r a t u r e ( r e f . 2,3,13,14) as has the p o s s i b i l i t y of e x p e r i m e n t a l y d i s t i n g u i s h i n g /3 p a r t i c l e s from background ( r e f . 2,3,14,18) Only the q u e s t i o n of the t h e o r e t i c a l e x i s t e n c e of an e f f e c t w i l l be c o n s i d e r e d i n t h i s work. 6 1.3 O u t l i n e C hapter 2 i s a d i s c u s s i o n of the motion of an e l e c t r o n , (or c h a r g e d p a r t i c l e ) , under the i n f l u e n c e of the e.m. f i e l d of a l a s e r . The c l a s s i c a l s o l u t i o n i n c l u d e d i s u s e f u l f o r b u i l d i n g i n t u i t i o n and the quantum s o l u t i o n f o r the decay c a l c u l a t i o n t h a t f o l l o w s . Then the quantum c a l c u l a t i o n of the decay r a t e i s p r e s e n t e d ; c h a p t e r 3. I t b e g i n s w i t h g e n e r a l c o n s i d e r a t i o n s i n o r d e r t o shed some l i g h t on the b a s i c p h y s i c s . These c o n s i d e r a t i o n s i n v o l v e some phenomenology g o v e r n i n g the H a m i l t o n i a n , but then the d i s c u s s i o n c e n t e r s on u n p o l a r i z e d n u c l e a r t r a n s i t i o n s , unobserved f i n a l p o l a r i z a t i o n s of l e p t o n s , and unobserved n e u t r i n o momenta. These q u a n t i t i e s a r e summed or i n t e g r a t e d out of the decay r a t e e x p r e s s i o n e a r l y i n the d i s c u s s i o n e c o n o m i z i n g on l a b o u r . Induced decay i s then t r e a t e d i n a s i m i l a r manner. Chapter 4 b e g i n s w i t h the d e t e r m i n a t i o n of the k i n e m a t i c a l domains f o r the d i f f e r e n t i a l decay r a t e and c o n t i n u e s w i t h some d e t a i l of the n u m e r i c a l work. Chapter 5 i s a c o n c l u s i o n . 1.4 C o n v e n t i o n s Throughout t h i s paper the fundamental u n i t has been chosen t o be energy, MeV. Then o t h e r u n i t s a r e d e r i v e d u s i n g •R=1=c . E l e c t r o m a g n e t i c u n i t s a r e G a u s s i a n w i t h t h i s m o d i f i c a t i o n . For example t h e charge of the e l e c t r o n i s e=WjTbT , ( d i m e n s i o n l e s s ) . 7 The s t a n d a r d summation c o n v e n t i o n i n c l u d i n g the use of the m e t r i c t o map t e n s o r s i n t o t h e i r d u a l spaces ( r a i s e and lower i n d i c e s ) and the d i s t i n c t i o n between L a t i n and Greek i n d i c e s i s a l s o f o l l o w e d ; however, v e c t o r or t e n s o r n o t a t i o n w i t h o u t i n d i c e s i s a l s o commonly used. B r a c k e t s w i l l o f t e n e n c l o s e a t y p i c a l term of the t e n s o r they denote, f o r example (M*'* =M/l*1 . The d u a l s of g e o m e t r i c a l o b j e c t s w i l l be denoted by a s t a r i n componentless n o t a t i o n . For example a 2 - t e n s o r has a d u a l F*=GFG, where G i s the m e t r i c . The n a t u r a l i n n e r p r o d u c t f o r t e n s o r s of rank 2 and lower i s ( X , Y ) = t r ( X Y * ) . The 'square' s h a l l r e f e r t o the i n n e r p r o d u c t of a t e n s o r w i t h i t s e l f . Feynman r e p r e s e n t a t i o n f o r Y a l g e b r a i s chosen so t h a t D i r a c e q u a t i o n r e s u l t s resemble t h o s e of B j o r k e n and D r e l l . 2 More d e t a i l i s d i s p l a y e d i n appendix A. Throughout the t h e s i s m w i l l be the e l e c t r o n mass; o t h e r masses w i l l be s u b s c r i p t e d . W w i l l be the l a s e r f r e q u e n c y ( e n e r g y ) ; o t h e r e n e r g i e s w i l l be s u b s c r i p t e d . The d i m e n s i o n l e s s parameter g o v e r n i n g the whole e f f e c t i s \) — where e i s the e l e c t r o n c h a r g e , and cl i s the f i e l d s t r e n g t h of the v e c t o r p o t e n t i a l r e p r e s e n t i n g the l a s e r . 2 B j o r k e n and D r e l l , r e f 5, appendix A 8 I I . MOTION OF AN ELECTRON IN AN E.M. FIELD 2.1 C l a s s i c a l S o l u t i o n G o v e r n i n g e q u a t i o n The L a g r a n g i a n s L m = J"4(A-yJd3y or L„= mfoL/d?y , d e f i n e d up t o m u l t i p l i c a t i v e c o n s t a n t s , l e a d v i a a m i n i m i z i n g p r i n c i p l e of the a c t i o n S=flJt t o the E u l e r -Lagrange e q u a t i o n s and t o the e x p e c t e d r e s u l t f o r a f r e e uncharged p a r t i c l e a t x,namly a=o . The L a g r a n g i a n d e n s i t y =-\ jfo, FF*=—!j , formed from the square of the e.m. f i e l d t e n s o r , l e a d s i n a s i m i l a r manner t o the g o v e r n i n g e q u a t i o n s of the e.m. f i e l d s i n vacuum, M a x w e l l ' s e q u a t i o n s . The E and B f i e l d s can be r e p l a c e d by p o t e n t i a l f i e l d s A , (where F ^ =A/|tjD -AV)^), a L o r e n t z 4 v e c t o r . The g o v e r n i n g e q u a t i o n s a r e now second o r d e r i n the time d e r i v a t i v e , and the d y n a m i c a l v a r i a b l e s resemble those of p a r t i c l e mechanics. The source f r e e Maxwell e q u a t i o n s S^F/1*' =0 a r e a u t o m a t i c a l l y s a t i s f i e d . The dynamics a r e g i v e n by the wave e q u a t i o n i f the L o r e n t z gauge i s chosen. The L a g r a n g i a n d e n s i t y £ = ^ + X M - J ^ A / * i n c l u d i n g s o u r c e f i e l d s J=m fu Si dz ,where U = $ 0 , V ) i s the 4-v e l o c i t y , l e a d s t o M a x w e l l ' s e q u a t i o n s w i t h s o u r c e s and a l s o a c c o u n t s f o r the i n t e r a c t i o n of the s o u r c e s w i t h t h e i r own f i e l d s . Gauge i n v a r i a n c e of the L a g r a n g i a n i s guaranteed by t h e v a n i s h i n g d i v e r g e n c e of J and the i n s e n s i t i v i t y of the m i n i m i z i n g p r o c e d u r e t o a d d i t i o n s of a t o t a l d e r i v a t i v e t o s£ . Here, as a good a p p r o x i m a t i o n the A**" a r e assumed t o 9 d e s c r i b e o n l y the l a s e r and the 1M t o d e s c r i b e the m e c h a n i c a l e l e c t r i c c u r r e n t of the p o i n t s o u r c e , the e l e c t r o n . The E u l e r Lagrange e q u a t i o n s l e a d t o homogeneous wave e q u a t i o n s f o r each of the l a s e r e.m. f i e l d s . I f k i s the l i g h t l i k e p r o p a g a t i o n v e c t o r f o r the l a s e r e.m. f i e l d and € i s the s p a c e l i k e p o l a r i z a t i o n , then and s o l u t i o n s of the wave e q u a t i o n s a re f u n c t i o n s , -f , of S- k x The o r d i n a r y d e r i v a t i v e of a f u n c t i o n w i t h r e s p e c t t o i t s argument w i l l be denoted by " 1 ". The p i e c e of the a c t i o n i n v o l v i n g the e l e c t r o n i s S = -f (J£ + eAv )M Though w r i t t e n u s i n g the q u a n t i t i e s measured i n an i n e r t i a l r e f e r e n c e frame, the E u l e r Lagrange e q u a t i o n s ^- ( m U + e d ) - e V k = O can be r e a r r a n g e d t o g i v e the f a m i l i a r c o v a r i a n t L o r e n t z f o r c e e q u a t i o n T I I . 1 where U i s the r e l a t i v i s t i c 4 - v e l o c i t y as s t a t e d above. S o l u t i o n The g e n e r a l s o l u t i o n f o r the motion of t h i s c l a s s i c a l p a r t i c l e has been found from t h e L o r e n t z e q u a t i o n I I . 1 by I t z y k s o n and Zuber ( r e f . 15). Landau and L i f s h i t z ( r e f . 10 17) have a l s o found the s o l u t i o n ; they s o l v e the r e l a t i v i s t i c H a m i l t o n J a c o b i e q u a t i o n f o r H a m i l t o n ' s p r i n c i p l e f u n c t i o n and then the dynamic v a r i a b l e s , momentum and energy. The approach taken here b u i l d s i n t u i t i o n by f i r s t s o l v i n g the L o r e n t z e q u a t i o n f o r the s i m p l e i n i t i a l c o n d i t i o n s of a p a r t i c l e a t r e s t a t the o r i g i n . Then the a c t i o n i s c a l c u l a t e d t o c o r r o b e r a t e the l i t e r a t u r e and t o f a c i l i t a t e c o mparisons w i t h the quantum m e c h a n i c a l s o l u t i o n of the next s e c t i o n . In t h a t case one e x p e c t s 4 ,=e L , where S i s the a c t i o n of a t r a j e c t o r y i n phase space, t o be the p r o b a b i l i t y a m p l i t u d e t h a t t h a t t r a j e c t o r y i s r e a l i z e d by a quantum m e c h a n i c a l p a r t i c l e . Next the l a b frame motion i s d e s c r i b e d and then L o r e n t z t r a n s f o r m a t i o n s a r e used t o show the e f f e c t s of d i f f e r i n g i n i t i a l c o n d i t i o n s . F i n a l l y the energy of the p a r t i c l e - i s r e l a t e d t o p h y s i c a l parameters of the l a s e r . C h o o s i n g the z a x i s of the l a b frame as the d i r e c t i o n of p r o p a g a t i o n of the e.m. f i e l d i s r e a l i z e d by s e t t i n g k= (u>, 0,0 ,u>) . For the purpose of i l l u s t r a t i o n f can be chosen as the p l a n e wave f= CLom^ and f o r l i n e a r p o l a r i z a t i o n as 2^=(0,1,0,0). T h e r e f o r e the l a s e r f i e l d s have the u s u a l c h a r a c t e r i s t i c s F F = 0 > I F l = I 6 I F * F ¥ — O > B • B —O F ^ i y = 0 > t-K = B K - O We choose the l a b frame i n i t i a l c o n d i t i o n s x(0)=v(0)=0, 11 then I M 0 ) = ( 1 , 0 ) . The L o r e n t z e q u a t i o n I I . 1 becomes a) i £ = LUV Art $ U, C) = O I I .2 where V = i s a parameter g i v i n g the s t r e n g t h of the i n t e r a c t i o n r e l a t i v e t o the mass of the e l e c t r o n . Comparison of I I . 2 a ) and d) g i v e s 0°= U 3 i e . 0° - U 3 =C , a c o n s t a n t , which e q u a l s 1 from i n i t i a l c o n d i t i o n s . S u b s t i t u t i n g t h i s and Ue + U2 = U°-U* i n t o I I . 2 b a l l o w s i t to i n t e g r a t e d . W ith U1 now, e q u a t i o n s a) and d) can be i n t e g r a t e d . U s i n g the i n i t i a l c o n d i t i o n s I I . 2 becomes ex.) U ° = /av~7uut + 7 b) U' ~ - y cur 2. The 4-momentum i s I I .3 I n s p e c t i o n of shows t h a t i t can be w r i t t e n as where U (00 ) i s a c o n s t a n t depending on i n i t i a l c o n d i t i o n s , ^ A i s a component p a r a l l e l t o A, and c = - § M a 2 j^y * s a n o s c i l l a t o r y p a r t p e r p e n d i c u l a r t o A . T h i s d e c o m p o s i t i o n i s 12 unique s i n c e the i n i t i a l c o n d i t i o n s of the p a r t i c l e , and the d i r e c t i o n s of p o l a r i z a t i o n and p r o p a g a t i o n of the l a s e r f l u x , a r e a l l f i x e d i n p u t p a r a m e t e r s . Thus the a c t i o n = -rv» v(«)x + fdL. (eA-u+g& tf) I I .4 T h i s i s a u s e f u l e x p r e s s i o n t o compare t o the quantum m e c h a n i c a l s o l u t i o n t o come. To b e t t e r v i s u a l i z e t h e m o t i o n , P t h a t i s , i n the l a b o r a t o r y frame, we must i n t e g r a t e P from I I . 