Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Hebron, John Stephen
Master of Science - MSc
University of British Columbia
After deriving and discussing the solution to Dirac's equation for an electron in an external electromagnetic plane wave, the electron field operator is expanded in terms of this solution. The usual electron field operator in the four point theory of nuclear beta decay is replaced by this new field operator, and a formula is obtained for the new beta decay rate. This formula is found to agree with the formula stated by Becker et al. Arguments are presented which indicate that the decay rate will not be significantly enhanced. These arguments are supported by a quantum mechanical model consisting of a square well with barriers through which an electron in a quasi-stationary state tunnels, the rate of tunnelling being enhanced by an external linear electric field.
LASER ENHANCEMENT OF NUCLEAR BETA DECAY by JOHN STEPHEN HEBRON B.Sc.(Honours In Astrophysics), The University Of Calgary, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1982 © John Stephen Hebron, 1982 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: December, 1982 i i Abstract After deriving and discussing the solution to Dirac's equation for an electron in an external electromagnetic plane wave, the electron f i e l d operator i s expanded in terms of t h i s solution. The usual electron f i e l d operator in the four point theory of nuclear beta decay i s replaced by t h i s new f i e l d operator, and a formula i s obtained for the new beta decay rate. This formula i s found to agree with the formula stated by Becker et a l . Arguments are presented which indicate that the decay rate w i l l not be s i g n i f i c a n t l y enhanced. These arguments are supported by a quantum mechanical model consisting of a square well with barriers through which an electron in a quasi-stationary state tunnels, the rate of tunnelling being enhanced by an external linear e l e c t r i c f i e l d . i i i Table of Contents Abstract i i L i s t of Tables v i L i s t of Figures v i i Acknowledgement v i i i I. INTRODUCTION 1 1.1 The Results Of Becker,et.al 1 1.2 How A Decay Rate Can Be Modified: Phase Space Vs Matrix Element E f f e c t s 1 1.3 An Overview Of This Thesis 8 1.4 Conventions And Units 9 11 . NUCLEAR BETA DECAY 11 2.1 Calculation Of Decay Rate Using Feynman Rules 11 2.2 Derivation Of Feynman Rules 19 2.3 What Is The Phase Space Of The Electron? 23 I I I . THE VOLKOV SOLUTION 25 3.1 Minimal Coupling Procedure 25 3.2 The Second Order Dirac Equation 26 3.3 Solution Of The Second Order Dirac Equation For An Electromagnetic Plane Wave ....26 3.4 Solution Of The F i r s t Order Dirac Equation 31 3.5 Normalization Of The Volkov Solution 33 3.6 E f f e c t i v e Mass Of A Volkov Electron 38 3.7 How To Express The Volkov Solution In The Form Used By Becker,et.al 39 IV. LASER ENHANCEMENT OF NUCLEAR BETA DECAY 45 4.1 Expansion Of Electron F i e l d Operators In Terms Of Volkov Solutions 45 4.2 The New' Expression For The S Matrix 46 4.3 Evaluation Of The Decay Rate 48 4.4 Comparisons To The Free F i e l d Decay Rate 60 4.5 Analysis Part 1: What Is Going On? 6 1 4.6 Analysis Part 2: How Large Is The New Decay Rate? 65 4.7 What Happens If We Increase The Electronic Phase Space In Free F i e l d Decay? 68 4.8 A Speculative Calculation Of The Acceleration Distance Of The Electron 72 4.9 The Transverse Acceleration 75 4.10 The Matrix Element Effect 77 V. A QUANTUM MECHANICAL ANALOG OF BETA DECAY AND ITS PHASE SPACE ENHANCEMENT 80 5.1 The Model. Is It Valid? 80 5.2 Matching The Neutrinoless Model To Real Beta Decay 81 5.3 How To Obtain The Decay Rate: A Summary 83 5.4 Numerical Results 85 5.5 Immersion Of The Model In An E l e c t r i c F i e l d : The Enhanced Decay Rate 89 5.6 Numerical Results 94 5.7 How Big Must The E l e c t r i c F i e l d Be To Match The Results Of Becker,et.al.? 98 5.8 A S l i g h t l y Modified Model 99 5.9 Discussion 102 VI. CONCLUSION 103 BIBLIOGRAPHY 105 APPENDIX A - FREE FIELD SOLUTIONS TO THE DIRAC EQUATION .107 THE DIRAC EQUATION AND MATRICES 107 DELTA FUNCTION NORMALIZATION 108 BOX NORMALIZATION 108 NOTATION 109 PROJECTION OPERATORS 109 APPENDIX B - USEFUL IDENTITIES 110 . IDENTITIES INVOLVING SPIN SUMMED NUCLEAR MATRIX ELEMENTS 110 IDENTITIES INVOLVING GAMMA MATRICES 112 APPENDIX C - DECAY OF AN ELECTRON FROM A ONE DIMENSIONAL SQUARE WELL 113 C1 . INTRODUCTION 113 C2. THE MODEL: SOLVING THE SCHRODINGER EQUATION 113 C3. THE STRUCTURE OF THE MATRICES WHICH RELATE THE COEFFICIENTS 114 C4. THE REFLECTED AND TRANSMITTED WAVES 118 C5. RESONANCE 122 C6. NOTATION. FUNDAMENTAL EQUATION 123 C7. ASSUMPTIONS USED 123 C8. ORGANIZATION OF THE EXPANSIONS 127 C9. FIRST APPROXIMATION: £ =0 128 C1 0 . SECOND ORDER APPROXIMATION 1 30 C1 1 . M M AT RESONANCE 135 C12. SOLUTION AT RESONANCE WHICH IS A PARITY EIGENSTATE 137 C13. THE n = 0 RESONANCE 142 C14. THE DECAY OF A RESONANCE AND A COMPLEX ENERGY EIGENVALUE 144 G15. A PROOF THAT THE REAL PART OF THE COMPLEX DECAY ENERGY IS THE RESONANT ENERGY 146 C16. ASIDE: THE RELATIONSHIP BETWEEN THE DECAY RATE AND THE WIDTH OF THE RESONANCE 147 C17. CALCULATION OF THE COMPLEX S REQUIRED FOR DECAY 147 C18. ASIDE: WHAT HAPPENS TO M 2 1 ? 150 C19. OBTAINING THE DECAY RATE FROM THE COMPLEX S ...151 C20. CALCULATION OF NUMERICAL VALUES FOR BETA DECAY .152 APPENDIX D - ELECTRIC FIELD ENHANCEMENT OF ELECTRONIC DECAY FROM A SQUARE WELL 154 D1 . INTRODUCTION 154 92. THE MODEL: SOLVING THE SCHRODINGER EQUATION 154 D3. THE STRUCTURE OF THE MATRICES WHICH RELATE THE COEFFICIENTS 157 334. THE REFLECTED AND TRANSMITTED WAVES 160 D5. THE FIRST APPROXIMATION 163 J>6. OBTAINING THE DECAY RATE USING THE FIRST APPROXIMATION 168 07. THE SECOND APPROXIMATION 169 1>8. OBTAINING THE DECAY RATE USING THE SECOND APPROXIMATION 172 V APPENDIX E - THE SQUARE WELL IN THREE DIMENSIONS 174 APPENDIX F - TYPOGRAPHICAL ERRORS IN THE REFERENCES 178 F l . BECKER ET AL 178 F2> DESHALIT AND FESHBACH 179 v i L i s t of T a b l e s I. A Table of the F u n c t i o n I(x) 71 I I . Well Parameters f o r *H 86 I I I . Well Parameters" f o r '|F 88 IV. Well Parameters f o r '*C 89 V. The Dependence of the Decay Enhancement Upon the D i s t a n c e Over Which the E l e c t r i c F i e l d i s A p p l i e d . .94 VI. The Dependence of the Decay Enhancement Upon the F i n a l Energy of the E l e c t r o n 95 V I I . The Dependence of the Decay Enhancement Upon the S t r e n g t h of the E l e c t r i c F i e l d 96 V I I I . The Decay Enhancement f o r *F 96 IX. The Decay Enhancement in the M o d i f i e d Model 101 v i i L i s t of Figures 1. Phase space enhancement 4 2. Model of f decay 6 3. Matrix element e f f e c t in model of decay. (a) and (b) should have the same decay rate 7 4. Feynman Diagram for Free Beta Decay 12 5. A Linearly Polarized Electromagnetic Plane Wave 28 6. Graph of F i n a l Electron Energy vs. Laser Intensity Obtained by Becker,et.al 67 7. V i r t u a l p Decay 78 8. Square Well Used as Model of Nucleus 83 9. The Potential Well in an E l e c t r i c F i e l d 90 10. The Shape of the Well for a Very Strong E l e c t r i c F i e l d 100 11. The Modified Model 102 12. The Potential Well 114 13. The n=0 Resonance 144 14. The Potential Well in an E l e c t r i c F i e l d 155 15. The Potential Well in 3 Dimensions 174 v i i i Acknowledgement I would l i k e to thank my supervisor, Dr. Nathan Weiss, for suggesting t h i s topic, for a s s i s t i n g me, for knowing when an idea i s only half-baked, and generally, for being a concerned and conscientious human being. I am also indebted to Dr. Douglas Beder who read over my thesis and offered useful suggest ions. I. also appreciate the stimulating conversations I had with Mark Shegelski and Ira Bl e v i s . Although my attempted solution of the three dimensional square well in an external e l e c t r i c f i e l d resulted in f a i l u r e a f t e r three months, I am thankful to Dr. William Unruh for helping me with t h i s . F i n a l l y , I thank N.S.E.R.C. for f i n a n c i a l support. DEDICATION To Miriam, who made sure I didn't work on thi s during our honeymoon. X Beware of the Ann Arbour Ambassador! 1 I. INTRODUCTION 1.1 The Results Of Becker,et.al. About one year ago, Becker et a l . 1 , published a l e t t e r on experimental s i t u a t i o n in which the most powerful laser within the c a p a b i l i t i e s of today's technology i s shot at a nucleus which i s beta unstable. Their c a l c u l a t i o n shows that 3H, which has a p~ h a l f - l i f e 2 of 12.33 years, w i l l decay 2 x 10" times faster in the laser. That i s , i t s new half l i f e w i l l be a mere 5.4 hours. S i m i l a r l y , the decay of '^ F, which has a p* h a l f - l i f e 3 of 109.8 minutes, is enhanced by a factor of 18. Its new half l i f e w i l l be 6.1 minutes. 1.2 How A Decay Rate Can Be Modified: Phase Space Vs Matrix Element Ef f e c t s "Why," the reader might ask, "should a laser have any e f f e c t at a l l upon the decay rate?" To answer t h i s one should note that a l l decays have the following form: where i s the decay rate. One starts in the i n i t i a l state /i> 1 Becker et a l . See "References". 2 Lederer and Shirley, pg.1. 3 i b i d , pg.24. the laser enhancement of nuclear beta decay. They envisage an T R A A J S I T / © r > / WA-rrt/x PHASE S?ACE (1.2.1) 2 at time t« . Using the time evolution operator U ( t ^ f t j _ ) , we time evolve the state to time tf , and ask for the amplitude that the state has decayed into the state (f>. This i s c a l l e d the tr a n s i t i o n matrix. It generally contains £ functions as a result of energy-momentum conservation. Squaring and di v i d i n g by the time i n t e r v a l y i e l d s the rate of decay into a p a r t i c u l a r f i n a l state | f ^ . IO m&Y consist of many p a r t i c l e s , each of which has a pa r t i c u l a r momentum. To obtain the rate of decay into any f i n a l state with p a r t i c l e momenta ly i n g between kf and k £ + dk~, (k* f — k f, k^ , . . . ) , one must multiply by the number of states in t h i s momentum i n t e r v a l . The number of states i s given by SdV where § i s the density of states in momentum space. d3lT i s the volume element in momentum space and i s c a l l e d the "phase space" factor. One may integrate eq.(1.2.1) to obtain the t o t a l rate of decay into a l l states, within the volume in momentum space. This volume is referred to as the phase space. Consider a f i n a l state in which we have an electron of momentum k c. If we choose to l o c a l i z e our system within a box of volume VB , then R= 7 ^ . . The phase space for the electron i s ka. d |keld/ig where d/i eis the s o l i d angle subtended by ke . (In general, the t r a n s i t i o n matrix squared w i l l contain a factor of to cancel the V_ in so V- does not matter. One therefore l e t s V„-»fl0. One also l e t s t^-zco and t(-*oo because w i l l also be cancelled by a factor in the t r a n s i t i o n matrix.) One notices that one may adjust the phase space and therefore the decay rate merely by adjusting JkJ . If |k e| l i s limited in size then the decay is said to be suppressed by phase space; i f 3 Jk e) 2'<0 i s required by energy conservation then the decay cannot happen at a l l . To answer our o r i g i n a l question, there are two general ways in which one may affect a decay. One may do something to the system which af f e c t s 1) mainly the phase space, or 2) mainly the matrix element. A pure phase space enhancement i s easy to understand. Only the energy-momentum conservation delta functions in the matrix element are changed, allowing the t o t a l energy of the f i n a l p a r t i c l e s to be higher. There i s a greater volume in momentum space and hence a greater number of f i n a l states accessible by the decay. If there are more states available, then the decay w i l l happen more read i l y . As an example of t h i s , consider the following scenario: A free neutron decays into a proton, electron, and neutrino. There i s an energy E 0 = m„ - trip - me available for this decay such that E B+ E„ = E 0 , (neglecting the proton's r e c o i l energy), where E e and Ev are the kinetic energies of the electron and neutrino, respectively, and the m's are the masses of the p a r t i c l e s . 0<E e<E o. The maximum momentum of the electron i s k0 and the electron's phase space i s a sphere of radius k<, i l l u s t r a t e d in FIG. 1(a). 4 Figure 1 - Phase space enhancement Suppose now that we have some way of increasing E 0 ; that i s , suppose we could increase the neutron's mass or decrease the proton's mass. The new maximum momentum of the electron i s \iQ>.k» and the new phase space i s shown in FIG. 1(b). The neutrino's phase space i s correspondingly increased, and we see, by the above arguments, that the decay rate i s increased. A matrix element e f f e c t i s harder to understand ph y s i c a l l y , and i t i s t h i s e f f e c t which must be understood i f one i s to comprehend the laser enhancement of B decay. Although Becker et a l . ' claim that the phase space i s increased, i t i s the claim of this thesis that t h i s i s not the reason for the decay enhancement. The laser a f f e c t s the decay by imparting momentum to the decay electrons. This occurs as an electron absorbs and emits photons, and i s accelerated over a c l a s s i c a l distance, by absorbing a net amount of photons. Equilibrium 4 Becker et a l . pg.1263. 5 se t s i n and the e l e c t r o n reaches a maximum speed when the a b s o r p t i o n of photons i s balanced by the emission of photons. At f i r s t glance, one might reason that the momentum of the e l e c t r o n i s higher, t h e r e f o r e the phase space i s l a r g e r , c a using a decay enhancement. Let us do more than g l a n c e , by c o n s i d e r i n g the f o l l o w i n g s c e n a r i o . Suppose a decay e l e c t r o n i n a l a s e r f i e l d reaches e q u i l i b r i u m at momentum k a f t e r being a c c e l e r a t e d a c l a s s i c a l d i s t a n c e d, and we o b t a i n a decay r a t e c j . Suppose f u r t h e r that we choose to modify the experimental s i t u a t i o n by adding an extremely intense l a s e r f i e l d beyond d which i s v a s t l y more powerful than the f i r s t l a s e r f i e l d . The e l e c t r o n w i l l reach a new e q u i l i b r i u m at momentum k'>>k a f t e r being a c c e l e r a t e d a f u r t h e r d i s t a n c e d'. How i s the decay rate a f f e c t e d by t h i s ? Even i f k' = 1 0 , 0 0 0 x k, one would expect no measurable e f f e c t at a l l . The system has a l r e a d y decayed. I t does not care much about what happens a c l a s s i c a l d i s t a n c e away. Upon pondering the above c o n s i d e r a t i o n s , i t i s concluded that even i f the phase space i s v a s t l y i n c r e a s e d by the l a s e r , there i s something w i t h i n the matrix element which wipes out the e f f e c t of t h i s i n c r e a s e : something which accentuates what i s happening c l o s e to the nucleus and e l i m i n a t e s what i s happening f a r from the nucleus. T h i s i s the type of matrix element e f f e c t we wish to understand. I t may be q u i t e s u b t l e because we are working i n momentum space and not i n p o s i t i o n space, making i t d i f f i c u l t to speak of l o c a l i t y . A p e c u l i a r t r a n s p o s i t i o n i s that an i n t e r n a l energy i n c r e a s e manifests i t s e l f as a phase space e f f e c t wheras an e x t e r n a l a c c e l e r a t i o n m a n i f e s t s i t s e l f as 6 a matrix element e f f e c t . As an a i d to understanding t h i s matrix element e f f e c t , we c r e a t e a one-dimensional n e u t r i n o l e s s model of beta decay, c o n s i s t i n g of a square w e l l with b a r r i e r s through which a quasi-bound e l e c t r o n l e a k s . I t i s i l l u s t r a t e d i n FIG. 2. V i s the p o t e n t i a l and x i s the d i s t a n c e . V ST/TTt F i g u r e 2 - Model of jB decay The e f f e c t of the l a s e r i s modelled by a l i n e a r p o t e n t i a l which a c c e l e r a t e s the e l e c t r o n over a c l a s s i c a l d i s t a n c e , as i n FIG. 3 ( a ) . 7 Figure 3 - Matrix element ef f e c t in model of decay. (a) and (b) should have the same decay rate. In l i g h t of thi s model, l e t us re i t e r a t e the above discussion. Suppose a decay electron reaches f i n a l momentum k after being accelerated a c l a s s i c a l distance d, as in FIG. 3(a), and we obtain a decay rate W. Suppose further that we extend the region of acceleration as in FIG. 3(b), causing the electron to reach f i n a l momentum k/>>k after being accelerated a further distance d'. We expect the decay rate to be unaffected by t h i s , for reasons similar to those described above: the decay i s affected only by what happens l o c a l l y , and not by what happens at c l a s s i c a l distances. This model thus seems a very good model of the kind of matrix element effect discussed above. 8 1.3 An Overview Of T h i s T h e s i s In order t o c a l c u l a t e a t r a n s i t i o n matrix i n quantum f i e l d t heory, one expresses U(co,-oo) i n terms of the f i e l d o p e r a t o r s and uses p e r t u r b a t i o n theory. (In p r a c t i c e , one j u s t uses the Feynman r u l e s . ) T h i s i s done f o r o r d i n a r y p decay i n chapter 2. The e l e c t r o n f i e l d operator c r e a t e s and a n n i h i l a t e s e l e c t r o n s (and p o s i t r o n s ) in s t a t e s which are normalized plane wave s o l u t i o n s to D i r a c ' s equation. To c a l c u l a t e l a s e r enhanced beta decay, Becker et a l . r e p l a c e d the usual e l e c t r o n f i e l d operator by a f i e l d o p erator which c r e a t e s and a n n i h i l a t e s e l e c t r o n s i n s t a t e s which are normalized Volkov s o l u t i o n s to D i r a c ' s e q u a t i o n . The Volkov s o l u t i o n i s an exact s o l u t i o n to D i r a c ' s equation i n the e x t e r n a l p o t e n t i a l of a plane e l e c t r o m a g n e t i c wave. I t i s d e r i v e d i n chapter 3. The formula f o r l a s e r enhanced B decay o b t a i n e d by Becker et a l . i s d e r i v e d i n chapter 4. In chapter 5, we present the model of p decay i n t r o d u c e d i n the l a s t s e c t i o n . A d e t a i l e d account of resonance e f f e c t s i n the type of square w e l l used, appears i n appendices C and D, along with the c a l c u l a t i o n of the decay of a resonance, and how an e l e c t r i c f i e l d a f f e c t s t h i s decay. An attempt to extend t h i s to three dimensions i s o u t l i n e d i n appendix E. Appendix F i s a l i s t of the t y p o g r a p h i c a l e r r o r s n o t i c e d i n the r e f e r e n c e s . 