3 e over the l a b frame time t . 11.3 can be used t o s o l v e f o r f i n terms o f t . To w i t : • m l x - ± . ^ z u j z H ? w I I .5 if. a. »i2 By i n s p e c t i o n we see t h a t -t - ^ X i s an asymptote of I I . 5 and i n f a c t t h a t the two i n t e r s e c t t w i c e per p e r i o d w i t h p e r i o d Vu> f o r t , or p e r i o d ^jp- -jjj f o r -i . Thus t i s w e l l a p p r o x i m a t e d by 13 F i g u r e 2 d e p i c t s t h i s . rc/u> F i g u r e 2 - LAB TIME AND PROPER TIME We then n o t i c e t h a t momentum from I I . 3 has h a l f the fr e q u e n c y of the f i g u r e and thus i n t e g r a t i o n s over i n t e g r a l numbers of p e r i o d s w . r . t . T or t a r e e q u i v a l e n t f o r f i n d i n g the P R M S . 'RMS + 1£ 1 1 I I .6 One can check t h a t PfnS =m In any frame o t h e r than t h a t i n which the p a r t i c l e s t a r t s a t r e s t we have K - ' T P , - / 8 ' P . ) I I .7 14 where t=(1-/i,%) ,and /3'=^(0), the i n i t i a l v e l o c i t y of the e l e c t r o n . Thus i n any frame the motion of the e l e c t r o n i s a c o n s t a n t p l u s o s c i l l a t i n g terms. Though we have not i n c l u d e d r e r a d i a t i o n e f f e c t s , the e l e c t r o n a c h i e v e s a q u a s i s t a t i o n a r y momentum s t a t e . T h i s i s a t t r i b u t a b l e t o the o s c i l l a t o r y n a t u r e of the L o r e n t z f o r c e i n the z d i r e c t i o n , or one may r e g a r d t h e onset of the e.m. f i e l d as c o n t r i b u t i n g a net i m p u l s e t o the e l e c t r o n i n the z d i r e c t i o n . Our a p p r o x i m a t i o n of not i n c l u d i n g e l e c t r o n f i e l d e f f e c t s i s v i n d i c a t e d by comparing the magnitudes of the r a d i a t i o n damping f o r c e t o the a c t i n g e x t e r n a l f o r c e . The r a d i a t i o n damping f o r c e from energy c o n s e r v a t i o n c o n s i d e r a t i o n s 3 i n the l a b c o o r d i n a t e s i s The a c t i n g f o r c e i s the L o r e n t z f o r c e FL ~ wv The r a t i o of the two i s of o r d e r which i n our u n i t s i s about 2X10" 6/137 or 1 0 " 8 . To r e l a t e t h e s e c a l c u l a t i o n s t o r e a l consequences we must r e l a t e the e x p e r i m e n t a l l a s e r power t o t h e parameters h e r e . To ensure c o n f i d e n c e , the P o y n t i n g v e c t o r t h a t has been used w i t h g r e a t e x p e r i m e n t a l s u c c e s s " t o r e p r e s e n t the energy f l u x of the e.m. f i e l d s i s r e d e r i v e d i n the f o l l o w i n g c o m b i n a t i o n of n a t u r a l and G a u s s i a n u n i t s . We 3 J a c k s o n , r e f . 16, p784 * See Feynman L e c t u r e s , r e f . 11, I I 27-4 " A m b i g u i t i e s of the F i e l d Energy". 15 have a l r e a d y used fi=1=c t o r e f l e c t the freedom of c h o i c e of u n i t s . I n c l u d i n g s o u r c e s i n Ma x w e l l ' s e q u a t i o n s , and c h o o s i n g the Coulomb's law and Ampere's law c o n s t a n t s a p p r o p i a t e l y g i v e s the M a x w e l l ' s e q u a t i o n s i n the form b) P x B - m - HIT J at I I .8 The r a t e t h a t work i s done on a charge e when i t responds t o a f i e l d E w i t h a v e l o c i t y v i s P=E«ev=E«J C o n v e n t i o n a l l y J i s r e p l a c e d from I I . 8 b £ = £ • ? - - f c f J Then J J • Y > R i s i d e n t i f i e d as the r a t e of l o s s of energy from the e.m. f i e l d and V'(ExB)/4tr as the flo w of energy out of a p o i n t . I f 5=0 then one says t h a t S=fixB/4 TT i s the energy f l u x of the e.m. f i e l d . As mentioned e a r l i e r the t r u t h of t h i s a s s e r t i o n i s i t s e x p e r i m e n t a l s u c c e s s , but our purposes have been s e r v e d as we now have P o y n t m g ' s v e c t o r , S, i n our u n i t s . Time a v e r a g i n g the magnitude of S u s i n g the p r e v i o u s d e f i n i t i o n s of the f i e l d t e n s o r g i v e s I = < i s ) > - c l x u j x I ? r r 16 For a l a s e r of I=10 1 8W/cm 2, and 0> =2eV, we have Thus , , and V = . S3© From 11.6 and f o r V =1 E \ — 7.07 /TO FC«MS I I .9 R e c a l l t h a t the mass of the e l e c t r o n i s measured i n n a t u r a l u n i t s , Mev. 2.2 V o l k o v S o l u t i o n G o v e r n i n g E q u a t i o n The o r i g i n a l c o n c e p t i o n was t o use the wave f u n c t i o n of an e l e c t r o n i n a p l a n e e l e c t r o m a g n e t i c f i e l d i n the b e t a decay t r a n s i t i o n a m p l i t u d e . T h i s wave f u n c t i o n i s the V o l k o v s o l u t i o n ( r e f . 21, 1935) t o the D i r a c e q u a t i o n w i t h a m i n i m a l c o u p l i n g i n t e r a c t i o n . I t i s now r e d e r i v e d and then compared t o the c l a s s i c a l s o l u t i o n t o see t h a t i t i s r e a s o n a b l e . The H a m i l t o n i a n s of c l a s s i c a l mechanics i n v o l v e c a n o n i c a l , or s t a n d a r d , v a r i a b l e s which a r e r e p l a c e d by c a n o n i c a l o p e r a t o r s i n the s u c c e s i o n t o quantum mechanics. O r d i n a r y quantum mechanics uses these o p e r a t o r H a m i l t o n i a n s as the g e n e r a t o r s of time t r a n s l a t i o n s ( S c h r o d i n g e r ' s e q u a t i o n ) . I n c o r p o r a t i n g s p e c i a l r e l a t i v i t y (by f i n d i n g L o r e n t z c o v a r i a n t e q u a t i o n s ) i n t o quantum mechanics was a c c o m p l i s h e d by u s i n g r e l a t i v i s t i c H a m i l t o n i a n s . O r i g i n a l l y proposed t o overcome problems of i n t e r p r e t a t i o n r e l a t e d t o 17 the K l e i n - G o r d o n H a m i l t o n i a n or i t s square r o o t , the D i r a c H a m i l t o n i a n was found t o i m p l y the f e r m i o n s p i n - s t a t i s t i c s c o n n e c t i o n ( r e f . 5 ) . To d e s c r i b e a c h a r g e d p a r t i c l e i n an e x t e r n a l e l e c t r o m a g n e t i c f i e l d , c l a s s i c a l c o r r e s p o n d e n c e can be i n v o k e d t o j u s t i f y the use of the c o n j u g a t e momentum p-eA i n p l a c e of p i n t h e f i r s t o r d e r e q u a t i o n , the D i r a c e q u a t i o n . The same r e s u l t can be found from the c a n o n i c a l q u a n t i z a t i o n p r o c e d u r e u t i l i z i n g a L a g r a n g i a n . The L a g r a n g i a n i s formed from a term c a l c u l a t e d t o l e a d t o the D i r a c e q u a t i o n f o r t h e e l e c t r o n , a term t o d e s c r i b e the e l e c t r o m a g n e t i c f i e l d , and an i n t e r a c t i o n term. As i n the c l a s s i c a l case the c h a r g e d p a r t i c l e ' s own f i e l d may be n e g l e c t e d . The L a g r a n g i a n d e n s i t y i s £ = (?(*) (p'- vn) Y(x) - -if F V J - e. #<y) H>(x) and the E u l e r Lagrange e q u a t i o n s of motion i n v o l v i n g the p a r t i c l e are 11.10 . ( ^ * , A denotes o p e r a t o r s , and 1 i s i m p l i c i t ) F u r t h e r j u s t i f i c a t i o n ( i f i t i s needed) f o r t h i s g o v e r n i n g e q u a t i o n i s t h a t i t i s r e q u i r e d by the n e c e s s i t y of i n v a r i a n c e of the D i r a c e q u a t i o n under l o c a l gauge t r a n s f o r m a t i o n s . These t r a n s f o r m a t i o n s i n v o l v e the a d d i t i o n of the g r a d i e n t of a f u n c t i o n ^ 0 ( x ) t o the v e c t o r p o t e n t i a l and the s i m u l a t a n e o u s m u l t i p l i c a t i o n of the wave f u n c t i o n by a phase f a c t o r e whose argument i s t h a t gauge f u n c t i o n , p -eA 18 i s the c o v a r i a n t d e r i v a t i v e f o r t h i s group of t r a n s f o r m a t i o n s . S o l u t i o n A f t e r the models of Landau and L i f s h i t z and I t z y k s o n and Zuber we s o l v e f o r Y by f i r s t l o o k i n g f o r a second o r d e r e q u a t i o n . To t h i s end we m u l t i p l y by $-e# +m and f i n d I l i ^  e AA X* ) l - 1 V - O y^Jr^has a s y m e t r i c p a r t G^" and an a n t i s y m e t r i c p a r t ( e q . A . 6 ) . p r o j e c t s t o the a n t i s y m m e t r i c subspace of i t s range. That a n t i s y m m e t r i c p a r t of the above o p e r a t o r i s — • • » and i s an a d d i t i o n a l term t o t h a t which r e s u l t s from the m i n i m a l s u b s t i t u t i o n i n t o the K l e i n - G o r d o n e q u a t i o n . I t r e p r e s e n t s a s p i n i n t e r a c t i o n w i t h the e.m. f i e l d . Thus we have 11.11 Now, as i n the a n a l y s i s of the c l a s s i c a l c a s e , we choose an e l e c t r o m a g n e t i c p l a n e p o l a r i z e d p l a n e wave 19 where , The s p i n term becomes 2 U s i n g A.6 and 7 t h i s can be r e a r r a n g e d t o and s i n c e t h i s i s The momentum term (i e A)3" becomes s i n c e we can choose the gauge h^fi^is-O ( L o r e n t z gauge). The D i r a c e q u a t i o n 11.11 has now been t r a n s f o r m e d i n t o l-n-ZCeflb -t *M 7-w 2-c'e )? #'IV ^ O 11.12 and we can l o o k f o r p l a n e wave s o l u t i o n s of the form 11.13 S i n c e t h e $ = kx dependence i s c o n t a i n e d i n Q , which i s y e t t o be d e t e r m i n e d , we a r e f r e e t o e x t r a c t f a c t o r s of f from <p . Thus we may use a r e d e f i n e d c o n s t a n t v e c t o r p-z. yO'+Ak t h a t l i e s on the mass s h e l l : p*-=tryi z 20 i n 11.13 Upon s u b s t i t u t i o n of 11.13 i n t o 11.12 we f i n d the f i r s t term i s -Lp» ecf* f'6) ^  -f cc*'*f*x The second term i s - If. A-f y I 11.14 11.15 Thus 11.12 becomes 11.16 U s i n g p 2=m 2 and d i v i d i n g out the phase y i e l d s T h i s can be i n t e g r a t e d t o g i v e 11.17 21 11.18 where Qc i s an i n t e g r a t i o n c o n s t a n t . U i s a D i r a c s p i n o r as can be seen by s u b s t i t u t i n g i t back i n t o the g o v e r n i n g e q u a t i o n . F i r s t , eq. A.2 g i v e s t h a t ft? tfrO f o r r Zl and we can r e w r i t e the second e x p o n e n t i a l of (p as Thus the V o l k o v s o l u t i o n i s 11.19 where S has been so named because i t i s the c l a s s i c a l a c t i o n (see s e c t i o n 1.1). The f a c t o r i n b r a c k e t s and U r e s u l t s from the s p i n i n t e r a c t i o n term. The r e s t would have r e s u l t e d from the use of the K l e i n - G o r d o n e q u a t i o n from the o u t s e t . N o r m a l i z a t i o n , (f>0 = 1 , i s found by l e t t i n g a-> 0 l e a d t o the p l a n e wave s o l u t i o n . The f a c t o r e t S has the e f f e c t of g i v i n g the most c o n s t r u c t i v e p r o b a b i l i t y a m p l i t u d e s u p e r p o s i t i o n s t o tho s e t r a j e c t o r i e s f o r which 8 S=0 . That i s , t h e c l a s s i c a l t r a j e c t o r y i s the most l i k e l y one, as e x p e c t e d . C h e c k i n g now f o r t h e c o n d i t i o n s on U we have 22 D r o p p i n g terms w i t h y/=0 and u s i n g A.1,2,3 t o commute 1+ xp.n t o fche l e f t a l l o w s the c o n t i n u a t i o n which reduces t o I I .20 Thus «E>vF=0 i m p l i e s t h a t (j0-m)l> =0 , i . e . t h a t U i s a D i r a c momentum e i g e n s p i n o r . l t i s n o r m a l i z e d the same way ( r e f . 