9 1 .4 Conventions And Units The conventions used for the metric, the IT matrices and the Dirac spinors appear in appendix A, along with the plane wave solutions to Dirac's equation. These are the conventions of Bjorken and D r e l l 5 with the exception that the Dirac spinors are normalized to 2m rather than 1. Unless otherwise stated, a Greek index runs from 0 to 3 wheras an English index runs from 1 to 3. Summation convention i s used throughout, always summing an upper index with a lower index, meaning there i s an i m p l i c i t metric involved. Units used throughout t h i s thesis are those such that 'fi=c=1, where c is the speed of l i g h t and fi=h/2ir where h i s Planck's constant. This leaves us with only one independent unit, which we may c a l l MeV, (mega-electron-volt), or fm (Fermi I0",srm), or s (second) depending upon the circumstances. For example: energy has units of MeV, or fm-', or s~'; length has units of fm, or s, or MeV"1; time has units of s, or fm, or MeV-'. These units are inter r e l a t e d by the following: 6 *c = 197.322 MeV fm = 1 * = 6.58195 x 10"23" MeV s = 1 c = 2.997 x IO 2 3 fm s"' = 1 (1.4.1) The electromagnetic units are Gaussian modified by setting c = -n = 1. This makes charge dimensionless and e l e c t r i c f i e l d 5 Bjorken and D r e l l , appendix A. 6 Calculated using the data from Lederer and Shirley, appendix 1. 10 have dimensions of MeV1 or MeV fm"'. The charge of the electron Ft? i 11 I I . NUCLEAR BETA DECAY 2.1 Calculation Of Decay Rate Using Feynman Rules The Fermi theory approximates the weak interaction by coupling the lepton and hadron currents at a single point. This gives the Fermi interaction lagrangian 7 : (2.1.1) f^p i V© and Hi* are the f i e l d operators for the proton, neutron, electron and neutrino, respectively. Conventions for the y matrices are those used by Bjorken and D r e l l 8 (see appendix A). V s y*^- "h.c." stands for "Hermitian conjugate". g is the weak coupling constant. C^ and C A are the c o e f f i c i e n t s for the vector and a x i a l parts, respectively, of the i n t e r a c t i o n . (If the nucleons were leptons, which have no v i n t e r n a l structure, then one would find C v = C^ y i e l d i n g the pure 1-^5, or "V-A" interaction.) de Shalit and Feshbach 9 give the following values for the coupling constants: C vg = 1.4100 x 10" 4 < ? erg cm 3 = 8.802 x l0" SMeV fm3 = 1 . 1456 x l0~"Mev~ a C A A V = 1 « 2 4 8 - 0.010 The Feynman diagram corresponding to t h i s interaction i s shown in FIG. 4. 7 See de Sh a l i t and Feshbach, pg 834. 8 Bjorken and D r e l l , Appendix A. 9 de Shalit and Feshbach, pg.834. 12 Figure 4 - Feynman Diagram for Free Beta Decay The p's are the 4-momenta and the s's are the spins. Applying the Feynman rules of Bjorken and D r e l l 1 0 , one obtains the invariant amplitude ?^ for t h i s process: 7K=jfi UpC?t S,MMCN &)U,(?„&) Ue (feQ£(h Vs)lr» fa*) (2.1.2) \JL and I T are the 4-spinors of Appendix A. In looking at nuclear beta decay, one generally uses the n o n - r e l a t i v i s t i c approximation to the f i e l d operators for the hadronic current. This i s manifested in C£p and by l e t t i n g Lu->m, (^>|«m in eq.(A.3). We obtain: u? (?, st) y u« (p„ fJ Jz^Vj^ SSpS^ $ M o (PL) u?t?Ps.)Wsu„(%s„) ^JznJzZilXs. vJXs W P " (2.1.3) where n i s the Kronecker delta function. Note that only the timelike component of the f i r s t expression survives, and only 1 0 Bjorken and D r e l l . Appendix B. 13 the spacelike components of the second expression survive. Up to t h i s point, we have been ignoring the fact that the nucleons are within a nucleus. We may take care of the nuclear ef f e c t s by p a r t i a l l y ignoring the interactions between nucleons, and pretending that a neutron of momentum p„ = (m*,0) has merely been replaced with a proton of momentum p p = (m* , p R ) . The r e c o i l momentum, p R , s a t i s f i e s jp^J<<m* so the nucleus i s approximately unchanged. We take care of nuclear interactions to the extent that we approximate the nucleus by noninteracting p a r t i c l e s of e f f e c t i v e mass m* . The e f f e c t i v e mass varies from nucleus to nucleus, making i t energetically favourable for p~ decay (m* >m* ) in some nuclei and for p + decay (irip >m* ) in other n u c l e i . A l l we need to know i s m*-mjl which i s the experimentally measured decay energy. The actual values of m* and m£ w i l l cancel in the c a l c u l a t i o n . Replacing a neutron by a proton and changing nothing else, as in eq.(2.1.3a) i s equivalent to r a i s i n g the isospin of one nucleon. That i s , one applies the isospin r a i s i n g operator, ^ 'f* (r) , to the i n i t i a l nuclear state. Since we can raise the r 1 isospin of any neutron in the nucleus, one must sum over a l l nucleons. • ( f * acting on a proton w i l l contribute zero to the sum). In eq.(2.1.3b) the r a i s i n g of isospin i s accompanied by a spin t r a n s i t i o n in each nucleon operated on by +. With these considerations, eqs.(2.1.3) become: 14 UpOr, s P ) ^ i u ( ? * a . ) ^ J ^ J ^ < P I ZJ-vWl<> (2.1.4) j a n d | f ^ are the i n i t i a l and f i n a l nuclear states. r denotes a p a r t i c u l a r nucleon. Mp i s c a l l e d the Fermi matrix, wheras M & T i s c a l l e d the Gamov-Teller matrix. 1 1 i s used for y decay. One can also define which uses the isospin lowering operator to get f$ + decay. From th i s point on, we w i l l drop the ^ and ^ to obtain a general result applicable to JS~ or "p* decay. Inserting eq.(2.1.4) into eq.(2.1.2) y i e l d s : (2.1.5) Squaring, one obtains the following: (2.1.6) Here yu . and Sc denote that only the spacelike components of ytA. and 1 1 de Sh a l i t and Feshbach, pg.834. 15 OC survive. This notation i s used in order to avoid confusion about which Greek index to contract with which English index. Summing over s and using eq.(A.6), the trace becomes: S* . (2.1.7) Using the trace theorems of Bjorken and D r e l l 1 2 , one obtains: + 1 (2.1.8) Eq.(2.1.6) becomes: (2.1.9) Let us calculate the unpolarized rate for p decay. We assume that the i n i t i a l nucleus has a t o t a l spin , and average over a l l spin projections mj. There are 2J.+ 1 of these. Let us also sum over a l l possible f i n a l spin projections m^ . Thus, we need to evaluate —r-r £L 2 VYY\ 1*. This w i l l be denoted by ^ j " " ^ f ^ f 1 - l ^ z l ^ " * : t i s shown in appendix B that the following equations are true: 1 2 Bjorken and Drell,pgs.104-105. 16 YZ /A&T MF - O (a) 2 * - AW t - = j H I ALT * • * (c) (2.1.10) Eq.(2.1.9) i s consequently s i m p l i f i e d : (2.1.11) To obtain the decay rate from |??tf* one uses the standard formula from Bjorken and D r e l l 1 3 : (2.1.12) duUfi i s the number of decays per unit time of a neutron at rest producing an electron of momentum pe to "pB+d3p£ , a neutrino of momentum py to "p^+d^p^, and a proton of momentum p p to p p+d 3pp. The r e c o i l energy, u;p, i s approximately mp. Since the neutrons are within a nucleus, we must use their e f f e c t i v e masses, mp and m*. Integrating over the r e c o i l momentum, and setting 1 3 Bjorken and D r e l l , appendix B. 17 U>n-Wp=u>0, eq.(2.1.12) becomes: ( 2 . 1 . 1 3 ) We now integrate over neutrino momenta d 9 p = uiclto.d-^., (D v=solid angle). The angular contribution n u l l i f i e s the Pa-P^, term in eq.(2.1.11). We obtain: (2.1.14) (2.1.15) (2.1.16) Letting £ = ^ * £ 0=^e, eg. (2. 1.16) may be written as follows: (2.1.17) Now: 18 X (a) (t; (O (2.1.18) Eq.(2.1.18b) has been integrated with the help of a standard i n t e g r a l t a b l e . 1 * (2.1.19) The t o t a l decay rate i s : (2.1.20) 1 4 Selby, eq.168, pg.425. 19 Corrections to eq.(2.1.20) due to the e f f e c t of nuclear charge on the electron's wave function, are discussed in deShalit and Feshbach. 1 5 2.2 Derivation Of Feynman Rules The above c a l c u l a t i o n of beta decay i s based on eq.(2.1.12). In order to deal with laser enhanced beta decay, we must extend t h i s c a l c u l a t i o n into realms in which a b l i n d application of (2.1.12) can be expected to result in f a i l u r e . On account of t h i s , we now derive equation (2.1.12). Let S be the amplitude that i n i t i a l state |n(p„s n)^ becomes the f i n a l state Jp(pp s p ) ,e(p es c ) ,^(p yR^ after applying the time evolution operator U(t^,,t-) from t'1=-oo to tf = +oo . That i s : <Hr^)>e(p^e)tj/(7p^)\u(^r^)hC?n^))' ( 2 . 2 . D One may expand the time evolution operator as f o l l o w s 1 6 : « (2.2.2) Here P represents the time ordered product, and ^ £ ' M T i s the interaction Hamiltonian density. }^UT=~^.i where i s given by eq.(2.1.1). FIG. 4 corresponds to the f i r s t term in the sum of eq.(2.2.2). Eq.(2.2.1) becomes: 1 5 deShalit and Feshbach, pg.777. 1 6 See Schweber, pg.333. 20 ( 2 . 2 . 3 ) We now e x p a n d t h e f i e l d o p e r a t o r s i n t e r m s o f t h e d e l t a f u n c t i o n n o r m a l i z e d s o l u t i o n s t o t h e D i r a c e q u a t i o n , a s g i v e n i n a p p e n d i x A. ( 2 . 2 . 4 a ) ( 2 . 2 . 4 b ) ( 2 . 2 . 4 c ) ^ ( 2 . 2 . 4 d ) a a n d a + a r e t h e a n n i h i l a t i o n a n d c r e a t i o n o p e r a t o r s , r e s p e c t i v e l y , o f f e r m i o n s i n a D i r a c p l a n e wave s t a t e , g i v e n by e q . ( A . 3 ) i n a p p e n d i x A. b a n d b + , c o n v e r s e l y , a r e t h e a n n i h i l a t i o n a n d c r e a t i o n o p e r a t o r s , r e s p e c t i v e l y , o f a n t i - f e r m i o n s i n a D i r a c p l a n e wave s t a t e . T h e i r a n t i - c o m m u t a t i o n r e l a t i o n s a r e : . 21 l ^ U , kVv} = ^ff-W (2.2.5) With these expansions, eq. (2.2.3) becomes: _ x (2.2.6) The f i n a l matrix element cont r ibutes : <n> = Jw. s w ^ c f ^ - s ) rtjv-*) svf-*; s Y M J Therefore eq. (2.2.6) becomes: 4 u<> ft ^ y ' (Cv~^CM ft s0 £ fr- ^ (ft, R) (2.2.7) Upon integrat ing over x and using eq . (2 .1 .2 ) , one obta ins : 22 (2.2.8) Squaring(2.2.8) gives r i s e to: At f i r s t the 0) may seem problematic but one may treat i t by l o c a l i z i n g our system within a very large box of 4-volume VgT B, and confining our spacetime integrals to t h i s domain. It turns out that the answer i s independent of the 4-volume so we may choose i t to be a r b i t r a r i l y large. (2.2.10) Our f i e l d operators acquire the box normalization of appendix A, (Eq.(A.4)). As there are 4 f i e l d operators, |S/2 acquires a factor of £^^~J • Therefore: (2.2.11) The t o t a l rate of decay into a l l possible f i n a l states i s given by: 23 (2.2.12) § i s the d e n s i t y of s t a t e s i n momentum space: $= When eq.(2.2.1!) i s i n s e r t e d i n t o e q . ( 2 . 2 . l 2 ) , one o b t a i n s : (2.2.13) When the i n i t i a l neutron i s at r e s t U j , - * ^ and eq.(2.2.13) i s i d e n t i c a l to eq.(2.1.12). Q.E.D. 2.3 What Is The Phase Space Of The E l e c t r o n ? We see from eq.(2.2.l2) that the phase space c o n t r i b u t i o n , (as d i s c u s s e d i n the i n t r o d u c t i o n ) , i s cr p p d p e d , without the u/ s. The UJ's i n eq.(2.2.l3) are a c t u a l l y p a r t of the t r a n s i t i o n m atrix. F o l l o w i n g the development from eq.(2.1.12) to see what happens t o the phase space, we f i n d t h at -11 £ d£d -C2e i s the phase space c o n t r i b u t i o n f o r the e l e c t r o n , and ( f 0 ~ £ ) 2 i s the phase space c o n t r i b u t i o n f o r the n e u t r i n o , i n eq.(2.1.17). Suppose we take a given f r e e decay t r a n s i t i o n and a c c e l e r a t e the decay e l e c t r o n . The n e u t r i n o r e t a i n s i t s former momentum but the e l e c t r o n ' s new momentum w i l l be d i f f e r e n t : we expect to change only the ? £ dEd-*^ and not the ( & " £ ) * term. Let us keep t h i s i n mind when we see how a l a s e r a f f e c t s the decay. Another t h i n g to keep i n mind i s that t h i s argument a p p l i e s 24 f o r a given f r e e decay t r a n s i t i o n . There w i l l a l s o be decay t r a n s i t i o n s o c c u r r i n g i n the l a s e r f i e l d which w i l l not occur i n the f r e e case because the e l e c t r o n must be emitted with 'negative' k i n e t i c energy and then be a c c e l e r a t e d to p o s i t i v e k i n e t i c energy. (The emission of a negative energy e l e c t r o n i s r e a l l y the a b s o r p t i o n of a p o s i t r o n which comes from a v i r t u a l e ^ e - p a i r produced by a photon.) The n e u t r i n o i s thus emitted with £3, = £ 0- £t> £ 0 because f c<0 (or because i t gains energy from the p o s i t r o n ) . T h i s causes an i n c r e a s e i n the phase space of both the e l e c t r o n and the n e u t r i n o . Laser enhanced beta decay w i l l be c a l c u l a t e d by expanding the f i e l d o p e r a t o r s i n terms of the Volkov s o l u t i o n to the D i r a c equation, r a t h e r than the plane wave expansion of eq.(2.2.4a). The o b j e c t df the next chapter i s to d e r i v e and d i s c u s s the Volkov s o l u t i o n . 25 II I . THE VOLKOV SOLUTION In 1935, D.M. Vol k o v 1 7 published an exact solution to the Dirac equation for a p a r t i c l e in the external f i e l d of an electromagnetic plane wave. This solution i s rederived in t h i s chapter, following the basic procedure of Itzyckson and Zuber. 1 8 3.1 Minimal Coupling Procedure Let p denote the canonical momentum of quantum mechanics conjuate to position x" such that fx* ,pjj = i £^. In c l a s s i c a l electrodynamics, 1 9 one may show that a charged p a r t i c l e in an external electromagnetic f i e l d has a canonical momentum given by p=TT +qA, where nr* is the kinetic momentum, q i s the charge of the p a r t i c l e , and A i s the magnetic vector potential of the f i e l d . In quantum mechanics, the canonical momentum i s p^= i ^ . The correspondence p r i n c i p l e suggests that we make the following i d e n t i f i c a t i o n : The Dirac equation, eq.(A.1), becomes: This i s known as the "minimal" coupling p r e s c r i p t i o n . See "Bibliography" under the German s p e l l i n g : "Wolkow". Itzyckson and Zuber, pg.68. Jackson, pq.574. 26 3.2 The Second Order Dirac Equation We wish to f i n d solutions to eq.(3.1.2) when A^. i s the 4-vector p o t e n t i a l for an electromagnetic plane wave of i n f i n i t e extent. It i s easier to solve eq.(3.1.2) i f we use the second order equation obtained by multiplying eq.(3.1.2) by (i^-qA^-m). One obtains: (jr^Tt^-w^f- o (3-2-1) Using identity(B.14) of appendix B, we f i n d : ( 7 r z - i cr** T^TTu - f n ^ V -Q (3.2.2) and (3.2.3) Therefore: [G 3 - % A ) 2 - J - < r ^ />. -.»>*] f = 0 (3.2.4) This i s the second order Dirac equation. 3.3 Solution Of The Second Order Dirac Equation For An Electromagnetic Plane Wave The f i e l d of an electromagnetic plane wave i s characterized by i t s 4-momentum, k^t and i t s 4-vector potential A^= £>*f (k« x). £/*" i s the p o l a r i z a t i o n vector, defined such that £ 2=-1, k»£ = 0, 27 and of course, k 2 = 0. A s p e c i f i c form of w i l l be considered l a t e r . (For c i r c u l a r p o l a r i z a t i o n , write A/4=£ff, (k»x) + £^f a(k. x) with the above conditions on and .) It i s s u f f i c i e n t to note that in the frame in which £°=0, £ i s a vector of unit norm perpendicular to the d i r e c t i o n of propagation of the wave. Note also that: Using e q . ( B . 14): - Z £ <$^)^AI Therefore: Also: (i3-y/i;2= ifO-A* -LA.?) - -0 tCk-A'+l*)) ,3.3.3) With eq.(3.3.2) and eq.(3.3.3), equation(3.2.4) becomes: (3.3.4) How do we solve this? Let us look at the s p e c i f i c case of a l i n e a r l y polarized wave t r a v e l l i n g in the z d i r e c t i o n , to stimulate ideas about the general solution. The set up i s shown in FIG. 5. 28 7 • A' e -Figure 5 - A L inear ly Po lar ized Electromagnetic Plane Wave E is the e l e c t r i c f i e l d and B i s the magnetic f i e l d : •2A- - k ° ^ = - fr° To, f f f r - i - 0 , o J ) y (3.3.5) E = Let us look at eq. (3.1.2) and denote the operator l^C-q^-m by cK • Eq. (3.1.2) is simply: X V =0. (3.3.6) Now we inquire into whether or not solut ions of (3.3.6) are stat ionary s ta tes . A stat ionary state is an eigenstate of the operator i J 0 . With k^ given by eq. (3.3.5) we read i l y see that J[ioor)d] £0, and therefore Y' i s not a steady state so lu t i on . S im i l a r l y , one a lso f i n d s : [ i ^ ,}t /0 but [ i ^ = [i^, =0. This means that ^ is an eigenstate of de f i n i t e x and y momentum, .but i t is not an eigenstate of z momentum nor an eigenstate of de f i n i t e energy. But: 29 (3.3.7) Therefore y i s a n eigenstate of d e f i n i t e P 0 +P 3 • T h e 4-momentum of the solution can consequently be written as: ?/.= 2 ( ^ , 0 , = ?^+3A> (3.3.8) where p^ i s fixed and /i i s completely indeterminate. Now: p 2 = p ^ = ( P o +3 k a )2-p^2-p^2- (p 3 k 0 ) 2 .'. ?*S*Wj f - ^ - P ^ - P ^ - V + ^ f c C M * ) (3.3.9) One also sees that m^ i s indeterminate and hence i s not a state of d e f i n i t e mass. With these considerations in mind l e t us write the solution of eq.(3.3.4) in the form: y P 6 0 = e'-^PCk-x) (3.3.10) where p^ i s given by eq.(3.3.8) with ^ chosen such that p2=m2, where m i s the rest mass of the p a r t i c l e in a n o - f i e l d environment. This insures that w i l l go to the standard solution given in appendix A as the f i e l d A^ goes to zero. Upon substitution of eq.(3.3.l0) into eq.(3.3.4), we arri v e at: We solve t h i s equation by treating the 4-spinor )P as though i t were a function and formally integrating. We note: This y i e l d s : 30 \J L 2.1c? V -fc-p i fr-p / J y (3.3.12) where u i s the constant of integration. The i n t e g r a l i s an i n d e f i n i t e i n t e g r a l . We now check our formal procedure by substituting eq.(3.3.12) back into eq.(3.3.11). It i s v a l i d with the proviso that u i s a constant 4-spinor. The f i r s t part of (3.3.12) i s readily integrated to y i e l d : 1 k-t> J " ' 'V Now: £##X = -JtXXX -because k*A=0 = -k 2A 2 -from eq(B.15) = 0 -because k 2 = 0 Therefore, in expanding the exponential, only the zeroth and f i r s t orders remain. Eq.(3.3.12) becomes: i 2 k. p J ~/ I «"-p Ik-? J (3.3.13) 31 3.4 Solution Of The F i r s t Order Dirac Equation Eq.(3.3.13) i s the solution to eq.(3.2.4), the second order equation, for a r b i t r a r y u. Let us check eq.(3.1.2), the f i r s t order equation, to see what extra conditions must be imposed. Writing ^ in terms of eqs.(3.3.l0) and (3.3.12), one finds: I -J L zh--P \ k-P afr-p / J J (3.4.1) Substituting eq.(3.4.1) into the f i r s t order equation, eq.(3.1.2), one obtains: I I I lk-p \ k-P *.k.?JjJ 5 (3.4.2) Noting that &M. = Q and writing y in terms of eq.(3.3.13) now rather than eq.(3.4.1), our expression takes on the following form: 1 ^ *P * W ,3.4.3) Noting again that &k' = 0 and 0.fC=-h2j( we get: (3.4.4) Using eq.(B.l6) in appendix B to commute (14- . f t ^ ^ U past (cC-m), eq.(3.4.4) becomes: 32 (3.4.5) This agrees with the equation given in Itzyckson and Zuber, 2 0 and implies: (3.4.6) Therefore, the solution (3.3.13) to the second order equation i s a v a l i d solution to the f i r s t order equation, providing that u s a t i s f i e s eq.(3.4.6). That i s , u i s a Dirac spinor. To f i x the normalization, we know that ^ ( x ) must become'a solution of 4-momentum p of Dirac's free equation i f we l e t 0, in accordance with the discussion in sec. 3.3. Let us assume that the normalization does not depend, in any manner, upon A*. (This assumption i s checked in the next section.) Adopting the delta normalization of eq.(A.3) in appendix A, eq.