5 ) . These r e s u l t s a r e c o r r o b o r a t e d by l e t t i n g A-* 0 . The D i r a c e q u a t i o n i s r e c o v e r e d ; p l a n e wave s o l u t i o n s a r e r e c o v e r e d ; and n o r m a l i z e d D i r a c momentum e i g e n s p i n o r s a r e a l s o r e c o v e r e d . Thus the D i r a c e q u a t i o n i n c l u d i n g 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 s has been s o l v e d i n c l o s e d form. U s u a l l y the s t a t e of a system of u n p e r t u r b e d H a m i l t o n i a n s e v o l v e s i n H i l b e r t space c o v e r i n g a p a r a m e t e r i z e d t r a j e c t o r y when i n t e r a c t i o n s a r e i n c l u d e d i n t h e H a m i l t o n i a n . Our s o l u t i o n 23 which l o o k s a s y m p t o t i c a l l y 5 l i k e a momentum e i g e n s t a t e i s such a t r a j e c t o r y W r i t i n g our D i r a c e q u a t i o n as ( t ' J e -W)V=0 shows t h a t we have not found the s t a t i o n a r y s t a t e s ( e i g e n s t a t e s ) of : we r e a d i l y see t h a t a r e e i g e n s t a t e s of pt , p^ and p^  +p ? w i t h e i g e n v a l u e s p^  , p^ and P e + P , r e s p e c t i v e l y . Each t r a j e c t o r y l f f > 4 (x)= VVCL has a c h a r a c t e r i s t i c x and y momentum and f o r each the d i f f e r e n c e of energy and z-momentum i s a c o n s t a n t . However energy and z momentum do v a r y around average v a l u e s , as seen f o r the c l a s s i c a l m o t i o n . To compare w i t h the c a l c u l a t i o n s of Becker et a l a c i r c u l a r l y p o l a r i z e d l a s e r e.m. f i e l d i s used. T h i s i s a l i n e a r c o m b i n a t i o n of p l a n e p o l a r i z e d p l a n e wave s o l u t i o n s t o the wave e q u a t i o n s , t h a t d i f f e r i n t h e i r phase by a c o n s t a n t Hf/2. We have - a. ( <=>i * i , O ) , <T= ± 1 AZ = - a * 11.21 where i s t n e a z i m u t h a l a n g l e of p. Thus 5 The d i f f e r e n c e of the f u n c t i o n s d e c r e a s e s f a s t e r than any power f u n c t i o n as we l o o k t o i n f i n i t y f o r the argument. 24 where p=p + J^, K k . p x i s the component of momentum p e r p e n d i c u l a r t o k (and t o k ) ; p.k=p„u> i s the p a r a l l e l component. I t can be seen t h a t ^ i n v o l v e s etS , t h a t i s , s i n u s o i d a l v a r i a t i o n s m o d i f i e d by t r i g o n o m e t r i c f u n c t i o n s i n the f r e q u e n c y . T h i s i s r e m i n i s c e n t of B e s s e l f u n c t i o n s . In f a c t e has a F o u r i e r B e s s e l s e r i e s 6 ^ c f c ^ C f - T V ) _ ^ ecr»(r-^«/) J (?) A--OB In t h i s case z = v ^ t - ^ i s a d i m e n s i o n l e s s parameter dependent on the l a s e r s t r e n g t h and the p e r p e n d i c u l a r e" r e s p o n s e , and we have n =• - oc I I .22 The summation index n can now be i n t e r p r e t e d as the number of photons of energy oj t h a t have been e m i t t e d or absorbed by the e l e c t r o n w i t h a p r o b a b i l i t y a m p l i t u d e t h a t i s p r o p o r t i o n a l t o J n ( z ) . As w e l l t h e x dependence i s amenable t o i n t e g r a t i o n when the t r a n s i t i o n a m p l i t u d e i s c a l c u l a t e d l a t e r . The dependence of % ( L on the f a c t o r 1+ • • ^ can be is removed t o an exponent as f o l l o w s L e t •: then 1 + g ¥9 _ 1 + ea. . fa:(^/^>^^^f ) 6 Duff and N a y l o r , r e f . 9, p300 25 and s i n c e . ^ — A * " 7 we have I I .23 T h i s form of the V o l k o v s o l u t i o n agrees w i t h Becker e_t a l ( r e f . 2, e q u a t i o n s 2,3) A m o d i f i c a t i o n u s e f u l f o r a l a t e r argument i s t o d e f i n e and note t h a t I I .24 I I .24 a.) a l s o t h a t , as 4 - v e c t o r s then I I .25 and 26 v „ > ) = r \tf t° n.26 As a check we see a g a i n t h a t a. =0 causes the arguments of the B e s s l f u n c t i o n s t o v a n i s h . Because of A.6 and 7, o n l y 3^(p)=o,and the p l a n e waves are found a g a i n . 27 I I I . INDUCED BETA DECAY 3.1 Beta Decay H a m i l t o n i a n The L a g r a n g i a n f o r m u l a t i o n of c l a s s i c a l f i e l d mechanics r e q u i r e s t h a t the c o o r d i n a t e s Vf) and momentum Ttif) be r e a l or complex v a l u e d f i e l d s d e f i n e d on t h e f o u r parameter space known as s p a c e - t i m e . The mathematics t h a t d e v e l o p e d on t h i s s u b j e c t was f i r s t a p p l i e d t o wave phenomena, the p r o p a g a t i o n of d i s t u r b a n c e s i n e l a s t i c media. L a t e r i t was found t o p l a y an i m p o r t a n t r o l e i n some d r a m a t i c achievements of t h i s c e n t u r y , the t h e o r i e s of r e l a t i v i t y and quantum mechanics. In s p e c i a l ' r e l a t i v i t y the f i n i t e v e l o c i t y f o r the p r o p a g a t i o n of s i g n a l s s u g g e s t s t h a t f o r c e s o p e r a t e l o c a l l y as c o n t a c t phenomena between the n e i g h b o r i n g p o i n t s of space; the f i e l d f o r m a l i s m f o r f o r c e s embodies t h i s c o n c e p t . In quantum mechanics a complex v a l u e d p r o b a b i l i t y a m p l i t u d e f i e l d was used i n a new c o r r e s p o n d e n c e between the o b j e c t s of Mathematics and the o b j e c t s of P h y s i c s t o s u c c e s s f u l l y account f o r many w a v e - l i k e p r o p e r t i e s of m a t t e r . T h i s new c o r r e l a t i o n was drawn from the o b j e c t s of the L a g r a n g i a n f o r m a l i s m i n P h y s i c s and H i l b e r t space i n M a t h e m a t i c s . S p e c i a l r e l a t i v i t y and quantum mechanics have been merged i n R e l a t i v i s t i c Quantum Mechanics and Quantum F i e l d Theory. In the l a t t e r the wave p a r t i c l e d u a l t y a f f o r d e d by quantum mechanics t o m a t t e r i s complemented by q u a n t i z i n g the f o r c e f i e l d , g i v i n g i t p a r t i c l e c h a r a c t e r i s t i c s . For example the quanta of the Coulomb f o r c e a r e photons and the 28 quanta of the Weak f o r c e are the v e c t o r bosons. In QFT the c o o r d i n a t e s W(c) and momenta 7T(ty a r e o p e r a t o r v a l u e d f i e l d s t h a t s a t i s f y c a n o n i c a l commutation r e l a t i o n s ( f o r bosons and a n t i c o m m u t a t i o n s r e l a t i o n s f o r f e r m i o n s ) . S t a t e s of a system a r e r e p r s e n t e d by r a y s i n H i l b e r t space. They can be d e f i n e d u s i n g the e i g e n v a l u e s of complete s e t s of commuting o b s e r v a b l e s (or "good" quantum numbers i f the H a m i l t o n i a n i s i n t h i s s e t ) as i n d i c e s . C o n v e n i e n t bases f o r the H i l b e r t space a r e o f t e n found i n the e i g e n v e c t o r s of the p o s i t i o n X, momentum P, or number N, o p e r a t o r s . I t s h o u l d be remembered t h a t H i l b e r t space can o f t e n be a t e n s o r p r o d u c t of subspaces on which t h e s e o p e r a t o r s are e l e m e n t a r y . P r e v i o u s e x p e r i e n c e w i t h quantum mechanics l e a d s us t o w r i t e a quantum f i e l d t h e o r y H a m i l t o n i a n both i n terms of the c o o r d i n a t e f i e l d s <p(?) and Tt(x) , and i n terms of the c r e a t i o n and d e s t r u c t i o n o p e r a t o r s of quanta of the f i e l d . T h i s t e c h n i q u e has been u s e f u l i n the o s c i l l a t o r and r o t o r problems 7 . For the s i m p l e case of a r e a l K.G. f i e l d , the e x p r e s s i o n s are W- i. f^*L~ *"2<P*W + V Z PfcJ -+Trl(x) ) I I I . 1 I I I . 2 where k»k=m 2 and k e=w . From t h e s e and the c a n o n i c a l 7 Cohen-Tannoudji, r e f . 7, s e c t i o n s VB,and VIA. 29 commutations, we can d e r i v e the F o u r i e r t r a n s f o r m e x p r e s s i o n c - b * f i t s ? (" °> °:) L • ' 111. 3 These r e l a t i o n s a re o f t e n p r e s e n t e d i n a d i f f e r e n t o r d e r , but from t h i s p r e s e n t a t i o n , i t s h o u l d be c l e a r t h a t i f i n s t e a d of a K.G. H a m i l t o n i a n i n I I I . 1 , we had s t a r t e d w i t h the D i r a c H a m i l t o n i a n w i t h an e.m. i n t e r a c t i o n , as we d i d f o r t he V o l k o v s o l u t i o n , then i n the e x p r e s s i o n I I I . 3 we would f i n d the V o l k o v s o l u t i o n s i n p l a c e of the i k-x D ' A l e m b e r t i a n e i g e n f u n c t i o n s , e , and an a d d i t i o n a l s p i n sum g i v i n g 8 I I I .4 To c a l c u l a t e a b e t a decay t r a n s i t i o n a m p l i t u d e , terms l e a d i n g t o the g o v e r n i n g e q u a t i o n s f o r the p a r t i c l e s i n v o l v e d a r e i n c l u d e d i n the L a g r a n g i a n . S i n c e n u c l e a r decay i s mo d e l l e d by the decay of a n u c l e o n w i t h i n the n u c l e u s , the p a r t i c l e s i n v o l v e d a r e p r o t o n s , n e u t r o n s and n e u t r i n o s . These a r e c o u p l e d t o the quanta of t h e weak i n t e r a c t i o n , the v e c t o r bosons; as w e l l the e l e c t r i c a l l y c h a r g e d p a r t i c l e s a r e c o u p l e d t o the e x t e r n a l e.m. f i e l d . B oth p r o t o n s and e l e c t r o n s s h o u l d c o u p l e t o the e x t e r n a l 8 F o r m a l l y , the F o u r i e r t r a n s f o r m i s a d i s t r i b u t i o n or f u n c t i o n a l on the v e c t o r space of " t e s t " f u n c t i o n s . I t i s a map from the v e c t o r space t o the " f i e l d " named i n the v e c t o r space axioms ( u s u a l l y the complex numbers). Here the map i s between o p e r a t o r s p a c e s , so t h e s e a r e not u s u a l F o u r i e r t r a n s f o r m s , though t h i s i s what they a r e c a l l e d . 30 f i e l d , but t h e r e s p o n s e , or energy t r a n s f e r f o r the p r o t o n i s e x p e c t e d t o be reduced i n p r o p o r t i o n t o the mass of the p a r t i c l e by a n a l o g y w i t h c o l l i s i o n s i n c l a s s i c a l mechanics. As w e l l the p r o t o n remains i n the i s o l a t e d environment of the n u c l e u s . T h e r e f o r e , as a f i r s t a p p r o x i m a t i o n , the term J«A i s added t o t h e L a g r a n g i a n o n l y f o r t h e e l e c t r o n . T h i s c o u p l e d c u r r e n t form of the i n t e r a c t i o n has been p r e v i o u s l y d i s c u s s e d . I t i s not ammenable t o p e r t u r b a t i v e t e c h n i q u e s s i n c e i n the p r e s e n t a p p l i c a t i o n A i s l a r g e and h i g h e r o r d e r p r o c e s s e s may not be n e g l e c t e d . However, V o l k o v c o o r d i n a t e f i e l d s a r e the e x a c t s o l u t i o n s f o r D i r a c p a r t i c l e s and e l e c t r o m a g n e t i s m . So the weak i n t e r a c t i o n terms a l o n e may be r e g a r d e d as a p e r t u r b a t i o n . Thus we expand /3 decay t r a n s i t i o n a m p l i t u d e s i n power s e r i e s f o r the weak c o u p l i n g c o n s t a n t , g. For each term of a p a r t i c u l a r power, Feynman diagrams w i t h a f i x e d number of weak i n t e r a c t i o n v e r t i c e s can be used t o p i c t u r e mechanisms by which the decay i s e f f e c t e d . In p e r t u r b a t i v e e x p a n s i o n s the a m p l i t u d e f o r each p r o c e s s i s added i n a c c o r d a n c e w i t h the p r i n c i p l e of s u p e r p o s i t i o n . Each such a m p l i t u d e i s the p r o d u c t of a m p l i t u d e s f o r the components of the Feynman diagram ( i n momentum r e p r e s e n t a t i o n 9 ) i n a n a l o g y t o the 9 In p o s i t i o n r e p r e s e n t a t i o n , t h i s p r o d u c t t a k e s the form of a c o n v o l u t i o n of Green's f u n c t i o n s f o r the p r o p a g a t i o n of i m p u l s e s . T h i s i s noteworthy because i n making an a l g e b r a out of a v e c t o r space of f u n c t i o n a l s , the c o n v o l u t i o n p l a y s the r o l e of v e c t o r p r o d u c t . ( t h e 8 f u n c t i o n a l i s the i d e n t i t y ; the Green's f u n c t i o n i s the k e r n e l of the Green's f u n c t i o n a l which i s the i n v e r s e of a d i f f e r e n t i a l o p e r a t o r . ) 31 m u l t i p l i c a t i o n of c o n d i t i o n a l p r o b a b i l i t i e s . The a m p l i t u d e f o r a weak i n t e r a c t i o n boson t o prop a g a t e i s t h e " i n v e r s e " of the K.G. o p e r a t o r . S i n c e the bosons have a mass of 84 GeV ( r e f . 6)and and the e n e r g i e s t h a t c o n c e r n us a r e o n l y Kev, the K.G. o p e r a t o r and i t s i n v e r s e a r e dominated by the c o n s t a n t mass term 9 T h i s c o n s t a n t can be absorbed i n t o a r e d e f i n e d c o u p l i n g c o n s t a n t ; then the t r a n s i t i o n a m p l i t u d e i n v o l v e s a p o i n t c o u p l i n g of the f i e l d s and t a k e s on the form of the Fermi t h e o r y of be t a decay proposed i n 1934 ( r e f . 6 ) . N u c l e a r ^ decay i s then d e s c r i b e d by the decay of a nu c l e o n w i t h i n a n u c l e u s . The b i n d i n g energy d i f f e r e n c e s between the n u c l e o n and p r o d u c t s a l o n g w i t h a n g u l a r momentum c o n s i d e r a t i o n s have l a r g e l y a c c o u n t e d f o r s p e c t r a v a r i a t i o n s and the wide range of /$ decay h a l f l i v e s . T h i s s u c c e s s i s i n c l u d e d i n the " u n i v e r s a l i t y " of the t h e o r y . To l e a r n the b a s i c t e c h n i q u e s of /3 decay c a l c u l a t i o n s , the decay r a t e f o r a n e u t r o n was c a l c u l a t e d w i t h o u t Coulomb c o r r e c t i o n s . From p u b l i s h e d d a t a ( r e f . 8) of p a r t i c l e masses and weak c o u p l i n g c o n s t a n t s the h a l f l i f e was found t o be about 640 s e c o n d s ; i n agreement w i t h e x p e r i m e n t a l r e s u l t s ( r e f . 6, p l 3 4 ) . The c a l c u l a t i o n and r e l e v a n t weak i n t e r a c t i o n t h e o r y can be found i n many s o u r c e s (eg: r e f . 8) and i s o n l y o u t l i n e d h e r e . F i r s t we need t o d e v e l o p f u r t h e r . Our p r i m a r y i n t e r e s t w i l l be i n 3H decay which i s a n e u t r o n decay. To d e s c r i b e n e u t r o n decay 32 n—»p ++e~+ V the l o w e s t term of the p e r t u r b a t i o n e x p a n s i o n i s 0 and we a r e l e f t w i t h the f i r s t o r d e r term p i c t u r e d as and w r i t t e n as I I I .5 where ^ i s a p r o d u c t of the f i e l d o p e r a t e r s f o r the p a r t i c l e s i n v o l v e d . As such i t a c t u a l l y i n v o l v e s ®p-'4e'%' 9* 1 0 for - the p r o d u c t i o n of the p r o t o n , e l e c t r o n and a n t i n e u t r i n o and the d e s t r u c t i o n of a n e u t r o n . These are o p e r a t o r s on a H i l b e r t space where the s p i n o r and charge c h a r a c t e r s of these p a r t i c l e s a r e e x p r e s s e d i n 4 components. That T^j i s h e r m i t i a n e n s u r e s the u n i t a r i t y of the s c a t t e r i n g m a t r i x S and the c o n s e r v a t i o n of p r o b a b i l i t y d e n s i t y . G e n e r a l c o n s i d e r a t i o n s w i l l p r o v i d e some i n f o r m a t i o n about /$ decay, but f u r t h e r d e t a i l can o n l y be o b t a i n e d from e x p e r i m e n t . We expect t h a t the i n t e r a c t i o n H a m i l t o n i o n t h a t governs r e f e r e n c e frame independent p r o c e s s e s s h o u l d be ) L o r e n t z s c a l a r s , and w i t h our f o u r f i e l d s w r i t t e n each w i t h 4 components, t h e r e would seem t o be many ways t o c o n s t r u c t t h e s e i n v a r i a n t s . By a n a l o g y w i t h the c o u p l e d c u r r e n t 1 0 Another common p r a c t i c e i s t o denote the f i e l d s and the p a r t i c l e by the same symbol. 33 i n t e r a c t i o n of e l e c t r o m a g n e t i s m J-A, we c o u l d c o n s t r u c t l e p t o n and hadron p r o b a b i l t y d e n s i t y c u r r e n t s and c o u p l e them, or we c o u l d g e n e r a l i z e , c r e a t i n g c o u p l i n g s of d i f f e r e n t g e o m e t r i c a l o b j e c t s and f o r m i n g a l i n e a r combinat i o n . S i n c e a f t e r decay the r e m a i n i n g heavy p a r t i c l e i s n o n r e l a t i v i s t i c , we t a k e the n o n r e l a t i v i s t i c l i m i t f o r the nu c l e o n D i r a c s p i n o r s c o n t a i n e d i n these terms. The p s e u d o s c a l a r c o u p l i n g v a n i s h e s ; the a x i a l v e c t o r t o g e t h e r w i t h the v e c t o r , and the s c a l a r t o g e t h e r w i t h the t e n s o r each reduce t o and *V>«„ ( xfv"Xni C T <r' Xr)- l^r where q" =I l ) r l and l-v_A and Ls_ T a r e the l e p t o n c u r r e n t 4-v e c t o r c o f a c t o r s . In f u r t h e r a n a l y s i s some of t h e s e f a c t o r s c a n c e l so i t i s c o n v e n i e n t here t o g i v e them t h e s h o r t e r symbols used i n the l i t e r a t u r e ( eg: r e f . 5) and d e f i n e H = ( c , M n CA H&T ) i n . 6 The Hp and the ^ T terms a r e c a l l e d Fermi and Gamow-Teller c o u p l i n g r e s p e c t i v e l y . The c o r r e s p o n d i n g n u c l e a r s p i n changes a r e 0 and +1 or 0. For t h e s e n u c l e a r s p i n changes the e m i t t e d l e p t o n s p i n s 34 must oppose or a l i g n t o c o n s e r v e a n g u l a r momentum'. That the h e l i c i t i e s of the l e p t o n s h e l i c i t i e s a r e opposed i s c h a r a c t e r i s t i c of the V-A i n t e r a c t i o n . 1 1 The s p i n and h e l i c i t y d e termine the e m i t t e d l e p t o n momenta t o be c o r r e l a t e d i n the a l i g n e d sense or the a n t i a l l i g n e d sense r e s p e c t i v e l y f o r the two c o u p l i n g s . T h i s i s born out by /*-n e u t r i n o a n g u l a r c o r r e l a t i o n e x p e r i m e n t s and so the S-T terms may be d i s c a r d e d . The l a s t c o n s i d e r a t i o n f o r the form of i s t h a t the r e l a t i v i s t i c l i m i t of the l e p t o n D i r a c s p i n o r s a r e P a u l i h e l i c i t y 2 - s p i n o r s . The m i x t u r e of v e c t o r and a x i a l v e c t o r p a r t s f o r LV.A i s then I I I .7 i n o r d e r t h a t the c o r r e c t r e l a t i v i s t i c l i m i t of the h e l i c i t i e s be o b t a i n e d . The f a c t o r — X£ i s now u n d e r s t o o d as the p r o j e c t o r f o r r i g h t handed a n t i n e u t r i n o c h i r a l i t y . 3 . 2 Beta Decay Rate To f i n d t he d i f f e r e n t i a l decay r a t e t o a r e g i o n of phase space, we must m u l t i p l y the p r o b a b i l i t y d e n s i t y , which i s t he 'square' of t h e p r o b a b i l i t y a m p l i t u d e S, by the number of s t a t e s a c c e s i b l e , the d e n s i t y of phase space. To f i n d S 2 we f i r s t s u b s t i t u t e e x p r e s s i o n s I I I . 4 f o r the f i e l d s i n t o I I I . 5 . We can e a s i l y e v a l u a t e the momentum i n t e g r a l s and s p i n sums because of t h e d e l t a f u n c t i o n 1 T h i s i s shown a l g e b r a i c a l l y w i t h l i t t l e d i f f i c u l t y . 35 c o n t r i b u t i o n s from the i n n e r p r o d u c t s w i t h the c r e a t i o n and d e s t r u c t i o n o p e r a t o r s . The i n t e g r a l over the c o o r d i n a t e s x of the e x p o n e n t i a l f a c t o r s then r e s u l t s i n t h e energy momentum c o n s e r v a t i o n / f u n c t i o n . The f a c t o r s ffip/x are r e p l a c e d by ^ n t^ i e t r a n s i t i o n t o box n o r m a l i z a t i o n . The n u c l e o n masses have c a n c e l l e d and u s i n g I I I . 6 and I I I . 7 we have I I I .8 S q u a r i n g we f i n d ' 1 ( M l f = A tlM+LL* I I I . 9 MM* l o o k s l i k e a p r o j e c t o r f o r which a c l o s u r e - l i k e r e l a t i o n might e x i s t . S i n c e we a r e l o o k i n g f o r the u n p o l a r i z e d r a t e we sum an element of MM+ over i n i t i a l and f i n a l s p i n s t o f i n d (no sum on jx,V) ( M M ' ) I I I . 