(3.3.13) becomes: 2 0 Itzyckson and Zuber, pq.68. 33 (3.4.7) This i s the Volkov solution. 3.5 Normalization Of The Volkov Solution Let us check the assumption that the normalization does not depend upon A"-. Choose A^=0, and |A| =a, where |A| i s the amplitude of A, (for either linear or c i r c u l a r p o l a r i z a t i o n ) . Condition (3.4.6) does not preclude the p o s s i b i l i t y that u depends upon a. Therefore, l e t : u=f(a)u f S, where u ? s i s the Dirac spinor. (3.5.1) f(0)=1 I i s given by eq.(3.4.7) We need f(0)=1 in order to s a t i s f y the condition that eq.(3.5.l) becomes the free f i e l d solution when a=0. To choose f( a ) , we demand that the Volkov % ( * ) be ^-normalized. Before checking t h i s , l e t us f i r s t calculate the spin-averaged current density. It i s given by: 34 (3.5.2) Using eq.(3.5.l) and id e n t i t y (A.6): 1 V W * p - (.'• ' V ^ - p / \ ik'-p (3.5.3) Using the trace theorems of Bjorken and D r e l l 2 1 (3.5.4) Assuming A°=0: J 0 = M [ L + K ( f ± A _ j*£X) (3.5.5) This w i l l be hel p f u l in deriving the normalization. It i s desired to show that the following i s true: (3.5.6) This i s a very complicated expression. We write i t as follows: 2 1 Bjorken and D r e l l , pg.104. 35 (44 " w | 2 J d V '^K^M^JU'< * «. ^ ^ ^P 2./r p Jr p ( 3 .5 .7 ) I f p ^p', t h e n t h e e x p o n e n t i a l i s o s c i l l a t o r y a n d i t seems r e a s o n a b l e t h a t t h e i n t e g r a l i s z e r o . I f p=p', t h e n t h e e x p o n e n t i a l i s 1, a n d we a r e l e f t w i t h i n t e g r a t i n g j° o v e r a l l s p a c e , w h i c h g i v e s i n f i n i t y . The % n o r m a l i z a t i o n o f eq. ( 3 . 4 . 7 ) t h u s seems v e r y r e a s o n a b l e , b u t what i s t h e c o n s t a n t i n f r o n t o f t h e S? I f we were i n t e g r a t i n g : we w o u l d c o n c l u d e t h a t j ° ( 2 T T ) 3 = 1. B u t l i f e i s n o t t h i s s i m p l e . The e x p o n e n t i a l i s much more c o m p l i c a t e d t h a n Q l^?"f>)'xt Anc; e v e n i f we l o o k e d o n l y a t t h i s p a r t o f t h e e x p o n e n t i a l , t h e A t e r m i n j° m i g h t c o n t r i b u t e : R a t h e r t h a n t r y t o s o l v e t h e h o r r i b l y c o m p l e x e q . ( 3 . 5 . 7 ) , we w i l l i n s t e a d assume t h a t t h e l a s e r f i e l d i s n o t q u i t e i n f i n i t e i n e x t e n t b u t h a s a v e r y s l o w d a m p i n g . We l e t : 36 (3.5.8) where r i s the d i s t a n c e from the o r i g i n . Transforming A i n t h i s way i n the Volkov s o l u t i o n and s u b s t i t u t i n g back i n t o the Di r a c equation, we f i n d t h a t we int r o d u c e an e r r o r of order £Q~e>r&ue to the d i f f e r e n t i a l operator a c t i n g upon Since £<<1 , t h i s e r r o r i s n e g l i g i b l e , e s p e c i a l l y f o r l a r g e r . Let us say that A/4" damped i s of non-vanishing s i z e w i t h i n a very l a r g e volume V. Outside of t h i s volume A-* i s e f f e c t i v e l y z e r o . Then: (3.5.9). where ( f r ) stands f o r the f r e e f i e l d s o l u t i o n . When p=p', both i n t e g r a l s on the r i g h t hand s i d e of eq.(3.5.10) are f i n i t e . . I f the constant i n f r o n t of the & f u n c t i o n n o r m a l i z a t i o n of d i f f e r e d from the constant i n f r o n t of the & f u n c t i o n n o r m a l i z a t i o n of V(y01-***) ^ then the two i n t e g r a l s would d i f f e r by an i n f i n i t e number, making the RHS of eq.(3.5.10) i n f i n i t e . S i nce i t i s not i n f i n i t e , Y y ^ r " e ^ and \JJ(VOLK*V) n o r m a i i z e c i w i th the same c o n s t a n t . Hence f (a) = 1 . T h i s argument may be somewhat nonrigerous but lends credence to choosing f ( a ) = 1 . Another somewhat nonrigerous argument i s to assume that 37 because A / 4(|) i s o s c i l l a t o r y , a l l the k^if) terms w i l l only contribute t h e i r average values. We denote the average value by a bar, noting that A ' u(|)=0. Therefore only the A 2 term survives. The exponential in eq.(3.5.7) becomes: Zk-f> J Writing: 2k-f the exponential becomes: e x p j i (3.5.11) Integrating over x gives ( 27T) 3 £ 3 (p' -p) . The other factors in eq.(3.5.7) give j°. We obtain: *;,£. M % T M = s s. r j - ^ s i [" f r o m e<j. (3.5". 5.) J Assuming the laser beam i s propagating in the z d i r e c t i o n : J P° (3.5.1 2) Using eq . (3.5.11) and the properties of £ functions: 38 a?* F* P. p. ?.t'° *5C3y (3-5-i3) Here we have used: aft?)- l f«T w k. -Therefore: Therefore: f(a)=1 as deduced before. 3.6 E f f e c t i v e Mass Of A Volkov Electron What i s the significance of p ? Its sig n i f i c a n c e is shown by Landau and L i f s h i t z , 2 2 who calculate the kinetic momentum density of a p a r t i c l e in state Y£ . (Recall the d e f i n i t i o n of Tp" given before in SEC. 3.1.) They f i n d : (3.6.1) In taking the time average of the above, a l l of the terms lin e a r in A^ disappear leaving the A 2 term: 2 2 Landau and L i f s c h i t z , pg.124. 39 r ; w r t -~ r = r - £ 0 - l 3 . 6 . 2 ) We f i n d t h a t a l t h o u g h t h e p a r t i c l e i s n o t i n a s t a t e o f s t a t i o n a r y mass, we c a n d e f i n e an e f f e c t i v e mass m*, b e i n g t h e s q u a r e o f t h e t i m e a v e r a g e v a l u e o f t h e e x p e c t a t i o n v a l u e o f t h e k i n e t i c 4-momentum. 3.7 How To E x p r e s s The V o l k o v S o l u t i o n I n The Form U s e d By B e c k e r , e t . a l . B e c k e r e t a l . 2 3 u s e a p a r t i c u l a r f o r m o f Pt1 f o r c i r c u l a r p o l a r i z a t i o n g i v e n by t h e f o l l o w i n g : A - a ( o , c o s J r . X ) C Sin k-x , O ) ( 3 . 7 . , , • o* = +1, - 1 , c o r r e s p o n d i n g t o r i g h t - o r l e f t - h a n d e d c i r c u l a r p o l a r i z a t i o n , r e s p e c t i v e l y . U s i n g t h i s p a r t i c u l a r f o r m i n e q . ( 3 . 4 . 7 ) , w i t h q=e=-|e|, t h e c h a r g e o f t h e e l e c t r o n , we a r r i v e a t : 2 3 B e c k e r e t a l . p g . 1 2 6 3 . 40 j*- ? A + ( e a r 2 fop (3.7.2) Let us write the o s c i l l a t o r y part of I in terms of a Fourier s e r i e s . Let : (3.7.3) C h , the coe f f i c i en t s of the Fourier s e r i e s , are determined by: If Jo 7f Jo (3.7.4) Looking at a standard table of Bessel funct ion i d e n t i t i e s , 2 4 we f ind the fol lowing integra l representat ion: 2 4 Abramowitz and Stegun, eq .9 .1 .21 , pg.360. 41 (3.7.5) Therefore: (3.7.6) (3.7.7) Let us rewrite I in the fol lowing form: I=-p.x-^§£L Px [cos* c^fr.x -asm* a* lr-* ] 7>j- -px f (3.7.8) J =• - p - X +- ^ cos | ? ~ COS (ct-hcrk.x) (3.7.9) From eq.(3.7.7) and eq . (3 .7 .9 ) : (3.7.10) Noting that J „ (-<rz ) = (-a)" J „ ( z ) , and that : 42 we can take the sum over all n in eq.O.7.10) because: T h e r e f o r e , eq.(3.7.l0) becomes: This may be w r i t t e n as: H - -as (3.7.11) because the s i n e p a r t i s odd under s i g n change of n and c o n t r i b u t e s n o t h i n g . F o l l o w i n g the n o t a t i o n of Becker et a l . 2 5 we d e f i n e a parameter Y**, such t h a t : pit* _ r>+i P W ?<° _ 2- p <X> (3.7.12) A l i t t l e a l g e b r a w i l l show that : oC = ^ . Since' 2 = 2. = Uo-) c a n c e l s the z (-&) i n e q . ( 3 . 7 . 1 l ) , we are l e f t w ith: ao (3.7.13) C{1-- £ T„^ e-!'f?r-M,'>xe-i"^ In eq.(3.7.2) one may w r i t e : (3.7.14) Using eq.(3.7.13): 2 5 Becker et al.pg.1263. 43 (3.7.15) S i m i l a r l y : (3.7.16) Combining eqs.(3.7.16), (3.7.15), (3.7.14), (3.7.13) and (3.7.2), we obtain: k~ Or*0,0,Ay? (3.7.17) This i s the form of the Volkov solution used by Becker et a l . 2 6 How i s th i s to be interpreted? If we compare to the free f i e l d solution in appendix A, we seem to be summing over electrons of momenta p-nk. This corresponds to the emission and absorption of photons of 4-momentum k^. We sum over a l l emissions and absorptions to give the average momentum p. The emissions and absorptions are modulated by a phase factor & 2 6 Becker et a l . eq.(2) and eq.(3). 44 and some kind of e f f e c t i v e amplitude, V h ( p ) , for the emission absorption of n photons. 45 IV. LASER ENHANCEMENT OF NUCLEAR BETA DECAY 4.1 Expansion Of Electron F i e l d Operators In Terms Of Volkov Solutions In chapter 2, we calculated the rate of nuclear beta decay by expanding the f i e l d operators in terms of normalized free f i e l d solutions. See eq.(2.2.4). To calculate the rate of beta decay in an electromagnetic plane wave, ( i . e . a l a s e r ) , we merely expand the f i e l d operators in terms of the normalized Volkov solution (eq.(3.7.17)). The argument, z, of the Bessel function appearing in eq.(3.7.17) i s given by eq.(3.7.9): * = iSf i fj, ~ ea Tx (4.1.1) For a proton, m*>>pp, and therefore: h°™r (4.1.2) We see that Zp is about 10 of z for an electron. Noting that J„(0) =0, n / 0, and J a ( 0 ) = 1, we see that n = 0 gives the only s i g n i f i c a n t contribution in eq.(3.7.17) for a proton. (J„_,(0) and J M +,(0) would contribute for n = +1 and -1 respectively, but they are m u l t i p l i e d by -^^^ which is small i f z is small). This becomes the free f i e l d solution. Therefore, one need only consider the Volkov solution for the electron. The f i e l d operator Ye, expanded in eq.(2.2.4a), i s now expanded as follows: 46 •+ a«^'p<».rt>c/e contribution (4.1.3) T h i s i s obtained from eq.(3.7.17), knowing t h a t y = V ' V and tfytf*. Ape^ e now c r e a t e s an e l e c t r o n i n Volkov s t a t e ) ^ ^ , as opposed, to which c r e a t e s an e l e c t r o n i n a f r e e s t a t e The a n t i p a r t i c l e c o n t r i b u t i o n i s not needed e x p l i c i t l y f o r t h i s c a l c u l a t i o n . 4.2 The New Expre s s i o n For The S M a t r i x With t h i s s u b s t i t u t i o n f o r Y£ , eq.(2.2.6) becomes: r r ~ r> *. ^ /Vr) 6^—-/ I—; ) r ver jr r »prir* i v W H I n*oJ J J J J J z ^ ^ ^ r ^ (4.2.1) Here, e (pfej£e ) i s the Volkov e l e c t r o n . As i n sec.2.2, the matrix between i n i t i a l and f i n a l s t a t e s 47 cont r ibutes : <"> = ^ . s . *\ s, ^ « . C 3 * - P.) S3Cf--*J f »- & (4.2.2) Integrat ing over the momenta in eq . (4 .2 .1 ) : X (4.2.3) Integrat ing over x: 2 L c v 2o>„ 2.0/ (4.2.4) Z L e t P e + P „ + Pp" P„ = 4 • 48 From the de l ta funct ions , one sees that a l l of the cross terms disappear leaving us with: (4.2.5) 4.3 Evaluation Of The Decay Rate Following the methods of sec .2 .2 , |S| 2 becomes: And the decay rate becom-?'.;: (4.3.1 ) 49 Now a l l that remains to do i s to e v a l u a t e |7?lJ2« T h i s i s done below by squaring eq.(4.2.4b) and u s i n g the second p a r t of eq.(2.1 .6): L i J (4.3.2) Here a g a i n , jZ. and Sc denote t h a t o n l y s p a c e l i k e components of j*. and o< s u r v i v e . Let us now eva l u a t e the t r a c e i n eq.(4.3.2): (4.3.3) T h i s r e p l a c e s eq.(2.1.7). 50 + 2 4-4 e a . Using the t r a c e theorems of Bjorken and D r e l l 2 7 i n an arduous marathon of a l g e b r a , we a r r i v e a t : [ rpJ^ h - fro, kjj( £tl it < t,„yj)J 4 (4.3.4) We have assumed k= (k° ,0,0,k 9 ) = (u>, O,0,U J ) . 2 7 Bjorken and D r e l l , pg.104. 51 If we now insert eq. (4.3.4) into eq. (4.3.2) we a r r i ve at the fol lowing express ion: E l^P* 4 - 3 u ^ ^ ) T ^ ) f i c v r i / i p i z ( u J e u , > + ^ . : p j ^ I 1 +" I C „ H (wu/,.- k* • Pv) IA1.rI1 + Pv- Al.? k • A l . r + M „ k • 4-^continued — 52 ^—» continued) (e<0* f fr-pe ( 4 . 3 . 5 ) 53 This horrendous expression may be compared with eg.(2.1.9). If we l e t a->0, only the f i r s t term in eq.(4.3.5) survives, for n = 0, y i e l d i n g eg.(2.1.9). The bulk of eg.(4.3.5) disappears when we take the unpolarized t r a n s i t i o n matrix. Using the notation of sec.2.1 we wish to evaluate: [Tr^f 2^ -]>* ^ j t y J 2 • As in sec.2.1, we use the following i d e n t i t i e s : &T*MGT = O (4.3.6) We obtain: 54 Z7i*l - ea + (4.3.7) If we follow the procedure in sec.2.1 we see that upon integrat ing over the d i rec t ion of neutrino momentum, the Pe*P^ , , k-p^ , and p^(S' hcosf +5^sinY/), terms contr ibute zero. Dropping these terms from eq . (4 .3 .7 ) , we obta in : 55 (4.3.8) Let us deal with eq.(4.3.1) in the same manner with which eq.(2.1.12) is dealt with. Integrating over the proton momentum, approximating ups m*, using the e f f e c t i v e masses of the protons and neutrons within the nucleus, and setting u^ -u^=^ # we obtain: (4.3.9) Of course: d 3p v= p^d (p^l d i l ^ = duy3/l y . Integrating over div,, and &flv we obtain: du^ = — - — 7 . r Y " - l l * ^d2L\fP[ I2 (4.3.10) Using eq.(2.1.15) in eq.(4.3.!0) we obtain: 56 1 00 We note from the delta functions in eq.(4.3.9) that: - UJ 0 - + /i co (4.3.11) As lo»j,>0, we should write ^(ujy) in diu,. to ensure t h i s . (4.3.12) Inserting eq.(4.3.8) into the above, we obtain: 87T-4-(4.3.13) Let us nondimensionalize a l l of the energies by di v i d i n g by the electron mass: m« rue (4 .3 .14 ) We obtain: 57 d » * . = £ X 7 ^ 0 l ^ A U M G , | W > (4.3.15) (4.3.16) Following Becker et a l . 2 8 we wr i te : From eq . (3 .7 .2 ) : UJe — }Oe ^ leaf- UJ We V»e 2 k . p e <^e (4.3.18) Now: : . f r . p e = (JOVT)9 ( e - ^ e a - / ' c o s e ) Eq . (4 .3 . !8 ) becomes: 2 8 Becker et a l . e q . d ) and eq. (5c ) . 58 In eq . (4 .3 .16 ) : e g - e g . _ v (4.3.20) Eqs. (4.3.15) and (4.3.16) become: (4.3.22) +0-<0x„] (?e'5c«sy* ?.wc.-»y^ + (4.3.23) From eq.(3.7.12) and eq . (3 .7 .8 ) : 59 ?C0 COS V =S/7) * = -It Px $/h Y = cos = ? e (4.3.24) ?JL Using the Bessel function i d e n t i t i e s : 2 9 -X+i = (4.3.25) and the expression obtained by multiplying these two together: 2 (4.3.26) along with: (4.3.27) we obtain eq.(4.3.23) in the following form: 2 9 Abramowitz and Stegun, eq.9.1.27, pg.361. 60 4.4 C o m p a r i s o n s To The F r e e F i e l d D e c a y R a t e E q u a t i o n s ( 4 . 3 . 2 2 ) a n d ( 4 . 3 . 2 8 ) a r e i d e n t i c a l t o e q u a t i o n s ( 5 a ) a n d (5b) o f B e c k e r e t a l . 3 0 They a r e t o be c o m p a r e d w i t h e q . ( 2 . 1 . l 7 ) , t h e d e c a y r a t e w i t h o u t t h e l a s e r f i e l d . We f i n d t h a t : Z(€9-e)2 i n e g . ( 2 . 1 . 1 7 ) i s r e p l a c e d b y : r\-~oo i n e q . ( 4 . 3 . 2 2 ) . T h a t i s : ( 4 . 4 . 1 ) 3 0 B e c k e r e t a l . p g . 1 2 6 3 . 61 4.5 Analysis Part 1; What Is Going On? What do these equations mean? Looking at the discussion in sec.2.3, we expected the ^ 6 2 - l ' E term to change but not the (£."£) 2 term. At f i r s t glance, i t seems eq. (4.4.1) shows the reverse: a change in the phase space contribution of the neutrino. Let us examine th i s more c l o s e l y . Recall from eq.(4.2.1) that we are c a l c u l a t i n g the amplitude for producing an electron in a Volkov state . From eqs.(3.6.3), and (3.7.2) the electron in thi s state has an average ki n e t i c 4-momentum of p a= pe + j-^ —, and an e f f e c t i v e mass of EC = + (ea) 2= m2. (When V = 1 , m* = JPme.) Its energy i s tue - H + ^ ^k'^ ' w h e r e 1 0 , 0 = Jfez+ rn^' One can show that ='Jfe^W* ' Therefore p e , GJ, and m| behave as one expects momentum, energy, and mass to behave. These are the parameters governing the p a r t i c l e in the f i n a l state. Now, l e t us look at the delta function in eq.(4.3.9). We see that =0^+04-nu> . W « i s the energy available for the decay, and wv i s the energy of the neutrino created, leaving CDe-nuu to be the energy of the electron created. But we know that our f i n a l time evolved state must be a Volkov state with energy Cue . Therefore, the electron absorbs n photons aft e r i t is created so that i t s f i n a l state at t = 00 has energy Coe. (If n<0 then i t emitts )n| photons.) We also note that the electron i s created with 4-momentum p^ - nk. This can be seen from the delta functions in eq.(4.3.1). The 4-momentum available to give to the electron and neutrino is p =p -p , the 4-momentum of the f i n a l Volkov electron i s pe , and the 4-momentum of the neutrino 62 is a =p„ -pp ~pe +nk. Therefore, p0 -py = (p e ) w t^i =pe-nk, and (p c-nk) 2*m 2. Thus the electron i s created off the " e f f e c t i v e mass s h e l l " (p* ^ nu1 ) and must absorb n photons so that i t becomes on the e f f e c t i v e mass s h e l l (p* =m« ). We sum over a l l n but we w i l l find in sec.4.7 that the Bessel functions are maximal for a p a r t i c u l a r value of n. This means that most of the contribution to the decay rate comes from electrons which are created a given amount off the e f f e c t i v e mass s h e l l . Of course, the laser f i e l d does not extend to i n f i n i t y but is of f i n i t e s i z e . As the electron exits the laser f i e l d and a-»0 we might expect that u? e -»i^> e . Since our detector i s outside the f i e l d we might expect to measure electrons of energy u^ e. This assumes two things: 1) When the electron exits the f i e l d , a-»0 as though i t were equivalent to damping at " i n f i n i t y (discussed in sec.3.5) and does not a f f e c t anything else. (This i s the adiabatic approximation.> 2) There i s enough room inside the laser beam for the absorption of the n photons required to put the off-effective-mass-shell electron into a Volkov state. Becker et a l . 3 1 state that the small size of a focused pulsed laser packet may cause problems so one should take the laser packet to be as large as possible. This is brought up again in sec.4.8. 3 1 Becker et a l . pg.1265. 63 The range of n i s limited by the step function which dictates t h a t : 3 2 n> no = ^ f e + J£--eo) . (4.5.1) <x« V IAS ' It no<0, the electron can emit a maximum of (n a J photons, meaning that the electron comes out of the decay with energy to spare. If no>0, the decay would not occur in free space, but i t can occur in the laser f i e l d providing the electron absorbs a minimum number, n 0, of photons. Thus we see that the phase space available for decay i s increased. This i s not necessarily the cause of decay enhancement as was pointed out in sec.1.2 but we w i l l address this question l a t e r . We are now in the position to address our o r i g i n a l concern about why the phase space contribution of the neutrino seems to be changed. We note that eq.(4.4.1) does not compare the decay rates for a given decay t r a n s i t i o n : i t compares the decay rates for a given f i n a l energy mne£, of the electron (after i t leaves the laser f i e l d ) . In the free f i e l d case, a decay occurs producing an electron of energy m e£. In the laser f i e l d case, a d i f f e r e n t decay occurs, then the electron absorbs photons from the f i e l d to obtain f i n a l energy me€. Let us look at what happens for the same decay t r a n s i t i o n - "sameness" being s i g n i f i e d by the neutrino having the same energy in both cases, (because the energy of the neutrino s i g n i f i e s the energy of the electron as i t i s created, before i t is accelerated by the f i e l d ) . Let the electron in the free f i e l d case have energy 3 2 Becker at a l . eq.(6). 64 me£^>. Therefore, from eq. (2.1.17), we have: (4.5.2) ITiH We see that the energy of the neutrino i s governed by: Looking to eq. (4.3.28), we see By = £> + ~ , where wig the subscript "# " denotes " l a s e r " . We wish to have the neutrino with the same energy in both cases: IT), (4.5.3) One can also solve eq.