1 o I f i n s t e a d of P a u l i s p i n o r s we had the p o l a r i z a t i o n s t a t e s of a h i g h e r a n g u l a r momentum b a s i s i n the above e x p r e s s i o n , 1 2 P a r a p h r a s e d : the square of an i n n e r p r o d u c t i s the i n n e r p r o d u c t of a s q u a r e , and t h i s j u s t r e f l e c t s the c o m m u t a t i v i t y of the i n n e r p r o d u c t d e f i n e d i n the i n t r o d u c t i o n . 36 as would a r i s e i n a more g e n e r a l /i decay, then the summations would s t i l l g i v e t h a t MM+ i s d i a g o n a l and t h a t the (MM* ) ^ a r e e q u a l f o r ji=\,2,3. An argument from symetry would s u f f i c e f o r t h i s r e s u l t . An immediate consquence i s t h a t t h e r e i s no i n t e r f e r e n c e between Fermi and Gamow-Teller terms. The s p i n summed MM* i s not the i d e n t i t y because of the c o n s t a n t s and the reduced m a t r i x elements of the Wigner-E c k a r t theorem found a l o n g the d i a g o n a l . S i n c e MM* i s d i a g o n a l we need o n l y the d i a g o n a l elements of LL* i n I I I . 9 . S i n c e we have i n c l u d e d p r o j e c t o r s f o r the a n t i n e u t r i n o h e l i c i t y we here a l s o sum on p o l a r i z a t i o n s t o f i n d D i r a c p r o j e c t o r s . Then we l e t the n e u t r i n o mass approach 0, a s t a n d a r d t e c h n i q u e . A d i a g o n a l element of LL* i s de v e l o p e d t h e n , (no summation i m p l i e d a g a i n ) = 1(2. P^PS- erfe 6^) I I I . 11 P u t t i n g t h i s and I I I . 1 0 i n I I I . 9 , and t h e r e s u l t i n the square of I I I . 8 , and u s i n g (M^ T ) 2=^M| T g i v e s I I I . 12 37 I t i s now c o n v e n i e n t t o r e p l a c e the square of the £ f u n c t i o n by /•(()) / « ( ( P ) and then (2 t T ) " J o(0) by the s i z e of the 4-box V BT on which H i l b e r t space i s d e f i n e d (see appendix A ) . Then an i n t e g r a t i o n over the u n d e t e c t e d n u c l e a r r e c o i l 3 momentum i s e a s i l y performed e v a l u a t i n g the J 3 ( f p) . F u r t h e r s i m p l i f i c a t i o n i s a c h i e v e d by i n t e g r a t i n g over the unobserved momentum of the n e u t r i n o d 3 p w ~ LPVX olu) y c J y i Then the a n g u l a r i n t e g r a l e l i m i n a t e s the n e u t r i n o momentum dependence and c o n t r i b u t e s 4 IT. The r a d i a l i n t e g r a l e v a l u a t e s the l a s t S f u n c t i o n . C o l l e c t i n g t h e s e changes 111.12 i s now 6 I I I . 1 3 where U>„ i s the mass energy d i f f e r e n c e of the d e c a y i n g n u c l e u s , iov = Wa-We, and now a l l masses from n o r m a l i z a t i o n s and f i e l d o p e r a t o r s have c a n c e l l e d . The f a c t o r i n square b r a c k e t s i s c a l l e d t he n u c l e a r p a r t ( N P ) . The " m a r g i n a l p r o b a b i l i t y " of a decay w i t h 4-momentum Pfi , P^ =m2, i s the p r o d u c t of S 2 w i t h the d e n s i t y of s t a t e s i n t h e phase space. T h i s d e n s i t y i s f i n i t e i f the c o o r d i n a t e space i s d e l i m i t e d t o a box. I t i s Vg/(21T) 3 f o r each of the 3 f i n a l p a r t i c l e s and so a l l r e f e r e n c e t o a f i n i t e c o o r d i n a t e space c a n c e l s from the p r o b a b i l i t y e x p r e s s i o n 38 The decay p r o b a b i l i t y per u n i t time i s dw =dw'/T . I n t e g r a t i n g the d i r e c t i o n s of the e l e c t r o n s i m p l y c o n t r i b u t e s 4TT. The r a d i a l p a r t can be w r i t t e n as I I I . 1 5 g i v i n g i at x*- / * T i = — (OIG-U) 0) J W R ^ Ue oleue The t o t a l decay p r o b a b i l i t y i s The h a l f - l i f e i s ln2/w and when the n u m e r i c a l c o n s t a n t s a re i n s e r t e d and the s i m p l e Fermi and Gamow-Teller neu t r o n t r a n s i t i o n a m p l i t u d e s (from I I I . 1 0 ) a r e i n s e r t e d , the ne u t r o n decay h a l f l i f e i s r e c o v e r r e d ( t o about 1% a c c u r a c y ) . The e n e r g i e s of the l e p t o n s a re U)e e [m,ub] and 6 [0,w o-m]. 3.3 Induced Beta Decay" Rate The m o d i f i e d r a t e can be c a l c u l a t e d r a t h e r e a s i l y now t h a t the r a t e c a l c u l a t i o n has been s e t up as i n the l a s t s e c t i o n . U s i n g 11.23, S ( I I I . 8 ) i s m o d i f i e d t o where 3 9 I I I . 17 and if>= f j , t ^-hk+fj,-^ i n c l u d e s e x t r a terms from the V o l k o v s o l u t i o n . In s q u a r i n g S, eq A7 can be used t o show t h a t o n l y the d i a g o n a l terms of the double sum on n do not v a n i s h . As b e f o r e , the n u c l e a r s p i n s sums de t e r m i n e MM* t o be d i a g o n a l a l s o , and the t a s k i s t o e v a l u a t e Z I I I . 18 where jul i s not summed i n t h i s e x p r e s s i o n . U s i n g the i n t e g r a l we i n t e g r a t e the a n g u l a r p a r t of the n e u t r i n o momentum t o f i n d I I I . 1 9 S i n c e V n and Vn' have even numbers of K ' s , the term p r o p o r t i o n a l t o m v a n i s h e s . Commuting the y^'s towards each o t h e r g i v e s Here we see t h a t the Fermi and Gamow-Teller c o n t r i b u t i o n s remain i n t h e same p r o p o r t i o n s as f o r the no l a s e r c a s e . Now we have Tyt - tftr vy A jfe VjQ-t**) X* V„ 40 o r , r e i n s e r t i n g 11.23 f o r V h and c o l l e c t i n g s i m i l a r terms I I I . 2 0 The f i r s t term i s e a s i l y e v a l u a t e d u s i n g the s t a n d a r d t r a c e theorems as T, = H 3*ue I I I .21 The second term i s e v a l u a t e d by u s i n g t h e s e theorems and the a l g e b r a i c formula A.6 and r e c a l l i n g t h a t Y-k=Y*-k=0. We have The t r i p l e p r o d uct i s c y c l i c . R e c a l l i n g 11.24 f o r the complex p a r t s of Y g i v e s I I I .22 The t h i r d term i s f i r s t m o d i f i e d u s i n g s i n c e |?'ljr=0, t o f i n d 41 - ? O J ( - ^ . K Y * V - T ^  Y * * K - R - ^ £ * V * R ; But YxY* has o n l y a t h i r d component which i s m u l t i p l i e d by Thus Once a g a i n u s i n g 11.24 f o r Y g i v e s and Thus ^ * 111. 23 S u b s t i t u t i n g I I I . 2 1 t o 21 i n t o I I I . 2 0 g i v e s I I I . 2 4 T h i s a g r e e s w i t h John Hebron's T h e s i s ( r e f . 14, eq 4.3.8.) Now r e c a l l t h a t y i s the a z i m u t h a l a n g l e of the e" momentum. The v e c t o r ( uti f ; A*J* ¥ © )* i s j u s t t he d i r e c t i o n of the component of P p e r p e n d i c u l a r t o the l a s e r f l u x . Thus the y dependence i s a r t i f i c i a l 42 \ o I where & i s the p o l a r a n g l e . F u r t h e r n o t a t i o n a l c o n f o r m i t y and s i m p l i c i t y i s a c h i e v e d by d e f i n i n g energy measures as p r o p o r t i o n s of the e" mass and A£ and l a s e r s t r e n g t h parameter I t i s c o n v e n i e n t f o r the p r e s e n t p urposes t o average over l e f t and r i g h t c i r c u l a r p o l a r i z a t i o n s of the l a s e r whereupon the T dependent terms c a n c e l . We have Then u s i n g the B e s s e l f u n c t i o n i d e n t i t y A.5 and n o t i n g t h a t z= 5^- 6. (11.22) we have I I I .25 As i n the l a s t s e c t i o n the i n t e g r a l over the r e c o i l momentum e v a l u a t e s S3(i.p) i n S 2 from I I I . 1 4 . And t h e i n t e g r a l over the n e u t r i n o momentum's r a d i a l component e v a l u a t e s £ (iu>) . The f a c t o r g 2/81T f l a r i s e s from n o r m a l i z a t i o n s , the c o u p l i n g c o n s t a n t , and the n e u t r i n o a n g u l a r i n t e g r a l as i t d i d i n I I I . 14 . We have 43 i / A£V"«+» •* I I I . 2 6 where * P I I I . 2 7 and the & f u n c t i o n g a u r a n t e e s t h a t t ^ > 0 , a p h y s i c a l c o n d i t i o n . 44 IV. NUMERICAL DECAY RATE 4.1 Summary A co m p u t a t i o n was performed f o r the case of 3H decay w i t h the l a s e r c h a r a c t e r i s t i c s t h a t have a l r e a d y been used i n c h a p t e r 2, namely U) =2eV and l> = 1. The r a t i o of decay r a t e s w i t h and w i t h o u t the l a s e r was 1 .00+.01 . The e r r o r was i n t r o d u c e d m a i n l y by the use of a s y m p t o t i c f o r m u l a e f o r B e s s e l f u n c t i o n s . Other a p p r o x i m a t i o n s used t o s h o r t e n the co m p u t a t i o n were d e v i s e d t o not c o n t r i b u t e s i g n i f i c a n t l y t o the e r r o r . The c o m p u t a t i o n was a c c o m p l i s h e d on the U.B.C. computing c e n t e r ' s Amdahl computer (MTS o p e r a t i n g system) u s i n g UBC's c u r r e n t v e r s i o n of F o r t r a n . The D i s s p l a g r a p h i c s package s u p p o r t e d a t the c e n t e r was used t o p i c t u r e v a r i o u s s t a g e s of the c a l c u l a t i o n , namely the B e s s e l f u n c t i o n s of l a r g e argument and o r d e r , the summand of e x p r e s s i o n IV.10 and the d i f f e r e n t i a l decay r a t e . The program t h a t c a l c u l a t e s the d i f f e r e n t i a l decay r a t e and the t o t a l decay r a t e was then used t o e x p l o r e o t h e r l a s e r p a r a m e t e r s . The r e s u l t s a r e d i s p l a y e d i n t a b l e I I . Some t r e n d s were n o t i c e d t h a t c o u l d be v e r i f i e d by hand c a l c u l a t i o n s . For example i t was n o t i c e d t h a t the both the decay r a t e and the d i f f e r e n t i a l decay r a t e d i d not depend on the l a s e r f r e q u e n c y artd t h i s was s u b s e q u e n t l y v e r i f i e d a n a l y t i c a l l y from t h e f o r m u l a IV.10 . I t was a l s o n o t i c e d t h a t the number of photons absorbed, the range of n , was 45 independent of y and W and t h i s was a l s o v e r i f i e d d i r e c t l y . As w e l l the r a t i o of decay r a t e s w i t h and w i t h o u t the l a s e r was seen t o be independent of the l a s e r c h a r a c t e r i s t i c s , c o n s i s t e n t w i t h the p r o p o s i t i o n of no enhancement. The range of the n e u t r i n o energy was shown t o be u n a f f e c t e d c o m p u t a t i o n a l l y and a l g e b r a i c a l l y . The d i f f e r e n t i a l decay r a t e was seen t o be a smooth f e a t u r e l e s s f u n c t i o n of i t s p a r a m e t e r s . F i n a l l y the enhancement f a c t o r f o r b e t a deays o t h e r than 3H, c h a r a c t e r i z e d by o t h e r decay e n e r g i e s u)0 was seen t o be u n a f f e c t e d as summarized i n t a b l e I I . 4.2 K i n e m a t i c Domain The d i f f e r e n t i a l decay r a t e f o r an u n p o l a r i z e d l a s e r , o b t a i n e d by a v e r a g i n g l e f t and r i g h t c i r c u l a r p o l a r i z a t i o n e x p r e s s i o n s from I I I . 