(4.5.3) for £^. Taking n to be a large p o s i t i v e number, (n>n 0), we have £ A>£^, meaning the f i n a l energy for the electron i s higher in a laser f i e l d , for the same decay t r a n s i t i o n . Plugging into eqs.(4.3.22) and (4.3.28): DW<; = & £ TTT £ Ocvi>rivic.n^j> dTT ri=na 27i-*-I A t (4.5.4) J Comparing eq.(4.5.2) with eq.(4.5.4): 65 (4.5.5) This confirms what we expected. The phase space contribution for the neutrino i s unchanged, but the phase space contribution for the electron i_s changed. And we notice something else: there i s a c o l l e c t i o n of terms coming from the matrix element which modifies the decay rate. As was discussed in sec.1.2, i t is this matrix element ef f e c t which we claim to be important, and not the phase space e f f e c t . A cursory glance shows that the Bessel functions are maximal at some value of n corresponding to £&(n)> £p, and i t appears that the LHS<RHS giving r i s e to a greater decay rate with the laser f i e l d for a given decay mode. 4.6 Analysis Part 2: How Large Is The New Decay Rate? Although n goes to oo , the phase space i s not i n f i n i t e l y increased because the Bessel functions are maximal for n ~ z and thus the contributions to the sum for n>z f a l l off r a p i d l y . 3 3 Hence, we don't want n0>z otherwise i t i s not l i k e l y that the 3 3 Becker et a l . pg.1264. See also Becker et a l . Respond, pg.6 5 3 . 66 electron can absorb enough photons for £ decay to occur. Therefore, Becker et a l . 3 f l take n 0<z. (Note that nuclei which don't normally decay may be caused to have a s i g n i f i c a n t decay rate, providing that the decay i s forbidden by a r e l a t i v e l y small energy: 0<no<z. Becker et a l . looked only at decays which do occur but which have a small phase space, and are therefore subject to a large enhancement: nQ<0 but small.) It turns out that n0<z for a given € i f ^ i s within a ce r t a i n range ^, v,<jv^.i/ m a x. Setting n<>=z, one can solve for V to o b t a i n : 3 5 Vxss =z{ee.-i±[(e*-i)C£M>r} u . s . o This also means that for a given "P, there i s a range of £ at which the decay can occur such that £ m I n 4. £^ £ m i x . We know that when 4^ =0 (no laser f i e l d ) , the energy of beta decay varies such that 1 ^ £ 6 Zo • Therefore, not only does the f i n a l energy of the electron increase, but the range of possible energies may also increase. To see the effect of t h i s , Becker et a l . look at the decay of fH for which £ o=1.036 and thus n 0 is a small negative number because there i s not much phase space available for the decay. (If n e i s a large negative number, the effect of the laser i s not s i g n i f i c a n t ) . They obtain the graph shown in FIG.6: 3 6 Becker et a l . eq.(7) This is eq.(8) of Becker et a l . with the change noted in appendix F. Becker et a l . FIG.1. 67 e-1 J f / •0.3 Wirt / • j V max •0.2 / •0.1 / 1 / V ^^ 0.5 1. F i g u r e 6 - G r a p h o f F i n a l E l e c t r o n E n e r g y v s . L a s e r I n t e n s i t y O b t a i n e d by B e c k e r , e t . a l . A l l o w e d v a l u e s o f £ a r e i n t h e s h a d e d r e g i o n . T h e r e f o r e , n o t o n l y i s t h e f i n a l e n e r g y i n c r e a s e d , b u t t h e a l l o w e d r a n g e o f e n e r g i e s i s i n c r e a s e d i n s i z e . A s s u m i n g n o=0, ( w h i c h means t h e f r e e d e c a y p r o d u c e s a n e u t r i n o o f momentum 0 a n d an e l e c t r o n o f momentum 0 ) , i n t e g r a t i n g e q . ( 4 . 3 . 2 2 ) , a n d t a k i n g t h e sum, B e c k e r e t a l . o b t a i n e d t h e enhancement o f t h e t o t a l d e c a y r a t e s f o r H a n d f o r ',,8F. F o r e x a m p l e , t h e y o b t a i n e d an enhancement o f 2 x 1 0 " f o r and an e n h a n c e m e n t o f 18 f o r '/F. when *=1. T h e r e a r e some p r o b l e m s w i t h t h i s . R e i s s 3 7 p o i n t e d o u t 3 7 R e i s s , p g . 6 5 2 . 68 that -n o=O(10") or 0 ( 1 0 s ) . But Becker et a l . 3 8 respond by p o i n t i n g out that the e r r o r i n assuming n o=0 i s only a few p e r c e n t . Both G e r s t e n 3 9 and R e i s s f t 0 p o i n t out that the experimental d i f f i c u l t i e s of o b s e r v i n g t h i s e f f e c t are s e r i o u s . In p a r t i c u l a r , the atomic e l e c t r o n s w i l l s h i e l d the l a s e r beam from the nucleus, c r e a t i n g a need f o r i o n i z a t i o n . T h i s t h e s i s w i l l not concern i t s e l f w ith the experimental d i f f i c u l t i e s of l a s e r enhancement of nuclear Ji decay. We are i n t e r e s t e d o n l y i n whether or not t h i s e f f e c t can, i n p r i n c i p l e , be observed, given s u f f i c i e n t technology. 4 . 7 What Happens If We Increase The E l e c t r o n i c Phase Space In Free F i e l d Decay? I t seems s u r p r i s i n g that the decay enhancement i s so l a r g e . I t was c l a i m e d i n sec.1.2 t h a t the enhancement i s a matrix element e f f e c t and not a phase space e f f e c t . But l e t us see what k i n d of an answer we get i f we assume i t i s a pure phase space e f f e c t . That i s , we are about to c o n s i d e r what happens when we a c c e l e r a t e the e l e c t r o n but leave the matrix element alone i n the formula f o r f r e e decay, eq.(2.1.20). We w r i t e eq.(2.1.20) f o r the t o t a l f r e e f i e l d decay r a t e i n the f o l l o w i n g form: 27T3 27j + j V L J ( 4 . 7 . 1 ) Becker et a l . Respond, pg.653. Gersten and Mittleman, pg.651. R e i s s , pg.652. 69 i (4.7.2) I (£«) i s evaluated in eq.(2.1.19), but i t i s more useful right now in the form of eq.(4.7.2): ' (4.7.3) (the subscript " f " stands for "free") Let us pretend that the effect of the laser is merely to accelerate the created electron such that i t gains an extra energy of m,A. That is,, in order to have a f i n a l energy of , the electron i s created with energy then accelerated to energy £^ . (The subscript "^" stands for "laser".) This means that the energy of the neutrino as per the previous discussion leading up to eq.(4.5.3), i s : £ w= £o - (£*-A) . (4.7.4) Looking back at how eq.(2.1.l7) was derived, we see that " I " in eq.(4.7.1) should actually read: J \ (4.7.5) rather than eq.(4.7.3). In obtaining eq.(2.1.17) we merely wrote Ev =£,-£> which was v a l i d for the free f i e l d case and gives eq.(4.7.3). For the laser enhanced decay, we should write f> as given by (4.7.4). Also, the upper l i m i t of the integration i s now EQ+A because SJI goes from 1.to £0+A . For the laser enhanced decay, I becomes: 70 (4.7.6) That i s , increasing the phase space of the electron by A is equivalent to increasing £», the energy available for the decay, by A . This i s precisely the scenario for a pure phase space eff e c t described in sec.1.2 and i l l u s t r a t e d in FIG.1! Comparing eq.(4.7.1) with eq.(4.7.6), one obtains: u s e r U, r ( - f ^ e ) I (Co) (4.7.7) From eq.(2.1.19): ICx)-3© 4- V 4- V (4.7.8) How big i s A? We assume A i s due to the absorption of n photons: ^ me (4.7.9) Assuming n to take the value at which the Bessel functions J n ( z ) are maximal, n « z . From eq.(4.3.27): (4.7.10) assuming we average over angles and £ isn't too close to 1. 71 For: y-l , A - 1 • , (4.7.11) That i s , the most probable energy t r a n s f e r i s m e A ^ j M e V . Becker et a l . " 1 c l a i m the most probable energy t r a n s f e r i s about 1MeV. Of course, these are j u s t order of magnitude e s t i m a t e s , so one may take A ~ 1 or 2 . (4.7.12) Let us c o n s i d e r the case of ?H d i s c u s s e d by Becker et a l . They s t a t e £ o = 1.036. From eq. (4.7.8), one o b t a i n s I (f„) = 1 . 9364x 1 0~6. From eq.(4.7.8), one a l s o o b t a i n s I (6o+2) = 5.1200, y i e l d i n g : I ( £ o + 1 ) = 3.5740x10'', and W r ( f r e e ) ?£2± = 1 .846x1 0 5 = 2.644x19 s A =1 A =2 The f o l l o w i n g i s a t a b l e of I(x) (4.7.13) X I ( x ) 1 o 1 .001 6 7700X10" 1 1 1 .01 2 1640x10"' 1 .02 2 4584x10" T 1 .03 1 0204x10 ' ' 1 .036 1 9364x10" ' 1 04 2 8045x10" ' 1 05 6 1494x10"' 1 06 1 1688x10" J 1 07 2 0 1 3 0 x 1 0 s 1 08 3 2254x10" 5 X K x ) 1 .09 4 8 9 1 0 x 1 0 s 1 . 10 7 1009x10" s 1 .20 8 3625x10" 4 1 30 3 5946x10" 3 1 40 1 0222X10" 2 1 50 2 3169X10" 1 1 60 4 5486x10"* 1 70 8 0 8 4 7 x 1 0 a 1 80 1 3359x10"' 1 90 2 0874x10~' 2 OO 3 1207x10"' X K x ) 2 .036 3 .5740x10" ' 2 20 6 .3019x10" ' 2 351 1.OOOO 2 40 1.1502 2 60 1.9483 2 80 3.1166 3 00 4 .7633 3 036 3 .5740 4 00 2.5301x10* 5 OO 8 .5903x10 ' Table I - A Table of the F u n c t i o n I(x) • 1 Becker et a l . pg.1263. 72 From table I we also see that i f the phase space i s large to begin with, (large E0 ), then the decay enhancement i s not very pronounced. Eg. Suppose £ 0 = 2 with A = 1 : w e 9 e t a n enhancement of 15.3. This demonstration was made with two assumptions: 1) the electron i s acceleraed i s o t r o p i c a l l y ; 2) the matrix element remains constant. In laser enhancement of p decay, both of these assumptions are f a l s e . In addition, there are other sub t l e t i e s such as < £< £™« , where £m-h >1. Thus, the lower l i m i t of integration of I(£ 0 + A ) should be moved upwards. The purpose of this demonstration was to show how phase space e f f e c t s could d r a s t i c a l l y change a decay rate. But this has l i t t l e to do with the laser enhancement of p decay because, as was claimed in sec.1.2, i t i s a matrix element e f f e c t . The argument presented in sec.1.2 was based upon the assumption that the electron i s accelerated over a c l a s s i c a l distance. This assumption i s j u s t i f i e d in the next section. 4.8 A Speculative Calculation Of The Acceleration Distance Of The Electron Presented below i s a very rough c a l c u l a t i o n of the distance over which the electron travels in gaining a d r i f t energy of 1MeV, due to the absorption of photons. This c a l c u l a t i o n ignores many sub t l e t i e s , (eg. the off-mass-shell nature of the electrons), and i s not assumed to be correct, but i t presents crude evidence in support of the claim that the electron i s accelerated over a c l a s s i c a l distance. This c a l c u l a t i o n also shows that the eff e c t of t h i s longitudinal acceleration, 73 emphasized by Becker et a l . , i s n e g l i g i b l e compared to the eff e c t of the transverse e l e c t r i c f i e l d in one wave-crest of the lase r . Becker et a l . * 2 envision an experimental s i t u a t i o n in which they have an I or Nd laser with ^ = Um, * H £ S . =1, a spot diameter of 10/im, and a pulse duration of 1ns. The c a l c u l a t i o n of laser enhancement of beta decay assumes a laser f i e l d of i n f i n i t e extent, so there i s no reason to assume that the 1MeV acceleration occurs over the a r b i t r a r i l y chosen lO^m. Becker et a l . " 3 state in their eq.(1): VR - 1S*I°" t l ^ ] I [ W e * ? ] (4.8.1) (This equation has been checked by the author of thi s thesis.) Setting V^= 1 and^\=1/<m, du=1.2eV), we fi n d : The Thompson scattering cross section"* for an electron, v a l i d for (jj «me , (1 . 2eV<<51 1 eV) , i s given by: a e= f gT= L6.S A? (4.8.3. Let I e be the power impinging upon the electron. From eq.(4.8.2) and eq.(4.8.3): I e=5.5xl0 6 MeV s"' (4.8.4) As a very rough estimate, we may say that the electron gains 1MeV in 1.8x10 7 s. Assuming i t i s t r a v e l l i n g at the speed of l i g h t , we find i t tra v e l s 55m before gaining 1MeV!! Becker et a l . Respond, pg.653. Becker et a l . pg.1262. Jackson, pg.681. 74 To obtain a better estimate of the distance over which acceleration occurs, we assume the electron starts at rest and i s accelerated by I e . We also assume I e remains constant u n t i l the electron has acquired i t s f i n a l energy of 1 M e V . This approximation w i l l tend to underestimate the distance because I e i s not constant. As the electron i s accelerated, i t sees the laser red-shifted, making the acceleration l e s s . The acceleration of the electron i s also retarded by the radiation emitted by the electron i t s e l f . At the f i n a l energy of 1 MeV, the absorption of radiation balances the emission of radiation. Let us naively ignore these complications, and calculate an estimate of the minimum distance t r a v e l l e d . The k i n e t i c energy as a function of time i s : = l e t ( 4 . 8 . 5 ) where V = ( 1 -V2 )"'\ V i s the vel o c i t y of the electron, and t i s in seconds. Solving for V : Integrating to obtain the distance t r a v e l l e d : • The upper l i m i t of integration i s determined by assuming the f i n a l energy is 1MeV=2me. d w i l l be in units of seconds. Letting "ft = i f ! + 1 : W g 75 Y Upon evaluat ing the in tegra l one f i nds : d=1 .5x10" 7 s = 45m ! This seems to ind ica te , indeed, that the acce lerat ion occurs over a c l a s s i c a l d is tance . 4.9 The Transverse Acce lerat ion The d r i f t acce lerat ion in the d i r e c t i on of the laser propagation vector was estimated in the las t sec t ion . Another thing to consider is the acce lera t ion due to the transverse e l e c t r i c f i e l d as the crest of one wave sweeps by the e l ec t ron . Let us look at the amplitude of th is e l e c t r i c f i e l d . From Jackson,* 5 the Poynting vector for a plane wave i s given by: I * AW (4.9.1) where E 0 i s the amplitude of the e l e c t r i c f i e l d . Le t t ing 1=8.3x10* MeV s"' fat"*, (from eq . (4 .8 .2 ) ) , we f i n d : E0 = J . 7 x lo~8KV/fi, (4.9.2) The force on the e l ec t ron , F, i s : * 5 Jackson, pg.272. 76 ' J | 3 7 (4.9.3) N o w : J ^ \y assuming the electron starts with zero momentum, and the force is applied for a time t. Let us assume that the force i s applied for a half-wavelength, -£^=0.5yiAm, (from sec.4.8). Actually, the average e l e c t r i c f i e l d over a half-wavelength i s ~~ times the peak e l e c t r i c f i e l d . Therefore, ? = S J U / o " ^ , * ( 4 . 9 . 4 ) Now, the time i t takes for t h i s half wavelength to sweep by is -t = d^r. = x / o " ' f s ,A 0 e. 3xfe»w/j (4.9.5) Inserting eq.(4.9.5) into eq.(4.9.4) y i e l d s P=1.04 MeV. Using E= ^P2+m2' , we a r r i v e at E=1.16 MeV. But the electron started off at rest with E=0.51 MeV. Therefore i t must have gained 0.6 MeV of kinetic energy. Using eq.(4.9.3): F 3/2. x/©"* A/.V/f*. ^ (4.9.6) Therefore, the transverse e l e c t r i c f i e l d causes the electron to gain about 0.6 MeV of kinetic energy, (coincidentally about the same amount which i s gained l o n g i t u d i n a l l y due to the absorption of photons), but t h i s i s 77 gained over a distance of This i s 1X10 8 nuclear r a d i i , and 4X10 3 atomic r a d i i . It i s safe to say that t h i s distance i s c l a s s i c a l . It i s interesting to note from eq.(3.6.3) that m* = J m2 + (ea) 2 , and from eq.(4.3.17) ea=Vm e. Setting V = \ , we find m\ = \~2tme. Therefore the Volkov electrons in thi s laser f i e l d are about .4 MeV off the mass s h e l l . This i s due to the transverse e l e c t r i c f i e l d j i g g l i n g the electrons, and i s consistent with the above c a l c u l a t i o n that the electrons gain 0.6 MeV in one pass of the e l e c t r i c f i e l d . 4.10 The Matrix Element Effect As was discussed in sec.1.2, the laser enhancement of beta decay i s not a phase space e f f e c t , because the electron i s accelerated over a c l a s s i c a l distance. The point i s that quantum processes in a system are not s i g n i f i c a n t l y affected by events occurring a c l a s s i c a l distance away from the system. To further enhance th i s point, l e t us consider the physical mechanism by which fi decay might be enhanced. In the model considered in sec.1.2, the wave function of the electron in the square well senses the changed environment outside the well, and t e l l s the electron i t can decay faster. In a real nucleus, electrons do not exist and hence have no wave function with which to sense the laser and t e l l the decay to go faster. But electrons do exist in the v i r t u a l sense, through processes such as the one i l l u s t r a t e d in FIG.7. 78 Figure 7 - V i r t u a l f Decay In t h i s process, three v i r t u a l p a r t i c l e s , e",p*", and are created, but they lack the required energy to become real p a r t i c l e s and thus recombine back to a neutron. Suppose the v i r t u a l p a r t i c l e s are created lacking an energy of AE. By the uncertainty p r i n c i p l e , they can only exist for a time At~ 2^f . If a laser can give the electron an amount of energy A E in a time At before the recombination occurs, then the decay can be made to happen. Fron sec.4.9, the transverse acceleration i s the dominant force in the laser. It imparts 0.6 MeV to the electron in 1.7xl0~,ir s. Setting AE=0.6, we fi n d At * v 10~2' s. Therefore, the time required to absorb t h i s energy is a m i l l i o n times too large. Simple c a l c u l a t i o n w i l l show that AE can be no greater than 7x10** MeV = 70eV. When AE has t h i s value, At = 9xl0"'* s and i t also takes the tangential f i e l d 9x1 0~'9 s to accelerate the electron by t h i s amount. Seventy eV i s not much of a phase space enhancement when we 79 consider that the electron coming out of ?H decay has a maximum energy of 1.86x10* eV. This i s a manifestation of the general argument that events occurring a c l a s s i c a l distance from the nucleus do not affect the decay. This again indicates that phase space e f f e c t s are not important, and i t seems very odd that Becker et a l . obtained an enhancement of 0(10*) for the p decay of ^H. What i s the physical reason for this enhancement? Is there something buried within the matrix elements which cause such a large enhancement? To try to answer these questions, a quantum mechanical model of f decay i s constructed in the next chapter in which we produce a matrix element e f f e c t which i s similar to the matrix element e f f e c t in laser enhanced f) decay. This model was introduced in sec.1.2. 80 V. A QUANTUM MECHANICAL ANALOG OF BETA DECAY AND ITS PHASE SPACE ENHANCEMENT 5.1 The Model. Is It Valid? An electron cannot be confined within a nucleus. But the nucleus gives i t the poten t i a l to e x i s t ; a latent, unrealized, v i r t u a l existence, which i s brought into being when the process of beta decay creates an electron out of the vacuum. In the following quantum mechanical model of beta decay we pretend that the electron can exist within the nucleus. This i s wrong, but n o n r e l a t i v i s t i c quantum mechanics forbids the nonconservation of p a r t i c l e number. The nucleus i s modelled by a one-dimensional square well with b a r r i e r s , through which the electronic wave function slowly leaks (see figure 8). In order to compensate for the existence of the electron inside the well, (the width of the well being the size of the nucleus), the well parameters w i l l be very strange, but we w i l l try to make them as r e a l i s t i c as possible. We ignore the neutrino. Remember that we are not trying to shed l i g h t on the physical mechanism of beta decay - we are merely trying to illuminate how a decay can be affected by a c l a s s i c a l acceleration of one of the decay products. Perhaps we are not constructing a model of decay but, rather, a model of t h i s type of matrix element e f f e c t . S t i l l , we need some c r i t e r i o n for deciding what the well parameters are, so we taylor the size of the well to the size of a nucleus and c a l l i t beta decay. 81 5.2 Matching The Neutrinoless Model To Real Beta Decay Following the discussion given in sec.1.2, the phase space for a neutrinoless beta decay i s where k e i s the electron's momentum, and kp i s the proton's momentum. The tr a n s i t i o n matrix w i l l introduce the energy momentum delta functions % (kp + k e - k M )xS( Uj, + ~ W» ) , where k^ i s the neutron's momentum, and the U J ' S are the corresponding energies. Integrating over the r e c o i l momentum k p leaves us with the decay rate dcu oc (u;a-u->e) k^ " dk e d & e , where cu0 i s the energy available for decay. Noting k edk e =UJedu>e, and integrating over dcue and d-Q.e , we f i n d : bU-r oc jfoOJ^ (5.2.1) where k 0 i s the momentum of the electron when i t has energy (jj0 i and W T i s the t o t a l decay rate. As we w i l l be using n o n r e l a t i v i s t i c quantum mechanics, we l e t U^~me, leaving us simply with: CU T « k0 (5.2.2) We w i l l f i n d that t h i s i s true also for the one dimensional model discussed in this chapter. What happens i f we increase the phase space for the decay in a manner similar to that discussed in sec.4.7? That i s , what happens i f we increase the energy £ 0 of the electrons to £„ +A ? We expect an enhancement of the t o t a l decay rate by J~^£—' Here £ 0 i s the n o n r e l a t i v i s t i c (kinetic) energy. For ^H, £ Q = .036. Taking A ~ 1 or 2 as in eq.(4.7.12) we find : 82 = 5 .36 A = 1 ( 5 . 2 . 3 ) = 7.52 This i s to be compared with eq.(4.7.13). We see that a phase space enhancement of 5.4 in our neutrinoless model w i l l correspond to a phase space enhancement of 1.8x10s when we include the neutrino, and a phase space enhancement of 7.5 in our neutrinoless model corresponds to a rea l phase space enhancement of 2.6x10s. Becker et a l . obtain an enhancement of 2x10" for the decay rate of ^H. If we assume t h i s i s s t r i c t l y a phase space e f f e c t , i t corresponds to an enhancement of 3.91 in our model. But neither the enhancement calculated by Becker et a l . nor the enhancement to be calculated for our model i s due to phase space. Therefore, we now invoke what we w i l l c a l l the phase space assumption. Assume that a matrix element e f f e c t giving a decay enhancement of 2x10" in the c a l c u l a t i o n of Becker et a l . corresponds to a matrix element e f f e c t giving an enhancement of 3.91 in the quantum mechanical model. This may be a shaky assumption but i t i s a sta r t i n g point for comparison. It i s abbreviated in (5.2.4): 2x10" enhancement of decay rate for rea l p decay V — > 3.91 enhancement of decay rate for model (5.2.4) 83 5.3 How To O b t a i n The D e c a y R a t e : A Summary The m o d e l i s d i s c u s s e d f u l l y i n a p e n d i x C. A b r i e f summary i s g i v e n b e l o w : The n u c l e u s c o n s i s t s o f a p o t e n t i a l w e l l o f t h e f o l l o w i n g f o r m : ^ ® A 3 ® C ® E F G H i K F i g u r e 8 - S q u a r e W e l l U s e d a s M o d e l o f N u c l e u s V, i s t h e h e i g h t o f t h e b a r r i e r s , -V e i s t h e d e p t h o f t h e w e l l , a n d b-a i s t h e w i d t h o f t h e b a r r i e r s . The f i v e r e g i o n s h a v e been l a b l e d © t o © , a n d t h e l e t t e r s A - K a r e t e n c o n s t a n t s u s e d t o m a t c h t h e s o l u t i o n f r o m one r e g i o n i n t o t h e n e x t r e g i o n . S c h r o d i n g e r ' s e q u a t i o n f o r r e g i o n { i s : A s s u m i n g E l i e s b e t w e e n -V 0 a n d V, , t h e s o l u t i o n s a r e : ( 5 . 3 . 1 ) 84 % = & e " x x * H e n x k H j 2 ( ^ E ' / M.^ J2Y»(V, - F ) U i H = Jli^ Ct + vS ( 5 . 3 . 3 ) The c o e f f i c i e n t s a r e d e t e r m i n e d by m a t c h i n g t h e wave f u n c t i o n s , a n d t h e d e r i v a t i v e s o f t h e wave f u n c t i o n s a t t h e f o u r b o u n d a r i e s : x=-b,-a,a,b. I f one c o n s i d e r s t h e c a s e o f a wave i n c i d e n t f r o m one d i r e c t i o n upon t h e w e l l , one f i n d s i n g e n e r a l t h a t p a r t o f i t i s t r a n s m i t t e d t h r o u g h t h e w e l l a n d p a r t o f i t i s r e f l e c t e d . T h e r e a r e c e r a i n r e s o n a n t e n e r g i e s a t w h i c h t h e wave i s c o m p l e t e l y t r a n s m i t t e d a n d none o f i t i s r e f l e c t e d . A t t h e s e e n e r g i e s t h e r e a r e m u l t i p l e r e f l e c t i o n s i n s i d e t h e w e l l a d d i n g c o n s t r u c t i v e l y t o y i e l d a huge a m p l i t u d e i n s i d e t h e w e l l . A q u a l i t a t i v e g r a p h o f t h e s o l u t i o n a t r e s o n a n c e a p p e a r s i n FIG.1 3 i n a p p e n d i x C. I t a l s o t u r n s o u t t h a t a s Vt-*oO , t h e r e s o n a n c e e n e r g i e s a s y m p t o t i c a l l y a p p r o a c h t h e e n e r g y l e v e l s o f an i n f i n i t e s q u a r e w e l l . We c o n s i d e r t h e c a s e o f an e l e c t r o n i n t h e l o w e s t r e s o n a n t s t a t e ( g r o u n d s t a t e ) a n d a s k how l o n g i t t a k e s t o d e c a y o u t o f t h e w e l l i f we remove t h e i n c i d e n t w a v es. I t i s shown i n a p p e n d i x C t h a t t h i s i s m a t h e m a t i c a l l y e q u i v a l e n t t o f o r m a l l y s o l v i n g f o r a " s t a t i o n a r y s t a t e " w i t h no i n c o m i n g w a ves. One ( 5 . 3 . 2 ) 85 obtains a complex energy "eigenvalue", the imaginary part of which y i e l d s the decay rate. It i s proven in appendix C that t h i s technique makes the rea l part of the complex energy equal to the resonant energy, as desired. The decay rate, W^., i s : J * L a X U \ ( 5 . 3 . 4 ) with the assumptions 0 V,» E. L ( 5 . 3 . 5 ) 3 i s a measure of the r e l a t i v e error of the c a l c u l a t i o n , i f we keep only the f i r s t term in e q . ( 5 . 3 . 4 ) . We note that the decay rate i s indeed proportional to k as discussd e a r l i e r . 5.4 Numerical Results The d e t a i l s of the cal c u l a t i o n s determining the values of the various parameters for the nucleus are discussed at the end of appendix C. The results are tabulated below in table I I . " 6 4 6 The data for was obtained from Lederer and Shirley, pg . 1 . 86 *H: E 0 = 0.018619(11) MeV Ti = 3.891X108 s x k = 1 .379x1 0"' MeV b = 1.731 fm J _ a x b-a b X (MeV) -2M.(b-a) E 0 + Vo V, - E 0 (MeV) ERROR 10" 10"2 I O " 3 I O ' * 10"5 io-' 4 . 1901x10'* 7.8154x10'' 2.4727x10'' 2 .9526x10_l 2.8029610 3 2.5798X10"1 2 . 3501x10'3 1 . 14x10~3 5.2189x103 1.5147x10* 1.1748x10' 1.1433x10' 1.1404x10' 1.1401x10* 2.7241x10' 8.4383x10'* 2.9334x10"M 3. 7478x10"" 3.8492x10"" 3.8580X10"3* 3. 8620x10"* 2. 1945x10** 6.5758x10 5 5.5388x10* 3.3322x10* 3.1560x10* 3.1399x10* 3. 1384x10* 3.1455x10* 2.6651x10'' 2.2448x10' 1 .3505X1010 1 .2791x10"-1 .2726x10'* 1 .2720x10" 7.2610x10 a 30% 3% 0 . 3% 0.03% 0.02% 0.06% 0.08% Table II - Well Parameters for *H We see that these numbers are consistent with assumptions 1, 2 and 3, with one exception: When — = 10~6, assumption 3 i s ax invalidated. The % error in Tx , as calculated using these parameters i s -2-X100% down to — = 10~4. Below t h i s , there i s error introduced because requires more s i g n i f i c a n t figures than i s reasonable, and has been truncated aft e r 5 s i g n i f i c a n t f igures. As discussed in the beginning of th i s chapter, the well parameters are very strange, for a number of reasons. F i r s t l y , the depth of the well i s 30,000 MeV 30 proton masses, and secondly, the height of the barrier is 10" to 10'3 proton masses. Note that i t i s not the height of the barrier alone that matters but both the height and the width. The model w i l l 87 work with a very t h i n very h i g h b a r r i e r of lO'^MeV and i t w i l l a l s o work with a b a r r i e r of 10 7 MeV which i s so wide i t takes up more than 3/4 of the w e l l . I t works because the b a r r i e r i s a n o n p h y s i c a l e n t i t y , the mathematics of which i s used to model decay. Is there anything which might i n d i c a t e how l a r g e the b a r r i e r should be? Perhaps one might argue that the width of the b a r r i e r i s the d i s t a n c e over which the weak i n t e r a c t i o n o c c u r s . In a very s p e c u l a t i v e manner, l e t the mass of the W-boson, (Mv) , which i s thought to mediate the weak i n t e r a c t i o n , be 10 s MeV, and l e t : b-o. ~ — - — ! — — - /. 7 7 x /o- 3 An lo^eV (5.4.1) b T h i s corresponds to the l a s t e n t ry in t a b l e II and g i v e s a b a r r i e r height of 7.26xlO , Z MeV. A s i m i l a r process i s done f o r '*F, the other example of Becker et a l . The r e s u l t s appear in t a b l e III below." 7 " 7 The data f o r JF was obtained from Lederer and S h i r l e y , pg.24. 88 E* = 0.6335(6) MeV Tj, = 109.8 min = 6.588X103 s k = 8.0464x10"' MeV b = 3.144 fm 1 <XM. b-a b X (MeV) V, - E 0 (MeVj ERROR 10"' 10"3 io-4 10 _ s 2 .8571X10'5 7.5495x10"' 2 . 1829x10"' 2.4877x10'' 2 . 3144X10"3 2 .0890x10"* 6.2660x10"* 2.5604x10s 8.0265x103 6.4344x10* 6.2889x10s 6.2757x10' 2.1974x10' 1 .7399x10"B 5.5582x10"" 6.9320x10'a 7 .0916X10"1' 7 . 1021x10"" 8.6904x10"* 1 .5828x10s 1 .5554x10* 9.9956x10' 9.5486x10' 9.5084x10'' 9.5164x10' 6.4147x10' 6 . 3037x10"* 4.0511x10" 3.8699x10" 3.8536x10" 4.7428x10'* 30% 3% 0.3% 0.09% 0.02% 0.05% Table III - Well Parameters for i,8F In th i s case, assumptions 1, 2 and 3 are v a l i d with two exceptions: When ~ = l 0~' assumption 2 i s invalidated, and when —- = 10~s assumption 3 i s invalidated. The l a s t entry in the table -3 i s again for b-a = 1.97x10 fm, the weak interaction distance given by eq.(5.4.l). Therefore: ~ — = 6.266x10 . This corresponds to a barrier height of 4.72x10 n MeV. Another nucleus which has a very long h a l f - l i f e i s 'g*C. The results for this nucleus appear in table IV below." 8 8 The data for j*C was also obtained from Lederer and Shirley. 89 cC : E„ = 0.156478(9) MeV Tl = 5730 yr = 1.8083x10" s 3. k = 3.9990x10'' MeV b = 2.892 fm 1 _ ax b-a b g-2M.(b-«0 E 0 + V 0 (MeV) V, - E 0 (MeV) ERROR 10" 10"2 IO" 3 10"* 10~s 10~ 6 2.2778x10"s 7 .9657x10'' 2.6585x10~v 3.2669x10"2 3. 1356x10'3 2 .9142x10"* 2 .6847x10"* 6.8230x10'* 3.3538X103 9.2932x10s 7.0530x10* 6.8441x10 s 6.8246x10' 6.8228x10* 2.9973x10' 9.7431x10"3S 3.5213x10'* 4.6318x10'* 4 . 7737x10"M 4.7885x10"M 4.7903x10** 9.2125x10'*' 2.7156x10s 2.0851x10* 1.2010x10* 1.1309x10* 1.1245x10* 1.1239x10* 1.1253x10* 1 . 1006x10T 8.4505X10 7 4.8674x10'' 4.5833x10" 4 .5573x10'3 4.5548x10,s 8 . 7905X10'1 30% 3% 0. 3% 0.05% 0.05% 0.06% 0. 1% Table IV - Well Parameters f o r '6*C As i n the p r e v i o u s two cases, the l a s t e n t r y i s f o r b-a _3 1.97x10 fm, the weak i n t e r a c t i o n d i s t a n c e given by eq.(5.4.1). T h e r e f o r e = 6.823x10 and we o b t a i n a b a r r i e r height of 8.79x10 1 2 MeV, compared to 7.26x1 o'ZMeV for *H and 4.72xlO , 2MeV fo r 'JF. 5.5 Immersion Of The Model In An E l e c t r i c F i e l d : The Enhanced Decay Rate The b a s i c model f o r beta decay has been developed above. It i s d e s i r e d to enhance the decay i n a manner sug g e s t i v e of the l a s e r enhancement of beta decay. The l a s e r enhances the decay r a t e by a c c e l e r a t i n g the e l e c t r o n over a c l a s s i c a l d i s t a n c e . Let us t h e r e f o r e immerse our model i n an e x t e r n a l l i n e a r e l e c t r i c f i e l d , which a l s o a c c e l e r a t e s the e l e c t r o n over a 90 c l a s s i c a l distance, i l l u s t r a t e d in FIG. The potential well takes on the form A c D 9. CD E F V ® Y w T © L <3> N r - c Figure 9 - The Potential Well in an E l e c t r i c F i e l d The e l e c t r i c f i e l d i s applied in regions (2) and (6) with c>>b. The shape of the well i t s e l f i s l e f t the same to minimize the Stark e f f e c t . Because the most probable energy transfer from the laser to the electron i s about 1 MeV, we assume Y = 1 MeV. In the nuclei we are interested i n , the energies of beta decay are smaller than 1 MeV so the wave function w i l l be a decaying exponential in region (7) , and ba r r i e r penetration w i l l occur only to the l e f t . This e f f e c t i v e l y cuts the decay rate in h a l f . Therefore, any decay with a decay rate greater than half of eq.(5.3.4), w i l l be considered enhanced. The solution to Schrodinger's equation has the following form: 91 X - CAiCx) + DftCsO 2 = < X - £ E ) ( 5 .5 .1 ) Ai and Bi are the Airy functions." 9 As before, the c o e f f i c i e n t s are determined by matching the wave functions, and th e i r derivatives at the 6 boundaries: x=-c,-b,-a,a,b,c. The d e t a i l s of this are discussed in appendix D. Also as before, the rate of decay i s the imaginary part of the complex energy "eigenvalue" obtained from solving for a "stationary state" with no incoming wave. The result is c r u c i a l l y dependent upon the arguments of the Airy functions at the four boundaries: x=-c,-b,b,c. These arguments are denoted by: -y,-z,-z,y for x=-c,-b,b,c, respectively: " 9 See Abramowitz and Stegun, pg.446. 9 2 -H - -de V i (*-«•) •6 YV4 ( 5 . 5 . 2 ) We f i n d that i f -~r«^, and the e l e c t r i c f i e l d Y/c i s not too strong, then y> > 1, z > > 1 , z > > 1 , "y>>i and we may use the asymptotic expansions of the Airy functions at these values. Appendix D shows that this y i e l d s the following r e s u l t : ( 5 . 5 . 3 ) •as expected. That i s : a weak e l e c t r i c f i e l d applied over a c l a s s i c a l distance does not a f f e c t the decay rate, even i f the electron gains 1 0 0 MeV, providing the distance i s large enough. Let us assume that the distance c i s large enough that y> > 1 and y > > 1 , but the f i e l d i s so strong that we cannot use the asymptotic expansions of the Airy functions at z and z. In appendix D we obtain the following: 93 (5.5.4) 94 5.6 Numerical Results The enhancement, Tj/gfz), i s tabulated in tables V to VIII along with the various parameters used. ?H: X = 2 .7241x10s MeV Y 1 MeV E/Y = 0 .018619 b 1 .731 fm b/c oc c z z y V 1?/g(z) >1x106 1x10' 1x10s 5x10* 1x10* 5x103 2.29x103 1x103 1 .73x10'' 1 . 73x10"5 3.46x10'5 1 .73x10'* 3.46x10'* 7.56x10"* 1 . 73x10"3 2.97x10"* 6.40x10"* 8.07x10"* 1 . 38x10~3 1 . 74x10~3 2 . 25x10'3 2 .97x10"3 2.97x10a 6.40x10' 4.03x10' 1.38x10* 8.69 5. 16 2 .97 5.53 1 . 19 7.52x10' 2.59x10' 1 .65x10"' 1 .00x10' 6.05x10' 5.53 1.19 7.50x10 2.54x10"' 1 . 59x10"' 9.22x10_l 5.02x10'2 3.03X102 6.52x10' 4.11x10' 1.41x10' 8.85 5.26 3.03 2.92x10* 6.28x10' 3.96x10' 1 .35x10' 8 . 53 5.07 2.92 1 .OO 1 .00 1 .05 1.11 1 . 48 1 . 75 2 . 18 Inval1d Table V - The Dependence- of the Decay Enhancement Upon the Distance Over Which the E l e c t r i c F i e l d i s Applied 95 2.7241x10s MeV 0.018619 MeV 1.731 fm 1 .00x1 0'° fm 1 .731x10"'° Y E/Y z z y y 1x10* 1 .86x10 2 .97x10"4 2 .97x10° 5.53 5.53 2 97x10 c 2 .97x10s 1 .00 1x10 s 1 . 86x10"* 6 .40x10"* 6 .40x106 1 . 19 1 . 19 6 40x10 4 6 .40x10s 1 .05 1x10 6 1 .86x10"g 1 . 38x10"3 1 38x10' 2 59x10'' 2.54x10"' 1 38x10^ 1 38x10T 1 .48 1x10* 1 .86x10""' . 86x10"'° 2 .97x10"' 2 97x10^ 6 05x10~l 5.02x10"2 2 97x10* 2 97x10' 2 .80 1x10 8 1 6 .40x10"3 6 40x10* 2 30x10'* 8.38X10"4 6 40x10' 6 40x10' 5.87 ^x^O', 1 .86x10"" 86x10"11 2 .64x10"* 2 64x10* 2 64x10"* -2 . 1x10"2 1 38x10 8 1 38x10 8 1 27x10' 1x10'° 1 2 .97x10"2 2 97x10? 5 20x10"1 -5. 1x10"a 2 97x10* 2 97x10 s 2 78x10' 1x10" 1 86x10"13 6 .40x10"2 6 40x10 1 11x10"' - 1 . 1x10"' 6 40x10' 6 40x10* 6 25x10' 1x10* 1 86x10"'* 1 . 38x10'' 1 38x10'' 2 39x10"' -2.4x10"' 1 s a x i o 9 1 38x10'' 1 46x10* 1x10'* 1x10'*" 1x10's 1x10" 1 86x10"'5 2 .97x10"' 2 97x10'' 5 14x10"' -5. 1x10"' 2 97x10'' 2 97x10" 3 67x10* 1 86x10"" 6 .40x10'' 6 40x10 q 1.11 -1.11 6 40X101, 6 40x10" 1 02x103 1 86X10'11 1 . 38 1 38x10" 2 . 39 -2.39 1 38x10'° 1 38x10'° 3 09x1O3 1 86x10*" 2 .97 2 97x10'° 5. 14 -5 . 14 2 97X101" 2 97x10'° 9 61x10s 2.9x10* 6 42X10-10 4 . 24x10"3 4 24x10" 3 45x10"z 1 .98X10"1 4 24x10' 4 24x10 > 3.91 Table VI - The Dependence of the Decay Enhancement Upon the Fin a l Energy of the Electron i n • X E b c b/c 96 X = 2.7241x10s MeV b = 1 .731 fm E = 0.018619 MeV Y (MeW °c c z Z y y °?/g(z) 1 x 1 0 " 1 x 1 0 ' ° 2 . 9 7 2 9 7 x 1 0 ' ° 5 . 14 - 5 . 14 2 9 7 x 1 0 ' ° 2 9 7 x 1 0 9 . 6 1 x 1 0 3 1 x 1 0 ' * 1 x 1 0 8 2 .97 2 9 7 x 1 0 8 5 . 14 - 5 . 14 2 9 7 x 1 0 8 2 9 7 x 1 0 * 9.61x10 3 I x l O * 1 1 x 1 0 3 2 . 9 7 2 9 7 x 1 0 3 5 . 14 - 5 . 14 2 9 7 x 1 0 3 2 9 7 x 1 0 3 9.61x10 3 1 x 1 0 * 1 x 1 0 ' 2 . 9 7 2 9 7 x 1 0 ' 5 . 14 - 5 . 14 2 9 7 x 1 0 ' 2 9 7 x 1 0 ' s 1 x 1 0 s 1 2 .97 2 . 9 7 5 . 14 - 5 . 14 2 9 7 x 1 0 ° 2 . 9 7 i n v a l i d 1 x 1 0 ^ 1 x 1 0 2 . 9 7 x 1 0 ~ 3 2 9 7 x 1 0 ' 6 0 5 x 1 o"^ 5 . 0 2 X 1 0 1 2 9 7 x 1 0 ' 2 9 7 x 1 0 ' 2 . 8 0 1 x 1 0 * 1 x 1 0 5 2 . 9 7 X 1 0 ' 3 2 9 7 x 1 0 * 6 05X10" 1 " 5 . 0 2 X 1 0 " 1 2 9 7 x 1 0 * 2 9 7 x 1 0 * 2 . 8 0 1 x 1 0 ' 1 x 1 0 * 2 . 9 7 x 1 0 " 3 2 9 7 x 1 0 ' 6 0 5 X 1 0 " 1 5 . 0 2 x 1 0 " * 2 9 7 x 1 0 ' 2 9 7 x 1 0 ' 2 . 8 0 1 1 x 1 0 3 2 . 9 7 x 1 0 ^ 2 . 9 7 € 0 5 x 1 0 " 2 5 . 0 2 X 1 0 " 1 3 . 0 3 2 . 9 2 i n v a l i d Table VII - The Dependence of the Decay Enhancement Upon the Strength of the E l e c t r i c F i e l d 18 p>. H = 2.1974x10s MeV b = 3.144 fm E = 0.6335 MeV Y (MeV) ex. ot c z "z y y ^ / g U ) 1 x 1 0 ' 6 1 x 1 0 ' * 1 x 1 0 ' 1 x 1 0 T 1 x 1 0 ' ° 1 x 1 0 ' ° 1 x 1 0 ' ° 1 x 1 0 ' ° 2 . 9 7 1 . 3 8 x 1 0 " ' 6 . 4 0 x 1 0 " 3 2 . 9 7 x 1 0 " 3 2 . 9 7 x 1 0 ' ° 1 . 3 8 x 1 0 9 6 . 4 0 x 1 0 ' 2 . 9 7 x 1 0 ' 9 . 3 4 4 . 3 5 x 1 0 " ' 4 . 2 6 x 1 0 " ' 1 . 8 9 - 9 . 3 4 - 4 . 3 x 1 0 " ' 3 . 8 5 x 1 0 " ' 1 . 8 7 2 . 9 7 x 1 0 ' ° 1 . 3 8 x 1 0 s 6 . 4 0 x 1 0 ' 2 . 9 7 x 1 0 ' 2 . 9 7 x 1 0 ' ° 1 . 3 8 x 1 0 ^ 6 . 4 0 x 1 0 ' 2 . 9 7 x 1 0 " -2 . 2 2 x 1 0 3 2 . 8 0 x 1 0 ' 1 . 3 0 1 . 0 2 Table VIII - The Decay Enhancement for '*F 97 ^?/g(z) i s the enhancement of the decay due to the e l e c t r i c f i e l d . °?/g(z) = 1.00 when the following conditions are met: ^/gOJ = / . ° ° -A»h : Ca) (fc) (5.6.1) Since we used the asymptotic expansions of the Airy functions for both y>>1 and y>>1, we need (from eq.(5.5.2)): cXc»1 for v a l i d i t y of solution O £ C ~ 1 0 : few percent error in asymptotic expansions <Xc~5: 0(10%) error in asymptotic expansions (5.6.2) Table V i l l u s t r a t e s the dependence of ??