2 6 , i s IV. 1 where 3"„ = 7„fr) £ D as b e f o r e i s the energy a v a i l a b l e t o the decay p r o d u c t s from the n u c l e a r p r o c e s s . I t i s the d i f f e r e n c e of n u c l e a r 46 e n e r g i e s , so t h a t f o r 3H (as f o r the no l a s e r c a s e ) , €. = 1 .0364 . I n s p e c t i o n of z shows t h a t i t i s a v e r y l a r g e number due t o the r a t i o ^~ , r e p r e s e n t i n g a p p r o x i m a t e l y the number of photons absorbed. T h i s shows t h a t a s y m t o t i c formulae f o r the B e s s e l f u n c t i o n s w i l l be needed. From t h e s e formulae i t can be seen t h a t the B e s s e l f u n c t i o n s v a n i s h v e r y r a p i d l y f o r n>z , which e s t a b l i s h e s an upper l i m i t of n=z f o r the summation i n IV.1 . Above t h i s l i m i t t h e r e i s n e g l i b l e c o n t r i b u t i o n t o the sum; an e r r o r e s t i m a t e w i l l be p r e s e n t e d below. A l s o the H e a v i s i d e f u n c t i o n , g u a r a n t e e i n g t h a t the n e u t r i n o energy i s p o s i t i v e , e s t a b l i s h e s a minimum f o r the summat i o n IV.2 Thus n m l n i s a l s o a l a r g e number. For each e n t i r e summation t o be n o n n e g l i b l e we a l s o r e q u i r e nm,-. t o be l e s s than z , g i v i n g c o n s t r a i n t s on the k i n e m a t i c domains i n &n>e and € t h a t we need t o c o n s i d e r . We have from t h i s c o n d i t i o n Now s e t S i n c e n m J /, >0 , both s i d e s of the i n e q u a l i t y a r e p o s i t i v e and we may square both s i d e s t o f i n d 47 IV.3 S i n c e the q u a d r a t i c e x p r e s s i o n i s t h a t of an u p r i g h t p a r a b o l a , the range of cooQ i s between i t s r o o t s one e£ Afr-L)±vJ&^7UT"^3/((<,-z)liV*) For t h e s e r o o t s t o be r e a l the d i s c r i m i n a n t must be g r e a t e r than 0. A f t e r some a l g e b r a i c c a n c e l l a t i o n s we have f o r the d i s c r i m i n a n t IV.4 and £ l i e s between the r o o t s of t h i s i n v e r t e d p a r a b o l a £ e .0+ t fA) * W c £ * / U t f - 0 J i v . 5 P l o t s of t h e s e domains f o r v a r i o u s v a l u e s of y a r e i n f i g . 3. From t h e s e p l o t s some of the e f f e c t s of the l a s e r a r e a l r e a d y e v i d e n t . F o r low l a s e r i n t e n s i t i e s g i v e n by low V the a v a i l a b l e e n e r g i e s f o r the e l e c t r o n a r e s t i l l low and the decay i s s t i l l a lmost i s o t r o p i c . But f o r l a r g e r l a s e r i n t e n s i t i e s the decay becomes more p r o b a b l e i n the f o r w a r d d i r e c t i o n and the energy i n c r e a s e s i n magnitude and e x t e n t . The h i g h e r energy e l e c t r o n s a r e more c l o s e l y a l i g n e d w i t h the d i r e c t i o n of the l a s e r f l u x . The n e u t r i n o energy domain can a l s o be found. S u b s t i t u t i n g n<z i n I I I . 2 5 f o r £ y g i v e s 48 S i n c e the B e s s e l f u n c t i o n s are maximized f o r z«n t h i s upper l i m i t a l s o r e p r e s e n t s the most p r o b a b l e £ v . From the a n a l y s i s o u t l i n e d i n appendix C we f i n d as i n u n m o d i f i e d decay. To show c o n c l u s i v e l y t h a t t h e r e i s no enhancement we w i l l have t o i n t e g r a t e the d i f f e r e n t i a l r a t e . The f i r s t s t e p i n a c c o m p l i s h i n g t h i s i s t o e v a l u a t e the sum i n e x p r e s s i o n IV. 1 . As a l r e a d y n o t i c e d fl,*in and 5fc a r e b oth v e r y l a r g e ; the r e q u i r e d a s y m p t o t i c e x p r e s s i o n s f o r the B e s s e l f u n c t i o n s a r e now p r e s e n t e d . 4.3 A s y m p t o t i c Formulae For B e s s e l F u n c t i o n s Four d i f f e r e n t f o r m u l a e , each w i t h v a r y i n g numbers of terms have been used t o a p p r o ximate the B e s s e l f u n c t i o n s needed h e r e . They a r e d e r i v e d i n Watson ( r e f . 22) and r e p r o d u c e d here w i t h the f o l l o w i n g c o n v e n t i o n s They a r e 1 3 f o r z. > n IV.6 n i s used here i n p l a c e of t h e u s u a l i/ because of the d i s c r e t e sum i n v o l v e d . 49 f o r r„t*)~ 4 rS)(|)"' + (S-«)^,^r(J)(4)1/3] IV.7 f o r 2 < n IV.8 and f o r t h e t r a n s i t i o n r e g i o n between 2>n and 177? = T&) ~ % t<x~,/$ /ceofa + 5 - i fan3,*) • ( J_v,+ T"y, ) IV.9 From t h e s e formulae i t can be seen t h a t the d i f f e r e n c e s between T n + n T n , and f o r l a r g e a r e n e g i g i b l e . Thus f u r t h e r s i m p l i f i c a t i o n of the d i f f e r e n t i a l decay r a t e g i v e s F i g u r e 4 i s a c o m p o s i t e computer drawn p i c t u r e of a r e g i o n of the summand of t h i s e x p r e s s i o n as a q u a s i c o n t i n u o u s f u n c t i o n of n u s i n g t h e f o u r formulae 50 above. S e r i e s of t h e s e p i c t u r e s were g e n e r a t e d f o r r a n g i n g from 4,000 t o 8,000,000 and ^=2-500 and ^ a -10000. From t h e s e s e r i e s the l o c a t i o n of key f e a t u r e s c o u l d be a b s t r a c t e d . F e a t u r e s such as t h e l o c a t i o n of the l a s t maximum or minimum b e f o r e f\ = 2 were found t o move a c c o r d i n g A ft f = z - >V - b ^ 3 where Kl^ i s the v a l u e of m f o r a f e a t u r e i n q u e s t i o n . For t h e s e f e a t u r e s was t y p i c a l l y 20 t o 200 and b needed t o be a c c u r a t e t o t h r e e d i g i t s . T h i s was o b t a i n e d u s i n g g r a p h i c a l a n a l y s i s . In t a b l e 1 "min, " and "min^" are the l o c a t i o n of the minima i n d i c a t e d i n f i g . 4 . The o t h e r f e a t u r e s a r e t r a n s i t i o n p o i n t s between the d i f f e r e n t f o r m u l a e i n use, though they were not always c r i t i c a l . F i g . 4 i l l u s t r a t e s t h a t the formulae o v e r l a p f o r an extended i n t e r v a l of n . Some u s e f u l bounds t o n o t i c e a re the ranges of some of the v a r i a b l e s f o r t h e case V-1. T h i s f a c i l i t a t e s the d e t e r m i n a t i o n of the t o t a l decay r a t e and i t s e r r o r . For V « 1 , 2 e [ 1 .7X1 0 5 , 3.0X1 0 5 ]. Then jta*/2e[0, .34] . For n e L, ], fa~/h 6 [.048,. 3 4 ] . 4.4 The Summation A b e t t e r p l a c e t o t r u n c a t e the summation i n IV.10 was seen from f i g . 4 t o be 2 + 100 or £+1.7 i n g e n e r a l . Then the r e m a i n i n g i n t e r v a l of to sees a d e c l i n e of the B e s s e l f u n c t i o n f a s t e r than e x p o n e n t i a l s i n c e the e x p r e s s i o n Y\ JtcL^M. ) ^ r a p i d l y changes from a slow i n i t i a l growth 51 t o n e a r l y e x p o n e n t i a l growth i t s e l f . An upper bound f o r the e r r o r i n c u r r e d by t h i s t r u n c a t i o n i s c a l c u l a t e d i n more d e t a i l i n appendix B. The e r r o r i n c u r r e d by r e s t r i c t i n g the domain t o t h a t on which nm,-B <z i s c a l c u l a t e d i n a s i m i l a r manner. The complementary c o n d i t i o n n,^ ,, g i v e s the complementary r e g i o n s of fan & and the same bounds on . From the c a l c u l a t i o n of appendix B i t can be seen t h a t the c o n t r i b u t i o n t o the t o t a l decay r a t e from th e s e r e g i o n s can a l s o be n e g l e c t e d . I f we had been a b l e t o do the complete sum on n and not had t o l o o k f o r the k i n e m a t i c domain s a t i s f y i n g n<z, then the e n t i r e £ domain would have been " a l l o w e d " . Even so i n v e r s e e x p o n e n t i a l f a c t o r s c o n t r o l the o u t l y i n g r e g i o n s and the summation c o u l d be found. The c a l c u l a t i o n used G a u s s i a n i n t e g r a t i o n on the l a s t two peaks w i t h enough p o i n t s t h a t the e r r o r was l i m i t e d by the a c c u r a c y of the a s y m p t o t i c f o r m u l a e . T h i s r e g i o n g i v e s the l a r g e s t c o n t r i b u t i o n t o the sum because of the square f a c t o r and must be a s s e s s e d c a r e f u l l y because of the d e l i c a c y of the a s y m p t o t i c f o r m u l a e i n v o l v e d . The main f o r m u l a t h a t g i v e s the p r o f i l e of t h e l a r g e s t l a s t peak of the summand i s the t r a n s i t i o n r e g i o n f o r m u l a JTR, IV.9 . T h i s f o r m u l a has no more terms but the e r r o r i s l i k e l y v e r y low a c c o r d i n g t o Watson. 1* The r e l a t i v e e r r o r i s 1* Watson p252 52 T h i s r e s u l t l e d t o an e f f o r t t o a c h i e v e 1% a c c u r a c y i n the f i n a l r e s u l t . Another r e g i o n of t h i s peak uses the f o r m u l a J L T , IV.7 . For 1% a c c u r a c y the second term becomes n e c e s s a r y f o r The maximum of X&h /i = .34 r e s t r i c t s t he l e f t s i d e ' s magnitude t o f a r l e s s than .01 . But near the minimum =0, the second term dominates and /O0O/2 ~ 1 Thus a second term was n e c e s s a r y f o r the l a s t two peaks b e f o r e n=z. A t h i r d term i s n e c e s s a r y f o r o n l y a s h o r t r e g i o n w i t h i n the peak b e f o r e JTR can be used and thus was n e g l e c t e d . For the JEQ fo r m u l a s i m i l a r arguments were used t o f i n d a l s o t h a t two terms s u f f i c e d over the range f o r i t s use. As can be seen from f i g . 4 or 5, the match up of the d i f f e r e n t f o r m u l a e was smooth. In t h i s way a m u l t i - e x p r e s s i o n f u n c t i o n was composed and s u p p l i e d t o the U.B.C. l i b r a r y G a u s s i a n i n t e g r a t i o n r o u t i n e t o sum the l a s t peak. As w e l l two terms of JGT were s u p p l i e d f o r a G a u s s i a n i n t e g r a t i o n of the second l a s t peak. There were s m a l l r e g i o n s where 1% a c c u r a c y of the summand was not g u a r a n t e e d , but these were r e g i o n s on which the 53 the summand was r e l a t i v e l y s m a l l . The t e c h n i q u e f o r summing the r e g i o n <n<i-, , where L, = B - A D , was a r r a n g e d t o t a k e advantage of the q u a s i s i n u s o i d a l n a t u r e of t h e summand. The sum i n q u e s t i o n from eq. IV.10 i s I = i (n- nM,„) ( r ) m i n - n + % £0) n = n^.;rt iv. 11 From IV.6 We may w r i t e the sum as an i n t e g r a l s i n c e the s t e p s i z e i s r e l a t i v e l y s m a l l , then a r r a n g e the terms i n o r d e r of s i z e i = (*)x-ir fli (*-«0f a to) i±±zzL j n - (Co f _ L r L ' r c C " - " » f ^ l " ' ^ ^ ! ? _ w t»-*)* 2 T i IV. 1 2 The f i r s t two i n t e g r a l s a r e e a s i l y p e r f o r m e d 1 5 and p r e s e n t e d as t h e y were a r r a n g e d f o r c o m p u t a t i o n 1 5 Though t r a c i n g s u s p e c t e d i n c o n s i s t e n c i e s i n the f i n a l r e s u l t back t o e r r o r s i n pre-1973 CRC i n t e g r a l t a b l e s was not so easy. CRC e d i t i o n 52, p A129, i n t e g r a l #220 54 We may n o t i c e t h a t I n c l u d i n g X x (which r e q u i r e s d o u b l e p r e c i s i o n a r i t h m e t i c ) c o u l d g i v e no more t h a n a 4% c o r r e c t i o n but i t was i n c l u d e d t o a c h i e v e 1% a c c u r a c y . 1 6 These e x p e c t a t i o n s were met i n the n u m e r i c a l c o m p u t a t i o n . T3 was e v a l u a t e d by an " a d a p t i v e b a l a n c i n g " t e c h n i q u e I t was f i r s t n o t i c e d t h a t /C«?*$=-/OJu*(2r>(.j&rf-fl)) a n d t h a t t h e f r e q u e n c y d i d not change a p p r e c i a b l y over each p e r i o d , but r a t h e r s l o w l y over l a r g e r changes i n h . T h i s was v e r i f i e d a n a l y t i c a l l y u s i n g elr* „3 and Then each s u c c e s i v e h a l f p e r i o d was a p p r o x i m a t e d by ( a m p l i t u d e ) x ( p e r i o d / 2 ) x ( i/l(>«Kiec( = (n-r»,) x(z x-M*) " V z * P a i r s of such a p p r o x i m a t i o n s f o r n e i g h b o r i n g h a l f p e r i o d s were f i r s t combined, then the s m a l l c o n t r i b u t i o n l e f t was a c c u m u l a t e d . The p e r i o d and the a m p l i t u d e were c o n t i n u a l l y 1 6 T r a c i n g p e r s i s t e n t problems t o the need f o r double p r e c i s i o n i n the e v a l u a t i o n of was a l s o e d u c a t i o n a l . 55 r e a s s e s s e d as s t e p s of T were made from L, back t o e i t h e r z-3000 or f o r n. From the d a t a a v a i l a b l e d u r r i n g the c o m p u t a t i o n t h i s c o r r e c t i o n was of o r d e r 1%. The l a s t i n t e g r a l I H was n e g l e c t e d . 4.5 D i f f e r e n t i a l Decay Rate The d i f f e r e n t i a l decay r a t e f o r a sequence of l a s e r i n t e n s i t i e s from V=.3 t o 5,(n=2eV, i s d i s p l a y e d i n f i g s . 6 t o 9. The r e l a t i v e s i z e s of the domains i s most e a s i l y seen on the p r e v i o u s p l o t f i g . 3. On t h e s e p l o t s the axes a r e changed so as t o a c h i e v e l a r g e s i z e . The r e s o l u t i o n of the c a l c u l a t i o n i s i n d i c a t e d by the number of l i n e s i n the p l o t . The same t r e n d s as n o t i c e d from f i g . 3 are n o t i c e d h e r e . A s e t of p l o t s (not i n c l u d e d ) f o r a sequence of W ' s s . z , e V showed no d i f f e r e n c e s , a r o u s i n g s u s p i c i o n s t h a t the photon energy d i d n ' t m a t t e r . Indeed t h i s i s e a s i l y v e r i f i e d . S t a r t i n g w i t h the e x p r e s s i o n IV.11 , we l e t n'=u;n. Then n^,;n = w ^ n d z'=tiJz a r e independent of u> . In t r a n s f o r m i n g t h e sum we d i v i d e by the new l e n g t h between summation p o i n t s , itf U> YY\X But f o r 1«n'<z' where Thus 56 T h e r e f o r e the to c a n c e l s , the n' may be r e l a b e l l e d , and we have . T - -L , £ (w- » C } a (( n U - n V r w + £0 ) J n (2) IV. 13 independent of U) . A d d i t i o n a l l y , the 1981 a r t i c l e by Becker e t a_l shows a d o u b l e d peaked spectrum f o r V =.3 and a s h o u l d e r i n the spectrum f o r V=.5 ( r e f . 2) T h i s i s r e p r o d u c e d here as f i g . 10. For comparison a h i g h r e s o l u t i o n p l o t has been o b t a i n e d from our s t u d y , a l s o f o r V=.3 . T h i s i s f i g . 11 and 12, wherin no c o m p l i c a t i o n s a t the s c a l e s i z e i n d i c a t e d by Becker et a_l can be o b s e r v e d . 4.6 T o t a l Decay Rate From I I I . 1 4 and 15 the t o t a l decay r a t e f o r ^ decay w i t h o u t a l a s e r i s xir3 L 30 where frjf s t a n d s f o r the n u c l e a r m a t r i x element f a c t o r . U s i n g double p r e c i s i o n a r i t h m e t i c and the v a l u e of € 0 = ^ =1 .0364 c o r r e s p o n d i n g t o Q=l8.60KeV f o r 3H decay, we f i n d x UJr - 9 m 2.013 Kio'6  T 2.TT3 57 For the l a s e r i r r a d i a t i o n case we use the e x p r e s s i o n s IV.10 and IV.11 f o r the d i f f e r e n t i a l decay r a t e The r a t i o of r a t e s i s ft- W laser = 1 .00+. 01 and t h e r e i s Wno Ltftr c l e a r l y no enhancement due t o the l a s e r i r r a d i a t i o n . T a b l e 3.2 c o n t a i n s the r a t i o of r a t e s f o r d i f f e r e n t v a l u e s of V ,U» ,and £ e a l s o . W i t h o u t the f u r t h e r use of double p r e c i s i o n , the l i m i t s of were d e t e r m i n e d above by nmin a n ^ z a t t a i n i n g 8 d i g i t s f o r V =5 and 9 d i g i t s f o r V =9 The l i m i t below was reached when n • became c l o s e t o 0 r f l l l l or n e g a t i v e f o r V-.3 .Then the a s y m p t o t i c formulae were no lo n g e r v a l i d . 4. 7 Program Notes T h i s s e c t i o n c o n t a i n s a b r i e f summary of the l a r g e r o r g a n i z a t i o n and f e a t u r e s of the main programs i n v o l v e d i n t h i s t h e s i s . The main program reads the £ and <eoo6 r e s o l u t i o n a l o n g w i t h the i n p u t parameters V , iv , and £ e , and t h e KJO^B bounds, and a format code f o r output from an i n p u t f i l e . The d e s i r e d r e s o l u t i o n and the fixnB bounds were based on t h e r e s e a r c h e r ' s e x p e r i e n c e f o r any g i v e n r u n . I f the /C<»© bounds chosen were t o o narrow the program enc o u n t e r e d n e g a t i v e square r o o t s i n c a l c u l a t i n g t he T h e r e f o r e the t o t a l decay r a t e i s and from t h e computer c a l c u l a t i o n we o b t a i n e d 58 summation l i m i t s nm!fl , and z, and ' c r a s h e d ' . I f they were t o o wide computer time ( m a i n l y i n p l o t t i n g ) was s a c r a f i c e d . Thus a p r o c e d u r e e v o l v e d w h e r e i n f i r s t a c o a r s e g r a i n e d c a l c u l a t i o n w i t h l a r g e one bounds was p e rformed, f o l l o w e d by a f i n e r g r a i n e d one w i t h narrower bounds. T h i s p r o c e d u r e was combined w i t h a s u c c e s i v e i n c l u s i o n of terms from IV.11 t o show the r a t e c a l c u l a t i o n converge on the numbers p r e s e n t e d i n t a b l e 3.2 The program i t s e l f was modular, each module h a v i n g been t e s t e d s e p a r a t e l y b e f o r e the e n t i r e program was assembled. The output of the program i n c l u d e d the t o t a l c r o s s s e c t i o n and the d i f f e r e n t i a l c r o s s s e c t i o n d a t a . T h i s d a t a was p l o t t e d i n the p e r s p e c t i v e p l o t s i n c l u d e d by another program t h a t c a l l e d P i s s p l a p l o t t i n g r o u t i n e s ( r e f . 2 4 ) . Because D i s s p l a i d e n t i f i e s the c o o r d i n a t e s of a f u n c t i o n v a l u e by i t s p o s i t i o n i n an a r r a y and F o r t r a n does not f i l l o v e r d i m e n s i o n e d m u l t i p a r a m e t e r a r r a y s c o n t i g u o u s l y i t was i m p o r t a n t t o r e s e r v e the e x a c t d i m e n s i o n s f o r the d i f f e r e n t i a l r a t e d a t a a r r a y . These numbers depended on the r e s o l u t i o n d e s i r e d and t h i s was most u s e f u l as i n p u t f o r a c a l c u l a t i o n . Thus the t r i c k of e x e c u t i o n time d i m e n s i o n i n g of a r r a y s u s i n g system commands a v a i l a b l e t h r o u g h the system g e n e r a l l i b r a r y was i n v o k e d . Another o p t i o n would have been t o s u p p l y the d a t a and i t s d i m e n s i o n s as p arameters t o a s u b r o u t i n e t h a t does read i n t o the a r r a y p r o p e r l y . 59 F i g u r e 3 - KINEMATIC DOMAINS 60 F i g u r e 4 - SUMMAND OF IV.10 62 63 F i g u r e DIFFERENTIAL DECAY RATE V = , 5 v^ZcV 3H a-64 F i g u r e 8 - DIFFERENTIAL DECAY RATE V-l,W-2eV»H ,OCT 12 65 F i g u r e 9 - DIFFERENTIAL DECAY RATE V=5,w=2eV,8H ,SEPT 3 66 F i g u r e 10 - ENERGY SPECTRUM OF BECKER e t a l e> /EM Jtin eue E Ln III I I • I I II I I l l J 1.1 1J 1.6 C FIG. 2. Logarithmic plot of the electron spectrum of *H decay for v = 0JJ. 0.5, 0.6, 1.0. The inset at the lower right shows the values for 1***1.1 with doubled scale of the abscissa. 61 ure 11 - DIFFERENTIAL DECAY RATE V=.3 to = Z eV 3 H 68 F i g u r e 12 - DIFFERENTIAL DECAY RATE $31 V=.3 w- 2eV 3 H 69 Ta b l e I - LOCATION OF FEATURES in • < (VI in ± • vri %\* • S I * to m O (»> o o-Cx. s vx* IK, vo ^ £ y-> Q s < OO OO < 6c M 0" bo &o o «a i o m Q O \ •> .? •3 0 m VP O fro tw 6o <n v/ O 0 0 0 0 0-O o 0 o o o vn O I *-8 o 0-i l l * § o 0 3 o o 0 o o 8 0 • • 70 T a b l e I I - NONENHANCEMENT I ( w/c« l) V U 3 x \oxl .3 2 I.OOH t x \ox% / 7 * lo'? 5" / .0/2 1 K \oxo 7 /. 000 / l.oioo l.iooo 1 / .003 V x lo1* / /.out g 1.006 71 V. CONCLUSION The main r e s u l t of t h i s t h e s i s i s t h a t the t o t a l decay r a t e f o r /3 decay i s unchanged by the presence of even i n t e n s e l a s e r f i e l d s . The n u m e r i c a l c a l c u l a t i o n s show t h i s f o r a range of the l a s e r i n t e n s i t y parameter V fe [.3,7] and e n e r g i e s Ui«[2,8]eV and a range of /I decay e n e r g i e s Q € [ 5 , 5 0 ] K e V . Furthermore t h e d i f f e r e n t i a l decay r a t e i s found t o be independent of w and the l i m i t s of the n e u t r i n o energy spectrum are found t o be unchanged ( a n a l y t i c a l l y and n u m e r i c a l l y ) by the presence of the l a s e r . These r e s u l t s were the s u b j e c t of c h a p t e r 4. In c h a p t e r 3 the d i f f e r e n t i a l decay r a t e was d e r i v e d as a f u n c t i o n of the e l e c t r o n energy and p o l a r a n g l e f o r ^  ~ decay i n the presence of an u n p o l a r i z e d i n t e n s e o p t i c a l l a s e r f i e l d . The t r a n s i t i o n a m p l i t u d e was d e r i v e d u s i n g the V o l k o v e l e c t r o n wave f u n c t i o n . A l t h o u g h the weak i n t e r a c t i o n and e l e c t r o m a g n e t i c e f f e c t s a f t e r c r e a t i o n of the e l e c t r o n a r e i n s e p a r a b l e i n t h e s e e x p r e s s i o n s the above r e s u l t s suggest t h a t the l a s e r f i e l d i s not a f f e c t i n g the /3 decay p r o c e s s i t s e l f , but i s o n l y a f f e c t i n g t h e subsequent e" m o t i o n . T h i