/g(z) upon c for a fixed Y of 1 MeV, using the parameters for ^ H. For c^1x10 6 fm, the ef f e c t of the e l e c t r i c f i e l d i s neg l i g i b l e because conditions (5.6.1) hold. Condition (5.6.1b) breaks down for C ^ 1 X 1 0 5 fm and the e l e c t r i c f i e l d has a s i g n i f i c a n t effect on the decay. For C < 3 X 1 0 3 fm the solution is no longer v a l i d because of condition (5.6.2). To correct t h i s situation in order that we see what happens for large enhancements, we must increase y while keeping c constant. This i s done in table VI for a constant c of l 0 , o f m = lO^m. The effect of the e l e c t r i c f i e l d f i r s t becomes s i g n i f i c a n t for Y=105 MeV, (where condition (5.6.1b) i s broken), and there is an order of 10" enhancement when Y=lO , 6MeV. Note that z f i r s t decreases, and then increases as Y increases. This stationary point i s due to the breaking of 98 condition (5.6.1a), so that both conditions in (5.6.1) are broken. Note that condition (5.6.2) i s not v i o l a t e d , meaning that the solution remains v a l i d . Table VII shows that the e f f e c t of the e l e c t r i c f i e l d depends only upon the e l e c t r i c f i e l d strength Y/c and not upon what the f i n a l energy i s or upon how far the electron i s accelerated. This remark must be q u a l i f i e d with the proviso that c not be too small otherwise our solution i s not v a l i d . Table VIII shows the e f f e c t of the e l e c t r i c f i e l d on the decay of '?8F. H i s chosen to be that of the l a s t entry in tables II and I I I . '77/g(z) i s not strongly dependent upon what we choose for a X, providing —- <<1 and J5?<<1. 5.7 How Big Must The E l e c t r i c F i e l d Be To Match The Results Of Becker,et.al.? From eq.(5.2.4) we wish to obtain an enhancement of 3 . 9 1 for ^H in order to match the real beta decay. From table VI, we find that we need a f i e l d strength of 2.9x10 MeV/fm/e. If we assume that the electron gains 1 MeV of energy, then i t must do - 12. so in a nonclassical distance of <l000fm=l0 m, where our solution i s not v a l i d , contrary to the large distance calculated for the laser enhanced beta decay. Even i f we assume the electron receives i t s 1 MeV over the diameter of the laser (lO^rn), we get a force of lO"'°MeV/fm which i s a factor of 3x10""' too small. We could, instead, assume that the decay enhancement i s due 99 to the transverse acceleration of the electron in the laser f i e l d . From eq.(4.9.3), the force i s given by Y/c = 3.2x10 MeV/fm. But even th i s i s a m i l l i o n times too small. 5.8 A S l i g h t l y Modified Model Perhaps we should modify the model in FIG.9 s l i g h t l y . If we look at the size of W =-£-Y, we see that when V= l06MeV/fm, C —— c W = 1 .731xl06MeV. But the well i s only 3x10'* MeV deep (see table I I ) , so we have a situation as i l l u s t r a t e d in FIG.10 below: V 4 CD. -V. Figure 1.0 - The Shape of the Well for a Very Strong E l e c t r i c F i e l d 100 The sharp drop at x=-b corresponds to an increase in phase space which i s not due to acceleration of the electron. We wish to look only at the e f f e c t of a gradual increase in the energy of the electron. Therefore, l e t us modify the model by s h i f t i n g V 2 up by W and s h i f t i n g V 6 down by W. That i s : V a -* V4' = ^x+W and V 6 ~ * V 6 " =-^ x-W (see FIG.11). The ca l c u l a t i o n proceeds in a similar manner to the ca l c u l a t i o n of the previous model, covered in appendix D. The only differences to the f i n a l result are to modify z,z,y,y as follows: (5.8.1) °4 Table IX i s e s s e n t i a l l y a reproduction of Table VI calculated with the new parameters. We see that t h i s new model af f e c t s the enhancement s i g n i f i c a n t l y only at very high f i e l d s . To get the desired enhancement of 3.91, Y/c i s s t i l l 2.9X10~3 MeV/fm. 101 3 H : y{ = 2.7241x10 s MeV b = 1.731 fm E = 0.018619 MeV / Y (/MeVJ c Y/C oC (X. c (**»-') 1x10" s 6 .40x10'* 6 .40x10 6 1x10"f 1 .38x10 1 38x10' 1x10~ 3 2 .97x10"3 2 97x10' 1 x 1 0 ^ 6 .40x10"3 6 40x10' 1x10"' 1 . 38x10"* 1 38x10 s 1 2 .97x10"2 2 97x10 8 1x10' 6 . 40x10" 1 6 40x10* 1X102" 1 . 38x10*' 1 38x10° 1x10 3 2 .97x10'' 2 97x10"* 1x10* 6 .40x10'' 6 40x10° 1x10 s 1 . 38 1 38x10'° 97x10° 1x10 6 2.97 2 1x10* 6.40 6 40x10'° . 1x10 s 2.9x10" 3 1 38x10' 24x10" 3 1 38x10" 4 4 24x10"* y ' y ' V /g(z') 6 .40x10° 6 40x10 1 .05 1 38x10' 1 38x10' 1 .48 2 97x10* 2 97x10' 2 79x10° 6 40x10' 6 40x10' 5.82 1 38x10 8 1 38x10 s 1 28x10* 2 97x10* 2 97x10* 2 76x10' 6 40x10* 6 40x10 ? 5 94x10' 1 38x10° 1 38x10° 1 28x10 a 2 97x10° 2 97x10° 2 76X107-6 40x10° 6 40x10° 5 95x10"°-1 38x10'° 1 38x10'° 1 28x10 3 2 97x10*° 2 97x10" 2 76x10 3 6 40x10" 6 40x10'° 5 94x10 3 1 38x10" 1 38x10" 1 28x10* 4 24x10' 4 24x10* 3.91 1x10 1x10 s 1x10' 1x10* 1x10'' 1x10'° 1x10" 1x10 , - L 1x10 1 3 1x10'* 1x10' s 1x10'" 1x10 8 2.9x10 7 1x10'° 1x10'° 1x10'° 1 x 1 0 ° 1x10'° 1x10'° 1x10'° 1x10'° 1x10'° 1x10'° 1x10'° 1x10'° 1x10'° 1x10'° 1x10'° 1 . 19 .57x10" .53x10" : . 19x10"? .57x10" ' . 5 3 x 1 0 * . 19x10"* . 57x10 ' 5 .53x10" ' . 19x10"' .57x10"* ,53x10" ' . 19x10"* .57x10"° Table IX - The Decay Enhancement i n the M o d i f i e d Model V F i g u r e 11 - The M o d i f i e d Model 102 An extension of thi s model of beta decay into 3 dimensions i s discussed in appendix E. It i s concluded that t h i s extension w i l l not strongly a f f e c t the conclusions that are to be obtained from the one dimensional model. 5.9 Discussion It was thought in sec.1.2 that the decay rate should only be affected by what happens l o c a l l y . And indeed, we see that the decay rate i s independent of how much energy the electron gains over a c l a s s i c a l distance. It i s dependent only upon the size of the e l e c t r i c f i e l d in the neighborhood of the well, and not what happens at large distances from the well. This i s therefore a matrix element e f f e c t and seems to q u a l i t a t i v e l y match what i s happening in the laser enhancement of beta decay. Quantitatively though, assuming the phase space assumption (5.2.4) to be correct, we f i n d that the size of thi s e f f e c t in our model i s much too small to explain the large enhancement of beta decay. It is d i f f i c u l t to see why the phase space assumption should be out by at least 6 orders of magnitude. 103 VI. CONCLUSION The Volkov solution to Dirac's equation was derived and found to agree with the form of the Volkov solution used by Becker et a l . 5 0 The electron f i e l d operator was expanded in terms of normalized Volkov states and thi s f i e l d operator was used in the 4-point theory of beta decay. The formula for the decay rate obtained was found to be in agreement with the formula stated by Becker et a l . 5 1 The numerical results of Becker et a l . were not checked. In p a r t i c u l a r , Becker et a l . obtained a 2x10" enhancement of the decay rate of ?H in a laser f i e l d of 10'8 W cm"1 , which imparts an average of 1 MeV to the decay electron. This numerical c a l c u l a t i o n was not checked, but physical and mathematical arguments were presented which delineate the p e c u l i a r i t y of t h i s r e s u l t . They are outlined as follows: 1) A very rough c a l c u l a t i o n shows that the acceleration of the electron occurs over a c l a s s i c a l distance. 2) It i s argued that only events which are microscopically close to the nucleus w i l l a f f e c t the decay of the nucleus. 3) This argument i s supported by a quantum mechanical model of beta decay in which the decay electron i s accelerated over a c l a s s i c a l distance. It i s found that the decay rate enhancement due to t h i s acceleration i s independent of how much energy the electron gains over a c l a s s i c a l distance. It i s dependent only 5 0 Becker et a l . eqs.(2) and (3). 5 1 Becker et a l . eq.(5). 104 upon the size of the accelerating f i e l d in the neighborhood of the nucleus. 4) The magnitude of the enhancement in the model i s much too small to explain the results of Becker et a l . Some of the assumptions involved in the above arguments are rough and very approximate; one might even say some of them are wrong. They may even be a couple of orders of magnitude wrong. But, i f we are to understand the laser enhancement of ^ decay in l i g h t of these arguments, assuming the numerical c a l c u l a t i o n s of Becker at a l . are correct, then these arguments must be many orders of magnitude wrong. This i s very peculiar. 105 BIBLIOGRAPHY 1 . Abramowitz, M. and Stegun, I.A. [Ed.] Handbook of Mathematical Functions. 9th p r i n t i n g . Department of Commerce - National Bureau of Standards: U.S.A. 1970. 2. Becker, W. et a l . "Laser Enhancement of Nuclear ? Decay". Phys. Rev. Letters 47, pgs.1262-1266, Nov. 1981 . 3. Becker, W. et a l . "Becker et a l . Respond:". Phys. Rev. Letters 48, pg.653. Mar, 1982. 4. Bethe, H.A. and Salpeter, E.E. "Quantum Mechanics of One- and Two- Electron Systems", Encyclopedia of Physics v o l . 35: Atoms 1 , S. Flugge rEd."] . Springer-Verlag. B e r l i n , 1957. 5. Bjorken, J.D. and D r e l l , S.D. R e l a t i v i s t i c Quantum Mechanics. McGraw-Hill: New York, 1964. 6. deShalit, A. and Feshbach, H. Theoretical Nuclear Physics Volume 1: Nuclear Structure. John Wiley & Sons: New York, 1974. 7. Gersten, J.I. and Mittleman, M.H. "Comment on 'Laser Enhancements of Nuclear "B Decay'". Phys. Rev. Letters 48, pg.651. Mar. 1982. 8. Itzyckson, C. and Zuber, J . Quantum' F i e l d Theory. McGraw-Hill: New York, 1980. 9. Jackson, J.D. C l a s s i c a l Electrodynamics 2nd Ed. John Wiley & Sons: New York, 1975. 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 Theoretical Physics, Vol.4, part 1. Pergamon Press: Oxford, 1971. Ledeirer, CM. and Shirley, V.S. [Ed.] Table of Isotopes 7th Ed. John Wiley & Sons: New York, 1978. 12. Merzbacher, E. Quantum Mechanics 2nd Ed. John Wiley & Sons: New York, 1970. 13. Messiah, A. Quantum Mechanics Vol.1. - John Wiley & Sons: New York, 1976. 10 1 1 14. Reiss, H.R. 'Laser Enhancement of Nuclear P Decay". Phys. Rev. Letters 48^ pg.652. Mar. 1982. 15. Schweber, S.S. An Introduction to R e l a t i v i s t i c Quantum F i e l d Theory. Harper & Row: New York, 1962. 106 16. Segre, E. Nuclei and P a r t i c l e s 2nd Ed. 2nd p r i n t i n g . The Benjamin/Cummings Publishing Company: Reading, Massachusetts, 1980. 17. Selby, S.M. [Ed.]. CRC Standard Mathematical Tables. 20th Ed. The Chemical Rubber Co.: Cleveland, Ohio, 1972. 18. Wolkow, D.M. "Uber eine Klasse von Losungen der Diracschen Gleichung". Z. Physik 94 pgs.250-260. ( 1935). 107 APPENDIX A - FREE FIELD SOLUTIONS TO THE DIRAC EQUATION THE DIRAC EQUATION AND MATRICES Dirac's free f i e l d equation for a Fermion of mass m: Conventions of Bjorken and D r e l l : 5 2 2 Bjorken and D r e l l , appendix A. 108 DELTA FUNCTION NORMALIZATION UL U. - - — 2 . loo (A.3) BOX NORMALIZATION "^PpsCx) is treated similarly ie. change jj-^? to in going J f^fpsJx-from g norm, to box norm. v i 8 (A.4) 109 v 0 „ i s t h e proper volume of a large box used for the purpose of normalization. Y " r " ~ v. ~ * v. NOTATION p i s the 4-momentum of the fermion: p = (u>,p) x i s time, space: x = ( t , x ) k»x = g A„ k^x v = i^t-k « x u s u+X° s i s the spin of the fermion u i s a 4-spinor; so is V % s i s a 2-spinor (A.5) for spin up along the z axis, for spin down along the z axis. PROJECTION OPERATORS It is shown in Bjorken and D r e l l , 5 3 that: 1 = (A.6) 5 3 Bjorken and D r e l l , appendix A. 110 APPENDIX B - USEFUL IDENTITIES IDENTITIES INVOLVING SPIN SUMMED NUCLEAR MATRIX ELEMENTS It i s desired to prove eq.(2.1.1Oa). Within M£t, acts upon isospin space and &J independently acts upon spin space. Let us seperate these two spaces: y- \ B. 1 ) .'. S'/VAli?= 2<f.SolT^h)|2 l$.><2,,./r-M/4:|So>-f ^ rr' Now: / a; ( ^ > - \cr^ > = O Al Q.E.D. (B.2) (B.3) (B.4) We w i l l now prove eq.(2.1.1 Ob). Using eqs.(B . I ) and (B.2), one obtains the following: 111 (as in eq.(B.3)) ( i f r ^ r ' then c(r) and O'(r') commute) = O (B.5) °Z x A£T = <B-6> At Q.E.D. F i n a l l y , we prove eq.(2.1.1Oc). From eqs.(B.I) and (B .2 ) : HInJ7" = £ ^ f t , £ < ( 7 i ^ i <r^ jcr-/ft-;/^ > /v. r r ' *V (B.7) If r ^ r', then cT^r) and (7J(r) act independently such that the arguments used in obtaining eq.(B.4) are applicable and we obtain zero. For r = r', cf^O^ = 3« Eq.(B.7) becomes: A» r > . ' >7)^ (B.8) Also, from eqs.(B.I) and (B.2): Using the above arguments, we see that r = r'. Now using eq.(A.2): _ - - r . „ O'OJ - g-j + i £ i s L ak (B.10) m * (B.11) The second part of eq.(B.1l) disappears by v i r t u e of eq.(B.3). 1 12 (B.12) Comparing eq.(B.8) with eq.(B.12), one finds that: Z! ot'^sr £ * A U r = J Q * r 2 | A^r I ^ ( B > 1 3 ) Q.E.D. IDENTITIES INVOLVING GAMMA MATRICES A useful identity similar to eq.(B.lO) i s : (from eq.(A.2)) ^^fV= C)*y- z V ^ V (B.14) From eqs.(B.14) and (A.2), i t i s obvious that: & gC - CXP- (B. 15) Another id e n t i t y based upon eq.(A.2) i s : (B.16) 113 APPENDIX C - DECAY OF AN ELECTRON FROM A ONE DIMENSIONAL SQUARE WELL " C1 . INTRODUCTION This appendix i s an account of the mathematical and physical d e t a i l s of the quantum mechanical model of beta decay considered in this thesis. We w i l l discuss how to estab l i s h a resonance in a square well with ba r r i e r s , what some of the properties of thi s resonance are, and how to make t h i s resonance decay. C2. THE MODEL: SOLVING THE SCHRODINGER EQUATION Consider a potential well of the following fo rm: © A 3 C Ti V ® E F <5> G H J -b Figure 12 - The Potential Well 1 14 V, i s the height of the b a r r i e r s , -V0 i s the depth of the well, and b-a i s the width of the b a r r i e r s . The fi v e regions have been labled (T) to (f), and the l e t t e r s A-K are ten constants used to match the solutions from one region into the next region. Schrodinger's equation for region i i s : dx1- (C2.1) Assuming E l i e s between -VQ and V( , the solutions are: 13 e-ik,x H e * * K e i f c « * (C2.2) k, and kj- are the same, but have been distinguished for mathematical reasons which w i l l become apparent l a t e r . G3. THE STRUCTURE OF THE MATRICES WHICH RELATE THE COEFFICIENTS The c o e f f i c i e n t s are determined by matching the wave functions, and the derivatives of the wave functions, at the four boundaries: x=-b,-a,a,b. There are 8 boundary conditions and 10 constants, leaving 2 a r b i t r a r y constants from which the 8 others are determined. Let: 115 1) where the (fi)'s are 2 x 2 matrices. (The i s used to s i g n i f y "matrix".) Rather than use brute force and drudgery, l e t us exploit the symmetries of eq.(C2.2). Define an operation denoted by "-<-+", such that *$f (a,b,c...) = f (-a, b, c . . . ) , *bf (a, b,c ...) = f(a,-b,c...), etc. That i s : x changes the sign of x in a function. From eqs.(C2.l) and (C2.2) i t i s obvious that j^ ic, ,HJ =0, where H i s the hamiltonian operator of eq.(C2.1). Thus the eigensolutions of (C2.1) can also be eigensolutions of IT, . Let *k>Y =«Y. Then (k, f V = o<f-Y = V . Thus, c* = ± 1 . Applying k, to eq.(C2.2), we f i n d : V,A- - B J 7 .B = ±A s k , c = i C > V,* = ±j) 5 h,Z - ± E This i s merely a fancy way of saying the following. Suppose we know the two constants in one region, say region (2), and we wish to find the two constants in the next region, say region (?) . The solution in region Q i s C,e x+C ae~ i K ," x. The 116 matrix (M,) w i l l dictate that the c o e f f i c i e n t of e , i . e . C, , w i l l be A, and the c o e f f i c i e n t of e~l^'x, i . e . C 2, w i l l be B. Suppose we wrote i t the other way around: C, e""^'x +C ae^' x ( i . e . l e t k,-» -k, ). The matrix (M, ) s t i l l must dictate that \k x the c o e f f i c i e n t of e 1 , in t h i s case C 2 , w i l l be A, and the c o e f f i c i e n t of e*^* , in t h i s case C, , w i l l be B. Applying % to the left-hand side and right-hand side of eq.(C3.1): The signs cancel, leaving: We also f i n d : and a l l other equations in eqs.(C3.l) remain the same. A similar analysis holds for X, i f , and ? 5 , y i e l d i n g : (C3.2) This leads to the following structure for the matrices: 117 (AI,W „ ^ X A 1 « YA)-L ^ <C3.3) Note a l s o : M u l t i p l y i n g t h e s e m a t r i c e s t o g e t h e r , we o b t a i n : (C3.4) (C3.5) (C3.6) T h u s , we r e a l l y n e e d t o know o n l y one component o f e a c h m a t r i x , w h i c h g r e a t l y s i m p l i f i e s t h e c a l c u l a t i o n . 118 C4. THE REFLECTED AND TRANSMITTED WAVES Matching the value of the wave functions, and their derivatives at the boundaries, one obtains: (C4. 1 ) From eq.(C3.4): Therefore: (A A], = T eifcViir"Ji • Js) (e *a"°+ £ ^"'J * ^-kj(cxCi -«2 fi-**-j> (C4.2) From eq.(C3.5): 119 T h e r e f o r e : (C4.3) From eq.(C3.6) — k, l k, xJj - Z i Si")i2ffQ X2- fr.H Iff fc*/Jl T h e r e f o r e : 120 e l 7 Cos IB.a- + -f — I K-Z^-Sl^i/v+lh-Z-h] / r , V , 22fa 3 -f-4-5fr, A * k, ] 4-+ ^ 4 * ^ £ - 4 QuiflL.* I L - J l f ) / S ^ 2 & < x ^ f (C4.4) Recall that k,= kj = k . Hence, eq . (C4.4) becomes: 0 (C4.5) where: (C4.6) 121 From e q . ( C 3 . 3 ) , M2l = "k, M. Applying k, to e q . ( C 4 . 4 ) , we obtain k, =~k^ =-k, and: ( C 4 . 7 ) These equations are of fundamental importance. To see t h i s we look at what i s going on p h y s i c a l l y . If we time evolve the state *P , . i t acquires a phase e~iE\ Thus A, E, and J are the amplitudes of waves t r a v e l l i n g to the right , wheras B, F, and K are amplitudes of waves t r a v e l l i n g to the l e f t . Let us consider the case where we shoot p a r t i c l e s from the l e f t only. Therefore K=0. A wave of amplitude A impinges at x=-b; a wave of amplitude B i s re f l e c t e d back from the well, and a wave of amplitude J i s transmitted through the well. The r e l a t i v e sizes of these amplitudes can be determined using e q s . ( C 4 . 5 ) , ( C 4 . 7 ) , and: A- M»T B = A l z . T ( C 4 . 8 ) Before we calculate the rate of decay of a p a r t i c l e from this well i t i s desirable to put the p a r t i c l e into a resonant state, as described in the next section. 122 C5. RESONANCE A resonant state i s a quasi-stationary state which maximizes the prob a b i l i t y that the p a r t i c l e i s inside the well and minimizes the pr o b a b i l i t y that i t tunnels out. This i s desirable because beta decay i s a weak process and the nucleus takes a r e l a t i v e l y long time to decay. A resonant state i s characterized by a transmission c o e f f i c i e n t of one, 5" meaning the incident wave passes through the well with none of i t being r e f l e c t e d . That i s , at resonance B=0, implying M^ j = 0 at resonance. Resonance occurs at energies for which there are multiple r e f l e c t i o n s adding constructively to y i e l d a huge amplitude inside the well. The energies correspond to quasi-stationary states which asymptotically approach the bound states in an i n f i n i t e square well as we l e t the barriers become a r b i t r a r i l y large. To see t h i s , l e t K-»oo and solve Mlt =0. The dominant „ term as X - » « 7 in eq.(C4.7) i s : Setting t h i s equal to zero implies: z * (C5.1) (n cannot be less than zero because E>0 from eq.(C2.3)) These are the energy lev e l s of an i n f i n i t e square well of width 2a. WE WILL CALL " K = £~2L " THE 0th ORDER APPROXIMATION TO K. 5 4 See the WKB treatment in Merzbacher, pg.129. 123 C6. NOTATION. FUNDAMENTAL EQUATION The s u b s c r i p t n w i l l d e n o t e a p a r a m e t e r c o r r e s p o n d i n g t o t h e n t h r e s o n a n t s t a t e w i t h n=0 b e i n g t h e g r o u n d r e s o n a n t s t a t e . A p p e n d i n g t h e p a r a m e t e r w i t h a s e c o n d s u b s c r i p t o f "0 " means t h a t we a r e t a k i n g t h e z e r o t h o r d e r a p p r o x i m a t i o n t o t h a t p a r a m e t e r . F o r e x a m p l e , K„ 0 = ** - was d e r i v e d i n t h e l a s t s e c t i o n . The e x a c t s o l u t i o n w i l l be w r i t t e n : ( C 6 . 1 ) where £n i s y e t t o be d e t e r m i n e d . THIS I S THE FUNDAMENTAL EQUATION USED I N APPENDICIES C AND D. We w i l l be s o l v i n g f o r t o v a r i o u s o r d e r s o f a p p r o x i m a t i o n , u s i n g t h e a s s u m p t i o n s d i s c u s s e d i n t h e n e x t s e c t i o n . C 7 . ASSUMPTIONS USED I t i s d i f f i c u l t t o s o l v e M = 0 e x a c t l y , s o we s o l v e a p p r o x i m a t e l y , w i t h t h e f o l l o w i n g a s s u m p t i o n s : 124 ASSUMPTION 3: P < Q / 1 ) ASSUMPTION 1: V, » E. X » Jc ASSUMPTION 2: V, » V0 =* (C7.D E,e, >> 1 [C*0>>i] Assumption 1 is j u s t i f i e d because we wish to have a high barrier in order that we have a slow decay as was discussed in SEC.C5. Assumption 2 i s j u s t i f i e d as follows. We wish to construct our model such that the n=0 state, (ground state), resonance has energy s l i g h t l y above zero so that i t has not much phase space and the decay i s slow (in analogy to decay). It may be interesting to also study how the decay depends upon n. From eq. (C5.1 ): J?, ~2 Ea . Using eq. (C2.3) : JE,+ v° ^ijto* v/0 ' . But E0<<Vo i f we demand that E 0 i s close to zero. .", ^/F A v»" ~ • .'. E,~3Vo • Hence, we need assumption 2 in order to ensure that E, ,E a ,E 3 , . . . « V , and thus the n=1,2,3,... resonances s t i l l constitute a slow decay for a f a i r l y large, range of n. We now discuss assumption 3. F i r s t l y , we desire to have e ^-^1, again because we want a slow rate of decay. Secondly e ~ 2 X ^""^ % 0 ( ~ T A ) i s necessary because we wish to use Ha as an expansion parameter, without worrying about the corrections to M. i t s e l f . This i s c l a r i f i e d as follows: 125 We star t b y noting that £ h a ~ O O ) from eq.(C5.1>, therefore from assumption 2: K a » 1 (C7.2) Rigerously speaking, n\an w i l l be used as the expansion parameter in th i s model, where: X^Ji^-E^y ; ETMo = Z i - V 0 ; H h o= frLLii!T (C7.3) See the notation discussed in SEC.C6. j We w i l l show below that £ = { -r •+ terms of lower order} Thus: and K ^ ^ A z r A - J - ) = E h . ( r - — ) <C7.4> E H - l i ( f l - ^ ~ ) - V 0 = ET M O- j £ _L . (C7.5) x*~J*» ^irJ. I f a ? * ' (C7.6) Thus >L~>l!fca with corrections of order ~ • . Henceforth, we w i l l use X h o in place of}i! hand note that t h i s approximation i s v a l i d providing we carry our expansions to order J y £ ji" anc^ n o 1 26 further. To further c l a r i f y t h i s , l e t us look at the case of S h. We w i l l show below that: Xla3 l IT- J This i s written as: S = -T ! 5— ^ —£— /i-fr^)+...7 l**.* >£<* »2.<*V n J J Actually, j__ ^ _j_ r + frfl)^M = + c / i ) Thus we expand £"„ only to O ^ ^ y ^ because in the next order corrections to X i t s e l f begin to become s i g n i f i c a n t . With these considerations in mind, l e t us complete the discussion of assumption 3. F i r s t , we note that; £~OC>tO (C7.7) and that: also: S f * 2 f f a ~ ° ( s k ) ( c 7 - 8 ) C O S l E a -v. i (C7.9) Below, i t w i l l be shown that (cos2£a + JL sin2"g:a)~ 0(e~^* ( t" a^ •2. ). Writing eq.(C4.7) in terms of the order of each term, we obtain: Now, from the discussion above, we w i l l have expanded the terms of order (1) down to order -7-.—TTT" . I f we wish to 127 ignore the l a s t term in , we had better make i t one order lower than t h i s . Therefore: e -** f t - ) < o which i s assumption 3. C8. ORGANIZATION OF THE EXPANSIONS With these assumptions in mind, we now wish to solve M2) =0, ignoring the e term. We obtain: x -n ( C 8 . D The expression cos2Ka + _§-sin2Ka, which we c a l l X i s very important. This is the expression which governs where the resonance i s . We see that the RHS of eq.(C8.l) i s of order , . Therefore l e t us f i r s t solve the equation: Co S 1~Ea- + JL Slh, IZCL •= CD (C8 2) This equation w i l l be solved for S„ and w i l l determine the energy of the resonance v i a the fundamental equation: K 0 = T£„0(1_S„) as stated in SEC.C6. Why do we wish to determine so precisely? Why isn't the zeroth order approximation £"„ =0 good enough? The answer to these questions i s that =0 i s quite f i n e , i f we only want to find the energy of the resonance. If X.<x = 10" then corrections to the zeroth order approximation to the energy of resonance w i l l appear only in the 4th decimal place. But, i f we wish to fi n d the decay rate of the resonance, 128 the story is d i f f e r e n t . We w i l l show that the decay rate can be found in two equivalent ways: 1) By adjusting the energy, to find the width of the resonance. 2) By solving for no incoming waves, which introduces a small imaginary adjustment to the energy. In either case, a very small adjustment to $ i s involved. That i s , the resonance i s extremely narrow. It i s so narrow that even solving eq.(C8.2) for S i s not good enough. (We w i l l c a l l t h i s the FIRST ORDER APPROXIMATION. ) We must solve eq.(C8.1) to obtain an expression for $ which i s good right down to the order of the decay adjustments. (We w i l l c a l l t h i s the SECOND ORDER APPROXIMATION. ) One can solve for the decay adjustments without knowing exactly what $ i s but i t i s interesting to see exactly where the adjustments f i t i n . C 9 . FIRST APPROXIMATION: £ =0 In order that we solve for eq.(C8.2) to order - 1 - , i t i s necessary that we expand the equation to order or ' ' — > because £», w i l l be of order - ~ — ( a s stated in sec.C7), and ? i s of order (Xa) . It may seem l i k e we are putting the cart before the horse, but a l i t t l e hindsight i s useful in recreating a path to the answer. From eq.(C6.1), and using the Taylor expansions of sine and cosine: 129 (C9.1) r „ ~ ~ rt "7 Sin 1 B^O. ~ The f i r s t term of sin2K ha i s of order — - — . To obtain eq.(C8.2) expanded to order > w e need to expand I" to order , because ^ from £ times from sin2K ha w i l l y i e l d - 1 ^ . From eqs.(C4.6), (C6.1), and (C7.3): F„ 06-0 * (h+07rr/-f-) (C9.2) Using the Taylor expansion: • • • = 1-x+x2-x3 +... (n*i)Tf ^^.Q- (C9.3) Inserting eqs.(C9.1) and (C9.3) into eq.(C8.2) and keeping terms i to order —- , we obtain: C o l l e c t i n g terms of l i k e power in £: (C9.4) Start by ignoring terms of order • O.YC Using the quadratic formula: Using the Taylor expansion *jHx - I + 1 * - + # 130 I Let: s-=-de.- ** + ° feb*) (c9-5) Substituting back into eq.(C9.4): (C9.6) The "1" denotes that t h i s i s the f i r s t order approximation to £\ Cl 0. SECOND ORDER APPROXIMATION We have solved equation (C8.2), which states 5*= 0, where we define: S» = coS2B M a+ I s i H 2 F h a (C10 . D We actually wish to solve eq.(C8.1) which states: J h = I f Sfh-ZB-a e - ^ - a ) (C10.2) Let us write the solution in the following form: § - o (C 1 0 . 3 ) 1 3 1 where [ - means that within the brackets i s contained a l l the terms of a l l orders needed to s a t i s f y Sh- 0 * o < e"ZX^"* a ^ contains the extra terms needed in order to s a t i s f y e q . ( C 1 0 . 2 ) . From assumption 3 , £ ' = 0 i s v a l i d to £ 0 ( 7 — — ) . Using eg. ( C 10.1 ) : S« ~ 2 — - ^ C O S i H ^ a e /^v°U^a;V(? ( C 1 0 . 4 ) To solve t h i s fore*, we f i r s t expand the LHS of e q . ( C l 0 . 4 ) . As with £, i t only makes sense to calculate <* to order ; — • — r : but not beyond. It w i l l also turn out, as with S that oc* is of order - - r — - — . Carrying ? to order <f„will y i e l d t h i s . Therefore carrying 5 to 2 orders beyond this w i l l y i e l d to o r d e r ; — ~ r i . Using the expansions ( C 9 . 1 ) and ( C 9 . 3 ) , we write t as: - ("+>rw 1 f o f a * » * * (CIO.5) Rewriting e q . ( C l 0 . 3 ) : When inserting e q . ( C l 0 . 6 ) into e q . ( C l 0 . 5 ) , terms involving powers of alone w i l l y i e l d zero. This leaves oC e and cross terms of thi s wi th S"s„=0. Only e -**' b' a J to the f i r s t power i s kept. We obtain: 132 4-**^ J v * * * * J (C10.7) + 2 C L ^ n o (C10.8) Using eq. (C9.6) and keeping terms to order in the J1 of eq.(C10.8): i 4 ' t(a^JVJ (C10.9) This i s the LHS of eq.(Cl0.4). Now for the RHS. Using eq.(C10.9): (C10.10) Dividing numerator and denominator by / - ^ - i L ? ) and usinq 133 the Taylor expansion: (l-x) = 1-x+x2-x3 + ... such that = (1+x)(1+x+x2+...) = 1+2x+2x2+..., and l e t t i n g : E. I f X* _ kE k& X * we obtain: - t + z r E _ jr. " k EC k I T 21?- JLI (C10. 1 1 ) We desire to carry J £ l t o order -?. ' t , . Multiply the numerator and the denominator of x by a 2kK: Note that aK i s of order one (from eq.(C5.l)) and from the discussion preceding eq.(C7.2), k i s of order "K. (or le s s , perhaps much less, when n=0). Therefore ak is of order one or less . 1 But the I a*X* c o n t r i b u t e s a term of order higher than we wish to We also see that x 2 carry 12L . Thus we write x = contributes only terms of neg l i g i b l e order. Thus: ? ~ 7 (C10.12) Of course, k„ depends upon but we w i l l leave i t as an im p l i c i t dependence and worry about i t l a t e r . We w i l l write the 134 e x p l i c i t dependence of K„ on g*H as usual from eq.(C6.1): (C10.13) Keeping only the terms which y i e l d terms up to the desired order of 7~~~rz in , one finds: 4 V. aJU« / (C10.14) Thus: — i + — f / ) + O / — (C10.15) Take eq.(Cl0.5) along with the expansion for cosZE^a given in eq.(C9.1), and insert into eq.(Cl0.4) f or £ . One gets: ( a * . . ) 3 J " (CIO.16) This i s the RHS of eq.(Cl0.4). Equating to the LHS in eq.(C10.9), one finds: o( 7 (C10. 17) a*. 0 4-Again using the expansion l + ac =1-x+x2-x3+... Plugging into eq.(C!0.3): (C10.18) 135 4-ho 4 J H* Thus, the resonances occur at: !Kh i s given by eq.(ClO.!9). (C10.19) = l^-7f (1 - &,), where £V, 2. a C11. Mn AT RESONANCE Now we have the energies at which M2| goes to zero. For interest sake, l e t us check to see what happens to M,j at resonance. We expect that [Mt( / 2 = 1 i f a l l of the wave i s transmitted, but what i s the phase of M/f ? -2.ttCfc-a) Ignoring the e term in M„ and substituting eq.(C8.1) into eq.(C4.5), we obtain: " 4 - "7 L ^ , From eq.(C4.6): (C11 . 1 ) (C11.2) (Cl1.3) Using the same argument used to obtain eq.(C!0.4), we have: 136 (C11.4) 1' ( C l l . 5 ) The expansion of cos2K„a has already been worked out, and appears in the square brackets of eq.(C10.16). Now from eq.(C4.6): I This was obtained using the Taylor expansion of -j and noting ak H i s of order 1 or smaller, as was noted preceding eq.(Cl0.12). Substituting t h i s into eq.(C11.6), one obtains: n . = (-<) e ( C l 1 . 7 ) Recall from sec.C8 that we have solved for Mz( =0 down to second order in ~~~• Thus we had better expand M„ only to 7—-—— also. M„ i s thus: ^ ^ [ x - r ^ i ^ o ^ ) ] ( C 1 I . B ) Using the Taylor expansions of cosx and sinx for small x, we immediately see that: 137 (C11.9) Indeed, we see that at resonance )Mh| 2 = 1 as expected. We also find (Cl1.10) From eq.(C3.3), the structure of (M) at resonance i s : \ C-0 e o o (C11 .1 1 ) 0 2. SOLUTION AT RESONANCE WHICH IS A PARITY EIGENSTATE Let us find an e x p l i c i t form of the solution (C2.2) at resonance, by solving for the c o e f f i c i e n t s at resonance. Using the matrix (M+) given by eqs.(C3.l), (C3.3), and (C4.1), one finds: , . ., , (C12. 1 ) Using the matrix (M3) given also by eqs.(C3.1), (C3.3), and (C4.1), one finds: 138 J (C12.2) Doing the same thing with (M 4 ) , a lso given by eqs . (C3 . l ) , (C3.3), and (C4.1): (C12.3) •2. J And of course, we already know that B=0, and A i s given by e q . ( C l l . l O ) . One may use (A,) given by e q s . ( C 3 . l ) , (C3.3), (C4.1) along with C and D above to ve r i f y that one obtains the same answer as : A = M f lJ, B = M^J, where M | ( and M 1 ( are given by eqs.(C4.5) and (C4.7). We not ice that th i s is not a state of de f i n i t e par i t y because we are shooting a wave from the l e f t , but none from the r i gh t . We wish to have a par i t y eigenstate because we want to 139 consider a symmetric resonance which can leak out of the well in both d i r e c t i o n s . Therefore, l e t us also shoot a wave from the right with amplitude K'. It i s ref l e c t e d with amplitude j ' , and transmitted with amplitude B'. A'=0. Recall that in an i n f i n i t e square well (the l i m i t i n g case of our model at resonance with a r b i t r a r i l y large b a r r i e r s ) , the sign of the parity i s the sign of n. We therefore construct our state of d e f i n i t e p a r i t y by l e t t i n g u'=(-1)" A. We also need: A'~ (-l)HK = o > B' - (- l)"T C'=(-0TH y c - < r G E'= (-0* F J F' = (-<)" e (C12.4) ^ ( - i ) " ] ) 5 H ' = ( - 0 M c T ' = ( - I ) H 8 i K ' = C - 0 H A where we write the solution to the Schrodinger equation: — . v)cJlkx X C c+ c O e _ x x + c 5 % — % = ( G ^ ' ) e ' n x + (H (j+x'Jcl'kx ( * 4 K ' ) c ~illx Inserting eq.(Cl2.4) into eq .(C12.5) y i e l d s : — -4 e£ < r x + X ti — ( E + C-O*F) eiK* + — ( < ? + ( - ' ) * ] > ) e " * * + - O 0 M e"1'** ( C l 2 . 5 ) (C12.6) 140 The c o e f f i c i e n t s A to J are given in eqs.(C11.l0) to (C12.3). We see that for even n, = 2(E+F)cosKx and for odd n, ^ = 2i(E-F)sinKx. "+" i s used to denote even n and even parity; "-" i s used to denote odd n and odd pa r i t y . Using E and F from eq.(Cl2.2), we f i n d : Sim E^a f TcoSR*,* (C12.7) 22. coSH„« If J + l + i As was done in eq.(C9.l) for the cosine and sine of 2K„a , we expand cosKa and sinKa in terms of S. We obtain: c o s s > = (-0 s [i-oo nr - j(4J3(*+03^ +• • • (C12.8) S i r , B.-« = L-^l-j. Mnr - [if + . . . J c o s a: = C-i) * f I - ^I1* • • • I Here the + and - again refer to p a r i t y . Using £ from eq.(CIO.9): 141 J Expanding _ L ~ i + s = 1 + _L o n e obta ins : • " yia-l-S COS Ignoring the e ~ ^ ^ " a ^ term in W4", we obta in : ( C 1 2 . 9 ) (C12.10) where K.ho i s defined in e q . ( C 6 . l ) . S im i l a r l y : ( C 1 2 . 1 1 ) At resonance we see a very pecu l ia r phenomenon: namely the standing wave inside the well has a huge amplitude compared to the incoming and outgoing waves. The incoming and outgoing waves have an amplitude of |J| wheras the wave inside the well has an amplitude of about Xae**" ^ Q(X a ) 5 times t h i s . This is what explains why the wave i s completely transmitted through 142 the barrier with no r e f l e c t i o n . The incoming wave at resonance has just enough energy to cause multiple r e f l e c t i o n s constructively i n t e r f e r i n g inside the well, setting up a standing wave of very large amplitude, as described in SEC.C5. The amplitude inside the well i s so large that the wave leaking out the other side has the same amplitude as the impingent wave. C1 3. THE n = 0 RESONANCE Let us look at the n=0 case, because t h i s is the resonance which w i l l be used to model beta decay. Note from eq.(C11.10): (C13.1) and (C13.2) , to ^ X.cJ M T7~ + 0 >f.0a (C13.3) At x = t a , we use the expansion of eq.(C12.8) to obtain: A (C13.4) Using eq.(C12.1l) in eq.(C12.6) for and one gets: 1 4 3 Yfw= 2-4(1 w i ) e r , u L t fc-i) - i n * 6*- 0 ( C 1 3 . 5 ) food)' A t the boundaries x - i b : ( C l 3 . 6 ) A qua l i t a t i v e graph of th i s so lut ion appears in FIG.13, ignoring the e"* 0 (i + i-£i) f a c to r s . C a l l th i s . 144 AT POINT 1 : V - 2 A AT POINT 2: e ^ ' ^ ' " ^ AT POINT 3: ? = ^ss. e * ~ f l > - a ) ^ Hoc F i g u r e 13 - The n=0 R e s o n a n c e Cl 4. THE DECAY OF A RESONANCE AND A COMPLEX ENERGY EIGENVALUE S u p p o s e we c r e a t e a r e s o n a n t s t a t e s u c h a s t h e one d e p i c t e d i n F I G . 1 3 w i t h t h e wave f u n c t i o n i n s i d e t h e w e l l n o r m a l i z e d t o one p a r t i c l e i n t h e g r o u n d s t a t e . Now we w i s h t o f i n d o u t how l o n g i t t a k e s f o r t h e p a r t i c l e t o l e a k o u t , so we s u d d e n l y s t o p s h o o t i n g i n c o m i n g waves a n d w a i t t o s e e what h a p p e n s . T h i s p r o c e s s w i l l be d i f f i c u l t t o c a l c u l a t e b e c a u s e o f t h e s u d d e n d i s c o n t i n u i t y i n t r o d u c e d when t h e i n c o m i n g waves were s h u t o f f . A p h y s i c a l s i t u a t i o n w h i c h i s a l m o s t e q u i v a l e n t t o t h i s b u t 145 vastly easier to calculate i s the following: Assume we started at time t = -oo with an i n f i n i t e supply of wave in the ground state of the well. One may think of t h i s as being an i n f i n i t e supply of p a r t i c l e s in the ground state, which have slowly leaked out. What went on at t = -co does not matter because i t s e f f e c t s have propagated to x =±co ; we are l e f t with a wave function normalized to 1 inside the well, joined smoothly to an outgoing plane wave outside the well. We want to calculate how long i t w i l l take for t h i s last p a r t i c l e to leak out. We assume the decay i s exponential, and write the time dependent state in the following form: V ( * . * J = e ^ V ^ n * , * . . ; (c,4•', It appears that the normalization of y(x,t) changes with time so that the number of p a r t i c l e s i s not conserved. But one must remember that x) i s normalizable inside the well only. The change of normalization inside the well means the p a r t i c l e is leaking out of the well. Outside the well, the normalization is i n f i n i t y , and e"1*^* x oo is s t i l l i n f i n i t y . This i s because there i s an i n f i n i t e number of p a r t i c l e s outside the well: the e merely changes the number of p a r t i c l e s per unit length. Because u->T is small, the state (C14.1) i s quasi-stationary. Let us demand that t h i s state be an "eigenstate" of the Hamiltonian, with "energy" E m e a n i n g i t s a t i s f i e s : J U (C14.2) One immediately finds a complex "eigenvalue": E' = E - itu. This is what happens when we demand that a non-stationary state be 146 s t a t i o n a r y . How can we use t h i s to c a l c u l a t e w^? We merely demand that M|,=0, meaning that we are s o l v i n g f o r an e i g e n s t a t e of the Hamiltonian with no incoming waves. But a " s t a t i o n a r y " s t a t e with' no incoming waves i s o b v i o u s l y not s t a t i o n a r y . From the above reasoning, demanding M,( = 0 w i l l y i e l d a complex energy E t h e imaginary part of which w i l l be o>T. C15. A PROOF THAT THE REAL PART OF THE COMPLEX DECAY ENERGY IS THE RESONANT ENERGY We must ins u r e that the r e a l p a r t of E' i s the energy of resonance and not some other energy. The proof of t h i s i s q u i t e simple: suppose we expand |M,,|2 to second order i n £ ( S i s d e f i n e d i n eq.(C6.1)) near the resonance as f o l l o w s . ' l C ? o The t r a n s m i s s i o n c o e f f i c i e n t i s d e f i n e d such that | J I2 = T J A j 2 . T h e r e f o r e T = - J rr • Resonance occurs when T=1. In g e n e r a l J M f ( / 2 > 1, so resonance occurs where | M „ | 2 i s a minimum. The c o n d i t i o n f o r a minimum i s - S L j M l l | 2 = 0. T h i s b i m p l i e s Sft-^" / where $ R denotes the value of £ at resonance. S e t t i n g aSB2-bS"R+c = 1, one f i n d s t h a t c = 1 + T h e r e f o r e : ' 4-a r - I, (C15.2) OR ~ - — Let us now solve f o r fta , the value of £ which y i e l d s |M|,J2= 0. Using- the q u a d r a t i c formula to so l v e f o r S i n I? a S 2 - b £ + 1 = 0, one f i n d s : 147 2 a (C15.3) Indeed, the real part of S i s 6^. Q.E.D. C1 6. ASIDE: THE RELATIONSHIP BETWEEN THE DECAY RATE AND THE WIDTH OF THE RESONANCE One w i l l note something interesting upon c a l c u l a t i n g Sifa, the value of £ at which T f a l l s to £ maximum. Setting |M(| 2 = 2 in eq.