s a s s e r t i o n i s a p p e a l i n g because i t i s c o n s i s t e n t w i t h the s i m p l e argument c o n c e r n i n g the mismatch of e n e r g i e s (wavelengths) p r e s e n t e d i n the i n t r o d u c t i o n . A d d i t i o n a l e v i d e n c e f o r t h i s a s s e r t i o n i s the s i m i l a r i t y ( c a l c u l a t e d ) between the m o d i f i c a t i o n of a f r e e e l e c t r o n ' s motion and a /$ decay e l e c t r o n ' s spectrum. In c h a p t e r 2 an e l e c t r o n i n i t i a l l y a t r e s t € =1, i s found 72 ( c l a s s i c a l l y ) t o a t t a i n an energy, £ =1.26 under the i n f l u e n c e of a p l a n e - p o l a r i z e d l a s e r f i e l d w i t h \) =1, la = 2eV. The c o r r e s p o n d i n g m o d i f i c a t i o n of the energy range i n the 3H/3" spectrum ( c a l c u l a t e d quantum m e c h a n i c a l l y ) i s from [1,1.0364] w i t h o u t a l a s e r t o £€[1.25,1.85] i n an u n p o l a r i z e d l a s e r f i e l d a l s o w i t h U = 1, o) =2eV. The agreement between the f r e e e" case and the case where the e l e c t r o n has been a p p o r t i o n e d a minimum of energy i s good; however, c l a r i f i c a t i o n of t h i s s i m i l a r i t y would be a c h i e v e d by an e f f o r t t o compare the l a s e r m o d i f i e d fi>~ spectrum w i t h a c a l c u l a t i o n of the e f f e c t s of a l a s e r f i e l d on the s t a t i s t i c a l ensemble of f r e e e" t r a j e c t o r i e s a r i s i n g from o r d i n a r y /$ decay. The n e u t r i n o energy l i m i t r e s u l t , a l t h o u g h a l r e a d y more p e r s u a s i v e of the a s s e r t i o n t h a t the l a s e r a f f e c t s o n l y the p o s t c r e a t i o n e l e c t r o n motion than the above e~ spectrum d i s c u s s i o n , a l s o might be augmented. One would examine i f the d e t a i l e d n e u t r i n o spectrum (not j u s t i t s l i m i t s ) i s unchanged by the p r e s e n c e of the l a s e r f i e l d . These f u r t h e r s t u d i e s c e r t a i n l y seem w o r t h w h i l e f o r the sake of the b a s i c p h y s i c s i n v o l v e d . At what i n t e n s i t y w i l l l a s e r l i g h t a f f e c t n u c l e a r p r o c e s s ? Have they been a f f e c t e d a t V =1 but so as not t o change /s decay r a t e s ? I s the s i m p l e e x p l a n a t i o n from r e l a t i v e magnitudes of the energy of the l a s e r f i e l d and the /5 decay the b e s t ? These a r e open q u e s t i o n s . 73 BIBLIOGRAPHY 1. B a i l i n , D. "The Theory of Weak I n t e r a c t i o n s i n P a r t i c l e P h y s i c s " Rep. P r o g . Phys. 3_4 pg. 491, 1 971 2. B e c k e r , W. e t a_l. " L a s e r Enhancement of N u c l e a r Decay". Phys. Rev. L e t t e r s £7,pgs. 1262-1266, Nov. 1981. 3. B e c k e r , W. e t - a l . "Becker e t al _ . Respond:". Phys. Rev. L e t t e r s 48, pg. 653. Mar, 1982. 4. Becker,W. e t a l "A Note on T o t a l C r o s s S e c t i o n s and Decay Rates i n t h e Presence of a L a s e r F i e l d " P h y s i c s L e t t e r s 94A, pg. 131, Mar. 1983. 5. B j o r k e n , J . D . and D r e l l , S.D. R e l a t i v i s t i c Quantum  Mec h a n i c s . M c G r a w - H i l l , New York, 1964. 6. Byrne, J . "Weak I n t e r a c t i o n s of the N e u t r o n " , Rep. P r o g . Phys. 45 pg. 115, 1982 7. Cohen-Tannoudji, D i u , L a l o e , Quantum Mechanics John W i l e y & Sons, New York, 1977 8. d e S h a l i t , A. and Feshbach, H. T h e o r e t i c a l N u c l e a r  P h y s i c s Volume 1: N u c l e a r S t r u c t u r e . John W i l e y & Sons, New York, 1974. 9. Du f f and N a y l o r , D i f f e r e n t i a l E q u a t i o n s of A p p l i e d  M a t h e m a t i c s , John W i l e y & Sons, New Yo r k , 1966 10. Enge I n t r o d u c t i o n t o N u c l e a r P h y s i c s A d d i s o n Wesley, T o r o n t o , 1966. 11. Feynman, R.P. The Feynman L e c t u r e s on P h y s i c s A d d i s o n Wesley, T o r o n t o , 1965. 12. Feynman, R.P. Theory of Fundamental P r o c e s s e s W.A. Benjamin, M a s s a c h u s e t t s , 1961 13. G e r s t e n , J . I . and M i t t l e m a n , M.H. "Comment on 'Laser Enhancements of N u c l e a r decay'". Phys. Rev. L e t t e r s 4IB_, pg.651. Mar. 1982. 14. Hebron, J . " L a s e r Enhancement of N u c l e a r Beta Decay" M.Sc. T h e s i s , UBC, 1982. 15. I t z y k s o n , C. and Zuber, J . Quantum F i e l d Theory. M c G r a w - H i l l , New York, 1980. 16. J a c k s o n , J.D. C l a s s i c a l E l e c t r o d y n a m i c s 2nd Ed. John 74 W i l e y & Sons, New York, 1975. 17. Landau, L.D. and L i f s c h i t z , E.M. R e l a t i v i s t i c  Quantum Theory,Course of T h e o r e t i c a l P h y s i c s , V o l . 4 , p a r t 1. Pergamon P r e s s , O x f o r d , 1971. 18. Reiss,H.R. "Laser Enhancement of N u c l e a r Decay". Phys. Rev. L e t t e r s 48, pg. 652. Mar. 1982 19. R e i s s , H.R. " N u c l e a r Beta decay induced by i n t e n s e e l e c t r o m a g n e t i c f i e l d s : B a s i c Theory " Phys. Rev. C27, pg. 1199 Mar. 1983 20. Ternov, I.M. e t a l "Change i n the Beta Decay P r o b a b i l i t y due t o the a c t i o n of an E l e c t r o m a g n e t i c Wave." JETP l e t t e r s , 3_Z» P9- 3 4 3 M a r « 1 983. 21. V o l k o v , "Uber e i n e K l a s s e von Losungen der D i r a c s c h e n G l e i c h u n g " , Z. P h y s i k 94 pg. 250, 1935 22. Watson G.N. B e s s e l F u n c t i o n s M a c m i l l a n , E d i t i o n 2, New York, 1944 23. S t a n d a r d M a t h e m a t i c a l T a b l e s E d i t i o n 21, The Chemical Rubber Co. C l e v e l a n d , 1973. 24. DISSPLA User ' s Manual, I n t e g r a t e d Software Systems Corp., San Diego, 1981. 75 APPENDIX A - NOTATION AND CONVENTIONS Mechanics For u n i t s we s e t fi=1=c . For a b a s i s of space-t i m e we use e^ , /*• ={0,1,2,3}. Then we have c o n t r a v a r i a n t c o o r d i n a t e s x=(t,x)=fc*)and the f l a t space c o v a r i a n t d e r i v a t i v e ^ = $/ix*. The 4 - v e c t o r v e l o c i t y i s u= $y = ( * , * v) , but the l a b frame v e l o c i t y i s v= 4 l (1,v) which i s not a L o r e n t z v e c t o r . The m e t r i c i s O The i n n e r p r o d u c t , t r ( X , Y * ) , f o r s c a l a r s i s xy*, f o r v e c t o r s i s x« y*=x°y^-xy*, and f o r m a t r i c e s i s X ^ v Y ^ y f E l e c t r o m a g n e t i s m The e l e c t r i c f i e l d E=E(x) and the magnetic f i e l d B=B(x) can be a r r a n g e d i n t o an a r r a y ' c O BY ** O -Sir '*Y 0 \ I known as the e.m. f i e l d t e n s o r because e x p e r i m e n t a l l y i t i s found t o behave as a sk e w - s y m e t r i c t e n s o r under L o r e n t z t r a n s f o r m a t i o n . The Hodge d u a l i s •V* D i r a c E q u a t i o n The energy momentum o p e r a t o r i n c o o r d i n a t e r e p r e s e n t a t i o n i s 76 For L o r e n t z c o v a r i a n c e of the D i r a c e q u a t i o n we f i n d A. 1 D e f i n e <r as A.2 In Feynman r e p r e s e n t a t i o n t h e s e a r e D e f i n e # as For a s p i n Vz p a r t i c l e and a n t i p a r t i c l e (f~ m) v (/> *) - ° Def i n e Some u s e f u l r e s u l t s a r e « *r p ^ * * rip If/" 4 . A.3 = ( tf.AW. q./.y) = ftx A.4 77 A.5 c A.6 Mathematics One u s e f u l p r o p e r t y of the d e l t a f u n c t i o n i s shown he r e . F i r s t , the s e t of ' t e s t ' f u n c t i o n s , C*9 f u n c t i o n s t h a t v a n i s h o u t s i d e of a f i n i t e r e g i o n R of space-t i m e , form a v e c t o r space. The d i s t r i b u t i o n s form the d u a l space. In p a r t i c u l a r the h f u n c t i o n a l has the p r o p e r t y We d e f i n e the i d e n t i t y t e s t f u n c t i o n I as u n i t y on R. Then (~ I X £. R Ifr) = t o For t e s t f u n c t i o n s d e f i n e d on sp a c e - t i m e we a r e i n t e r e s t e d i n a l s o else A.7 Some u s e f u l B e s s e l f u n c t i o n i d e n t i t i e s a r e ( r e f . 2 2 ) 78 A.9 79 APPENDIX B - ESTIMATE FOR JGT INTEGRAL The i n t e g r a l i s where ^ f e j , * = Jl-*Vr»* , n £ ( z ^ <*] , ^ 2UKU e[^'] . Break t h i s A i n t o two i n t e g r a l s l o o s 5 Z A t r a p e z i o d a p p r o x i m a t i o n f o r the f i r s t g i v e s r , ^ i K'O2" For the second use the boundary v a l u e as a maximum f o r the e x p o n e n t i a l f a c t o r . A change of i n t e g r a t i o n v a r i a b l e n-1.002z=x g i v e s gamma f u n c t i o n s and the e s t i m a t e T h e r e f o r e the e r r o r , i n c l u d i n g t h e cr>& range and the € range i s or .01 %. 80 APPENDIX C - NEUTRINO ENERGY F i n d i n g t h e maximum of the n e u t r i n o energy b o t h f a c i l i t a t e s u n d e r s t a n d i n g l a s e r i n d uced /9 decay and p r o v i d e s a check of the n u m e r i c a l work. From IV.6 the d i m e n s i o n l e s s n e u t r i n o energy (€ v = 0>v/mt) has a maximum of The r e q u i r e m e n t £v>0 gave V n ^ - E - M ^ a n d so we see t h a t Thus a maximum of i s a l s o a maximum of z-n • . The a n a l y s i s i s o u t l i n e d as f o l l o w s . For a s t a t i o n a r y p o i n t of £ y we have - 0 • v ^ ' - l ' ^ S - % - A£ d e v , . ^  - t — =. o • i ; = J ? 2 - ' /2^<5 From t h e s e e q u a t i o n s a q u a d r a t i c i n cos© i s found f o r which one r o o t i s a c c e p t a b l e ,000 & » c T h i s g i v e s t h a t & t = 1 and £ = 1 + V*/z . We have i m m e d i a t l y t h a t 0 < £ y < £ e - 1 and z-n m i„ < 2L ( e e - i ) 9300. The l a t t e r f a c t was n o t i c e d f i r s t from i n s p e c t i o n of the c o m p u t a t i o n a l d i f f e r e n t i a l c r o s s s e c t i o n d a t a . 81 The a n a l y s i s of s e c t i o n 4.2 can be g e n e r a l i z e d t o i n c l u d e and v e r i f y the r e s u l t s of appendix C f o r the n e u t r i n o . S t a r t i n g w i t h the domains a l l o w e d by p h y s i c a l c o n s i d e r a t i o n s we seek t o f i n d the s m a l l e r domains on which the d i f f e r e n t i a l decay r a t e i s n o n n e g l i g a b l e . S e t t i n g n<z from ( I V . 1 ) , d e f i n i n g ^ _ - a £ f g , - €^ - <L) + Vz and s q u a r i n g t o f i n d a q u a d r a t i c i n e q u a l i t y i n cos©, we f i n d F o r t h e s e l i m i t s t o be r e a l the argument of the r a d i c a l must be p o s i t i v e . The q u a d r a t i c i n e q u a l i t y i n £ g i v e s A g a i n f o r th e s e l i m i t s t o be r e a l we f i n d (*© - f y ) 1 >0 which has one a d m i s s a b l e s o l u t i o n 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0096047/manifest

Comment

Related Items