(C15.2) and solving for £, one obtains: £. = _k. i - i - (C16. 1 ) That i s , the imaginary part of S i s just the -J- width of the resonance at ~ maximum, which i s why tu,- i s sometimes c a l l e d the "decay width". This makes sense from the point of view of the uncertainty p r i n c i p l e : 4 E A t ~ 1 . But' 4E=2U) T /. 2U/ rAt~1 or — 2U*r, implying that the decay rate i s approximately the same At as the width of the resonance. C l 7 . CALCULATION OF THE COMPLEX S REQUIRED FOR DECAY Let us now calculate the value of S such that M„ =0. Recall from eq.(C8.l): £ E C o s l K a * - f S , M H a = £5 S^2 ffa e " ^ " ^ *• *>7 \ c i /. 1; -2M (b-a) , . Ignoring the e term in eq.(C4.5), M ( / i s just: 148 (C17.2) Plugging eq.(C17.l) into eq.(C17.2) yiel d s eq.(C11.8), reproduced below: (C17.3) We wish to modify £ to get Ml( =0. Let: \ ~1 J (C17.4) Plugging eq.(C17.4) into eq.(C17.2) y i e l d s : (C17.5) Setting M(/ =0, and using ^ from eq.(C4.6): To solve for , we extend eq.(Cl0.6) to: (C17.6) (C17.7) Using the method following eq.(Cl0.6), we merely replace <X by oc + ip> in eq. (C10.9) : From eq.(C17.4) and eg.(CIO.4): 149 6" 1 (C17.9) The f i r s t part of th i s expression has been given by eq.(C10.16). Equating eq.(C17.8) and eq.(Cl7.9), one finds analagously to eq.(C10.17); ot+i? = 2 l- 2Ji-^!rVif.,) 1v] and therefore: (C17.10) CXXMO L Inserting y from eq.(C17.6); _ 4-fe. ' r i-(C17. 1 1 ) (C17.12) Eq.(Cl0.19) for £ i s modified: 150 " Lax,, (ax..)1 (aXjPV II / UaXjVJ^oo [ i - — + - 1 — - A - - 2 o / — i — ) ] e ^ JL.fj - ^ - - J — A . fa'Prf-Ktp[-1— | L ~1 (C17.13) One may now see why we expanded to so many orders in eq . ( C l 0 . l 9 ) . The resonance i s so sharp that a very small change in £ (of o r d e r — — e =• J^^e) destroys the resonance. C18. ASIDE; WHAT HAPPENS TO Ma, ? S was o r i g i n a l l y calculated by setting M2, =0. Now that we have modified £ in eq . (C17. l3) to get M„ =0, what happens to Ma,? Plugging eq . (C l7.4) for S into eq . (C4.7) for M2f and ignoring the e term: Alw = -j-^iWJ (C18.1) From eq . (C l7.6) for 2f and eq . (C4.6) for ' In summary, at resonance we fi n d M,, = (-1) e *• X«» J , and Maj= 0. When we demand no incoming waves and demand that the state decay, then we find M(( =0, and = (— 1) . Thus for 151 decay (M) has the structure of (Mj) below: (A,) = ( ° (-°' K U-0" O J (CIS.3) We take A = K = 0, and B = (-1)"j. There i s a wave leaking out of the well equally in both directions with parity governed by the parity of n. The s i t u a t i o n i s as was described in sec.C14. C l 9. OBTAINING THE DECAY RATE FROM THE COMPLEX S Let us calculate LUT, the decay rate. In the d e f i n i t i o n of K (eq.(C2.3)) f E has been replaced with E ' = E —iu - v as defined in sec.C14. H'= /y2(^(E4 Vo-zuJ-r) = K(t + z'S1 (C19.1) where E R and Kx are the real and imaginary parts of Z. Therefore: ITR SI U J T = no ( C 1 9 . 2 ) Now K R i s just Xrt=E;M<7( ]-gK) where i s the real part of C. Ki= -K^ g x , where £ t is the imaginary part of £. U , T = *i_C^,-£„) (C19.3) too From eq.(C17.13): (C19.4) Writing ffHO = / j 2 w M ( E „ + V . ) ' (C19.5) one obtains-: 152 (C19.6) C20. CALCULATION OF NUMERICAL VALUES FOR BETA DECAY The numerical values of the relevant parameters for the nucleus considered by Becker et a l . are calculated as follows: We use the following parameters which come from a table of isotopes. 5 5 E 0 = O.O) 8 6 / 9 0 0 ^ e V T i / Z - 11.33 y** - 3.29/ * >°$ s (C20.1) We note that 77- = f where f i s the mean l i f e , and Tl.= 7rln2. We assume b i s given by: „ (C20.2) where A i s the atomic mass and b i s in Fermi's. This is the Rutherford formula for nuclear r a d i i . 5 6 The c a l c u l a t i o n goes as follows: f i r s t we specify the height of the barriers by saying what we wish - ~ to be. Remember that ^ is a measure of the percentage error in the 5 5 Lederer and Shirley. 5 6 Segre, pg.219. 153 c a l c u l a t i o n of CVTr assuming we keep on l y the f i r s t term i n e q . ( C l 9 . 6 ) . We assume o n l y the n=0 (ground s t a t e ) resonance i s e x c i t e d . Then we s p e c i f y what we wish - ( t h e r e l a t i v e w id th of the b a r r i e r s ) to be. Us ing the b c a l c u l a t e d i n e q . ( C 2 0 . 2 ) , we f i n d what a i s . That i s , we assume the ou te r d i s t a n c e b i s f i x e d by the r a d i u s of the n u c l e u s , and we vary a , the i nner d i s t a n c e to the b a r r i e r s . Then u s i n g : E ° 4 V ° * * 1 (C20.3) which i s the ground s t a t e energy of the square w e l l , we f i n d out what E e o + V a i s . We a l s o c a l c u l a t e E 0 o + V o from e q . ( C l 9 . 6 ) : E , . + v 0 = i l « t t a H . J (C20.4) If these are not c o n s i s t e n t w i th each o t h e r , then we modify -k"** u n t i l c o n s i s t e n c y i s a c h i e v e d . The numer i ca l r e s u l t s a re b t a b u l a t e d in chapter 5. 154 APPENDIX D - ELECTRIC FIELD ENHANCEMENT OF ELECTRONIC DECAY FROM A SQUARE WELL D1. INTRODUCTION This appendix i s an account of the mathematical and physical d e t a i l s of solving for the decay of a resonance in a square well with ba r r i e r s , which has been immersed in a lin e a r e l e c t r i c f i e l d . The f i r s t few sections of t h i s appendix p a r a l l e l the f i r s t few sections of appendix C, and i t i s recommended that one read appendix C before reading t h i s appendix. D2. THE MODEL: SOLVING THE SCHRODINGER EQUATION Consider a potential well of the form i l l u s t r a t e d in FIG.14. FIG.14 is to be compared with FIG.12. Again, the regions have been l a b e l l e d with numbers, and the l e t t e r s are the constants used for matching solutions from one region to the next. 155 Q A "6 c D 4-® 6-H •Y • -w a - y T O L N ? - c Figure 14 - The Potent ia l Well in an E l e c t r i c F i e l d The e l e c t r i c f i e l d i s appl ied in the region: -c<x<c. The shape of the well i t s e l f is assumed to remain the same because we do not want to introduce a Stark e f fec t in the resonance energies. As discussed before, the most probable energy t ransfer from the laser to the e lectron is about 1 MeV. Therefore we assume Y = 1 MeV. In the nucle i we are interested i n , the energies of beta decay are smaller than 1 MeV so the wave funct ion w i l l be a decaying exponential in region (l), and the bar r ie r penetrat ion w i l l occur only to the l e f t . Par i ty no longer commutes with H. We a lso assume that the acce lerat ion occurs over a c l a s s i c a l distance c>>b. C a l l t h i s : ASSUMPTION 4: c>>b In conjunction with t h i s , we a lso use assumptions 1, 2 and 3 of appendix C. Schrodinger 1 s equation for r e g i o n(T ) i s : 156 where: V0> = -T s V*,-Y Vf,j=VW = V, y (D2.1) V(i) - - v0 J In regions (|)and (gV Schrodinger' s equation has the form: Lett ing one obtains: (D2.2) (D2.3) j - 1 (D2.4) Equation (D2.4) is merely Airy's equation, 5 7 with solutions: Ai(x) and Bi( x ) . The general solution to eq.(D2.l) i s therefore: (assuming -V0< E < Y) 5 7 See Abramowitz and Stegun, pg.446. 157 X r , 4-3? = ^ X - ^ E J E e G t (D2.5) D3. THE STRUCTURE OF THE MATRICES WHICH RELATE THE COEFFICIENTS As before, the c o e f f i c i e n t s are determined by matching the functions and their derivatives at the bondaries. There are 12 boundary conditions, and 14 constants. But we know P=0, therefore we have only one a r b i t r a r y constant from which the other 12 are determined. As before, we write the matching conditions in matrix form: 158 A r > X E H '3^1 K , ( F E)»(*>(S, A l 5 -A/ T (D3. 1 ) (A) = ( A j ^ (M,) to (MA) above have the same structure as the (M,) to (M^ .) used in apppendix C. Using eq.(C3.3), we replace k, by k and replace ^ 5 by j f so that eq.(C3.6) for (M) becomes: (D3.2) From eq.(C4.1): - i H a -iffa as before. Of course, M, and w i l l be d i f f e r e n t : (D3.3) 159 = (A*)u ( M i l + (D3.4) l i One may e x p l o i t the symmetries of (Ma) to (Mj) i n the same manner that the symmetries of (M, ) to (M*.) have been e x p l o i t e d . We extend the operator d e f i n e d i n sec.C3 to i n c l u d e the f o l l o w i n g : That i s , the operator A i changes A i to Bi and Bi to A i . Using the methods of l a s t chapter, we f i n d : (D3.5) Using eq.(D3.5) i n eq.(D3.4), one f i n d s the f o l l o w i n g : /At - ( 14 Al)(M«Mk) (D3.6) 160 D4. THE REFLECTED AND TRANSMITTED WAVES Matching the V's and s at the boundaries yie l d s the following: (D4.1) The factor of 7Y appearing in Mt and M^ a r i s e s due to the Wronskian: 5 8 W{ Ai (x), Bi (x)} - rjf > Inserting eq.(D4 . l ) into eq.(D3.6): (D4.2) The (1-Ai) in eq.(D3.6) has become (1-Ai) in eq.(D4.2) because of the Wronskian: W{Bi(x),Ai(x)} = -W{Ai(x),Bi(x)}. Abramowitz and Stegun, eq.10.4.10, pg.446. 161 We now insert M2 and M3 from eq.(D3.3), together with M, and M4. from eq.(D4.2), into eq.(D3.2) to obtain M/; and M^. After performing the algebra in a manner similar to that done last chapter, we ar r i v e at: (D4.3) (D4.4) These equations are of fundamental importance, in t h i s appendix and are to be compared to eqs.(C4.5) and (C4.7). The various parameters are defined on the next page: 162 <* h k if >• V k © = Al(_-y)z;(-z)-u(-r)A;(-2) 0 = ®/ ®f © , (d) are obtained by replacing z with z, and y with -y. 163 D 5. THE FIRST APPROXIMATION How large are these parameters? Y~1 MeV, wheras E i s as low as 0.019 MeV for ^ H, so 0.01^ | £ 1. On the other hand, c w i l l be a c l a s s i c a l distance, so -fe.«i, and ctc»1. We w i l l therefore start by assuming: y>>1 ; y » 1 ; z~/ z ~ occ-^ >> 1 (D5.1) THIS IS OUR FIRST APPROXIMATION. With these assumptions, we may use the asymptotic expansions of Ai and B i : 5 9 (D5.2) These are the same expansions we use for z and !? also. But, for y the argument of the Airy functions i s p o s i t i v e , leading to the following asymptotic expansions: 5 9 Abramowitz and Stegun, pgs.448-449. 164 My)' f e 3 ' t-Jv (D5.3) I n s e r t i n g eq. (D5.2) and eq. (D5.3) i n t o eq. (D4.6) y i e l d s : © ~ -® r, 1 / 1 — - 1 I (D5.4) (D5.5) J 165 It i s useful to note from the d e f i n i t i o n s of y and y in eq.(D4.1), from z and z in eq.(D5.1), and the d e f i n i t i o n of <x in eq.(D2.5), that: where k=/2mE' as defined in defined in eq.(D2.5). Inserting eq.(D5.4) and eq.(D5.5) into eq.(D4.5) and using the i d e n t i t i e s (D5.6), one finds the following: k J (D5.7) (D5.10) (D5.6) appendix C, wheras k and are 166 (D5.11 ) ( b o , ) ^ (D5.12) 3 (D5.13) The upper sign corresponds to the unbarred parameters and the lower sign in brackets corresponds to the barred parameters. Inserting eqs.(D5.7) to (D5.13) into eqs.(D4.3) and (D4.4) for M M and Mjv one obtains: 1 6 7 x 7 ( D 5 . 1 4 ) x ("cos 'Z.tfa + Siv» 2 a j ^ ( D 5 . 1 5 ) ( D 5 . 1 6 ) 168 D6. OBTAINING THE DECAY RATE USING THE FIRST APPROXIMATION Using the methods of appendix C, l e t us solve for no incoming wave by l e t t i n g M,, = 0. This w i l l give us the resonance energy and the decay rate. We ignore the e - ^ ^ a ^ term in eq.(D5.14). Recall the d e f i n i t i o n of £ in eq.(CIO.I) and how we l e t t = s i n 2Ka + j e 2 W ^ " ^ i n eq.(Cl7.4). Let us s i m i l a r l y write ? in eq.(D5.14) as: \ ~1 ' (D6.1) and solve for X. It turns out that # i s complex: Y=Yn.+ z^i' y, _ S i * Pe - yt C e ? s £ > * '* -1 cosfr,+ £ * , V , * e ( D 6 ' 2 ) 9^ represents a very small Stark e f f c t in the resonance. ^ is 0(1), and represents a £1+0(-££")J m u l t i p l i c a t i v e correction to £. This w i l l modify the second term in the 6 part of b« given in eq.(C17.13). Hence the ef f e c t on the energy of the resonance i s n e g l i g i b l e . What about the decay rate? We see that the imaginary part of T i s : 1 (D6.3) Setting £ u = cos2Ka +^.sin2Ka = 0 at resonance, one finds: t S<h i B a = (D6.4) (using e+4 = ) £ Inserting eq.(D6.4) into eq.(D6.3): 169 lr - — e Comparing this to eq.(Cl7.6) for Y, we see that we have reduced exactly by a factor of 2, which reduces the imaginary part of S by a factor of 2, which reduces the decay rate UJT, by a factor of 2. That i s : elec't'"'c ) free (D6.6) This i s to be expected from FIG.14 because the p a r t i c l e can escape into only one d i r e c t i o n wheras before the p a r t i c l e could leak out in both d i r e c t i o n s . This also indicates that the e l e c t r i c f i e l d does not enhance the decay unless our parameters are such that i t is no longer v a l i d to take the asymptotic expansions of the Airy functions. This leads to our second approximation. D7. THE SECOND APPROXIMATION If we look at eq.(D4.l), we see that i t i s possible to make z«y and zf<<y i f we choose the e l e c t r i c f i e l d large enough and the decay energy small enough. Let us therefore assume that the asymptotic expansions for the Airy functions of y and y are s t i l l v a l i d but the asymptotic expansions for the Airy functions 1 70 of z and zf are not v a l i d . We w i l l c a l l t h i s our SECOND APPROXIMATION. We write the Airy functions of z and z in terms of the modulus and phase, 6 0 as follows: /4i(-0 = cc$ 9to) T>il~*) = rtfc) (D7 . D The same also holds for z. Substituting eqs.(D7.l) and (D5.2) into eq.(D4.6), one obtains: ® = fe J - ' ^ M ft) S/H(9fr> - j * f 7*y=js>r*M<v s;" »• V^WfO cosfef(i)-2! 1 s -_L i/,..,,, (D7.2) 0 _ _ / - - I © © ® (D7.3) 7 6 0 Abramowitz and Stegun, pg.449. 171 Inserting eq.(D7.2) and eq.(D7.3) into the parameters (D4.5), one finds: 1. ~V? (D7.4) @ rr J-JT 7 e%5 * ^ - i A|(Offv, j . - 4 i f " # + * N ^ > s ' , h ? A > + CT) W(*)coS%Ai C-i) & A/60 CosMi C-*jJ ^ A ) 6* ) S * £ / { i C-2 ) c t , i N 6 0 C c s ? /4 i (~ i ) J ( b a h } f f (D7.5) (D7.6) (D7.7) (D7.8) (D7.9) 172 (bar) ifV Jr /. o(. U <* y (D7..10) Inserting the parameters (D7.4) to (D7.10) into eq.(D4.3) for M„, one obtains: . J (D7. 1 1 ) D8. OBTAINING THE DECAY RATE USING THE SECOND APPROXIMATION As before, we l e t M.=0, and ignore the e term in eq.(D7.11). Plugging £ from eq.(D6.1) into the above, and solving for Y, one finds: 3 ^ (D8. 1 ) 173 (D8.2) We can compare th i s with tfj = C/*j in eq.(D6.2) and ve r i f y that g-»°9 as z>>1 and we use the asymptotic expansions of the Airy funct ions . In comparison with eq. (D6.5) , we see: - —7-7 e We therefore f i n d : (D8.3) (D8.4) as compared to eq.(D6.6) when we assumed z>>1. Using a table of Airy f u n c t i o n s , 6 1 we tabulate the function on 3<*> parameters used. in Tables V to VIII in Chapter 5, along with the various 6 1 Abramowitz and Stegun, pg.475. 174 APPENDIX E - THE SQUARE WELL IN THREE DIMENSIONS In appendices C and D, we assumed that the nucleus i s a one dimensional object. Let us extend our model into three dimensions. Consider a sph e r i c a l l y symmetric square well as shown in FIG.15: V -V4 A 8 M T K Figure 15 - The Potential Well in 3 Dimensions There are only 3 regions, and 6 constants to be used for matching from one region to another. Because of the spherical symmetry, the Hamiltonian commutes with L 2, and L,: the t o t a l angular momentum squared, and the z component of angular momentum, respectively. Separating out the angular parts of the Schrodinger equation, we are l e f t with the ra d i a l equation: ± ± f a IM!)- l(*+>)^h 2 Vn Q-V) ftp) - Q (E.1) where $ i s the angular momentum quantum number. Let %,,(r,9 , $ ) be the eigenstates of the Hamiltonian. We may write: 175 ( (9,$0)'s are the spherical harmonics. We write: £ (E.3) ^ ( 0 ) = 0 i s required in order to make the r a d i a l operator h e r m i t i a n . 6 2 We obtain: d r a ^ (E.4) We w i l l be interested only in the J? = 0 ground state: + 2W> [E~V)U. - O ; ltro) = 0 (E.5) The solutions in the 3 regions are: u+ •= ' c + y e X ) r UL $ — T C 1 ^ + K e ~ i k y Note that B=-A to s a t i s f y u(0)=0. Upon matching u and i t s derivative at the boundaries, we f i n d : (E.6) 6 2 See Messiah, pg.346. 176 + e^ ( b- a )0 +^)( c o S 2 a"l s r > , K a^" 4 e"^"^'-1'^058"-^*"^}"1 (E.7) As before, we s o l v e f o r no incoming waves ( i n t h i s case K/J = 0 ) to o b t a i n the decay r a t e and resonance energy. We f i n d : a- (E.8) where £ i s given by e x a c t l y the same expre s s i o n as befo r e . (See eg.(C17.13)). Hence the decay r a t e i s given by e x a c t l y the same » equation as b e f o r e . (See e q . ( C l 9 . 6 ) ) . That i s , i t i s p r o p o r t i o n a l to k as expected from eq.(5.2.2). The only d i f f e r e n c e between the one dimensional case and the three dimensional case (with no e l e c t r i c f i e l d ) i s t h a t K„ i s twice as l a r g e and hence V 0 i s 4 times deeper. To keep the same decay rate we need only make (b-a) a touch l a r g e r . If we now apply an e l e c t r i c f i e l d a l i g n e d along the z a x i s , the s p h e r i c a l symmetry i s broken, meaning L 2 no longer commutes with the Ham i l t o n i a n . One s t i l l has [ L 2 ,H] = 0, so one may separate out the ^ dependence of ^ . But then we are l e f t with a non-separable equation i n r and Q . I t can be made separable 177 i f we use paraboloidal coordinates: 6 3 *§=r+z, /fj=r-z. The solution may be written in terms of confluent hypergeometric functions of f and °1 . Unfortunately, the spherical boundary conditions are d i f f i c u l t to meet with confluent hypergeometric functions of parabaloidal coordinates, and the c a l c u l a t i o n was not completed. It i s expected that the effect of the e l e c t r i c f i e l d in three dimensions does not d i f f e r r a d i c a l l y from the effect in one dimension, simply because of the strong s i m i l a r i t y of the one and three dimensional cases in the free f i e l d decay. 6 3 Bethe and Salpeter, pg.1 1 3 . 178 APPENDIX F ~ TYPOGRAPHICAL ERRORS IN THE REFERENCES F l . BECKER ET AL. A) After eq(4), i t i s stated that the most probable energy transfer i s z o u ~ 1 0 6 u ^ . It should read z u j r v -vme ~ 1 0 6 V (eV). B) In eq.(5a), the t r a n s i t i o n rate i s proportional to m3. It should be proportional to m5. C) Add a right paranthesis to enclose the expression in eq.(5b). This was already pointed out by R e i s s . 6 4 D) Eq . ( 8 ) i s an expression for 2 £ a K -„ • It should a c t u a l l y read \)^r«<, w i n • E) In FIG.1 the subscripts ^ and should be interchanged. F) On pg.1265 i t i s stated that £„ = 2.29 for '*F. It should read 2.24. 6 4 Reiss, pg.652. 179 F2. DESHALIT AND FESHBACH A) B) Eq.(2.11), pg.778 reads Wl c It should read (mc^ . The l a s t term of eq.0 1.26), pg.828 reads: Eq.(l1.26) i s e s s e n t i a l l y the same as eq.(2.1.9) of t h i s thesis, the l a s t term of which reads: The difference i s that -4 should be replaced by +2 and one should insert a factor of Mp.
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Laser enhancement of nuclear beta decay
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