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All Graduate Theses and Dissertations Graduate Studies
5-1973
A Relativistic One Pion Exchange Model of Proton-Neutron Electron-Positron Pair Production
William A. Peterson Utah State University
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ii
ACKNOWLEDGMENTS
The author sincerely appreciates the advice and encouragement of Dr. Jack E. Chatelain who suggested this problem and guided its progress.
The author would also like to extend his deepest gratitude to
Dr. V. G. Lind and Dr. Ackele y Miller for their support during the years of graduate school.
William A. Pete r son iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS • ii
LIST OF TABLES • v
LIST OF FIGURES. vi
ABSTRACT. ix
Chapter
I. INTRODUCTION. • ...... • •
II. FORMULATION OF THE PROBLEM. 7
Green's Function Treatment of the Dirac Equation. • • • . • . . • • • 7
Application of Feynman Graph Rules to Pair Production in Neutron- Proton Collisions 11
III. DIFFERENTIAL CROSS SECTION. 29
Symmetric Coplanar Case 29 Frequency Distributions 44
BIBLIOGRAPHY. 72
APPENDIXES •• 75
Appendix A. Notation and Definitions 76
Appendix B. Expressi on for Cross Section. 79
Appendix C. Trace Theorems and "Y Identities 84 iv
TABLE OF CONTENTS (Continued)
Page
Appendix D. Phase Space Distributions for Neutron- Proton Electron- P ositron Pair Production . . • . • • 86
Appendix E. Computer Program PHASE . 111
VITA. 136 v
LIST OF TABLES
Figure Page
I. Glossary of symbols ...... •.•.•.. 1 2 vi
LIST OF FIGURES
Figure Page
l. Typical Feynman diagrams for nucleon-nucleon electron-positron pair production. • • • 5
2. Vertices for (a) spinor electrodynamics, (b) elec trodynamics of spin-zero boson, and (c) meson- nucleon scattering. • • • • • • • • • • • • • 15
3. Feynman graphs for process considered in this paper ••••••.••••• 17
4. Symmetric al geometry for cross section 30
5. Theorectical curves of differential scattering cross section for the symmetric geometry of Figure 4 • 34
6. Angular distribution of final proton in laboratory system with the directi~'S of incident proton with matrix for weights< 10 . • . • • • • • • . . 48
7. Angular distribution of electron in laboratory with the direction §f the incident proton with matrix for weights < I o- ...... 50
B. Angular distribution in final proton with the direc tion of the incident proton in center of momentum system with matrix for weights < I o-B...... 52
9. Angular distribution of electron with the direction of the incident proton in center of momentum sys- tem with matrix for weights < 1 o-B . . . . . 54
1 o. Angular distribution between pair in lg-bot·atory system with matrix for weights < 1 o- . . . . . 56
11. Angular distribution between pair in center of _ 8 momentum system with matrix for weights < 1 0 • 58 2 2 12. A = (p + p ) distribution of gamma ray with matrix for w;ights < 1 o-8 • . . . . • . . . . . • . . 60 vii
LIST OF FIGURES (Continued)
Figure Page 2 2 13. '-"A =-(nf- n . ) d1stn• •b ut10n. o f ;rO w1t• h matr1x• f or -8 1 < 10 •..•.•.•••••••••••• 62
14. Angular distribution between final proton-neutron in the laboratory system with matrix for weights < 1 o-8 ••••••••..•••.••••• 64
15. Angular distribution between final proton-neutron in center of momentum system with matrix for weights< l o-8 . • • • • • • • • • • • • • • • 66
16. Momentum distribution of electron in laboratory system with matrix for weights < 1 o-8 68
17. Momentum distribution of electron in center of_ 8 momentum system with matrix for weights < 10 • 70
18. Angular distribution of final proton in laboratory with the direction of incident proton without matrix. 87
19. Angular distribution of electron in laboratory with the direction of the incident proton without matrix • 89
20. Angular distribution of final proton with the direc tion of the incident proton in center of momentum system without matrix • • • • • • • • • • • •
21. Angular distribution of electron with the direction of the incident proton in center of momentum sys tem without matrix. • • . . • . . • • • • . 93
22. Angular distribution between pair in laboratory system without matrix • . • , . 95
23. Angular distribution betwe en pair in center of momentum system without matrix •. .• 97 2 2 24. !::,. = (p_+ p+) distribution of gamma ray without matrix ...... 99 viii
LIST OF FIGURES (Continued)
Figure Page
25. '-"A2 = - ( nf- ni )2 dtstn" "b uhon. o f .,.0 wtt• h out matrtx• l 01
26. Angular distribution between final proton-neutron in the laboratory system without matrix . . • . l 03
27. Angular distribution between final proton-neutron in center of momentum system without matrix l 05
28. Momentum distribution of electron in laboratory system without matrix • . . • . . . . • . • l 07
29. Moment um distribution of electron in center of momentum system without matrix • . . . . . l 09 ix
ABSTRACT
A Relativistic One Pion Exchange Model of
Proton- Neutron Electron- Positron
Pair Production
by
William A. Peterson, Doctor of Philosophy
Utah State University, 1973
Major Professor: Dr. Jack E. Chatelain Department: Physics
Proton-neutron electron-positron pair production c ross sections are calculated in the framework of the ps e udoscalar one- pion exchange model in a fully relativistic manner,
A computer program has been developed to evaluate invariants and Dirac traces for a given data point.
The c ross sections for symmetric coplanar events for labora- tory kinetic energies of l 0 to 250 MeV were calculated for pair angles
Frequency distributions were also calculated, at a laboratory energy of 200 MeV, using a random number generator to select data points. The frequency distributions are illustrated by curves.
It was noted that the inclusion of heavier bosons will not significantly improve the results at laboratory energies less than
200 MeV. ( 145 pages ) CHAPTER I
INTRODUCTION
The early investigations of scattering and absorption of gamma
rays showed that the interaction with heavy elements exceeded that
predicted by the Compton and photoelectric effect alon e . In 1932
Anderson observed the ejection of electron-positron pairs from lead
foil subjected to c osmic ray irradiation; this experiment provided
evidence for the Oppenheimer and Plesset (1933) interpetration of pair production on the basis of Dirac 1s electron theory.
The first studies of the problem of electron- positron pair pro
duction were theoretical. Some of the first papers were those of Furry
and Carlson (1933) and of Heitler and Nordheim (Heitler, 1960). In these papers a cross section was calculated by the use of the Born
Molter perturbation theory for relativistic particles . They found that for cosmic-ray energies the production of pairs is as important fo r
electrons as it is for gamma-rays . Another early calculation was
done by Bhabha (193 5 ). The Bhabha calcu lation uses the Weizsacker
Williams approximation to calculate the cross- sec tion for pair pro du ction by an e lectron scattering off a fixed electromagnetic field (the trident process). In the Weizsacker-Williams method the incident
particle is placed in a rest frame by the Lorentz transformation. 2
The scattering p roduced by the nucleus is treated as if its field of virtual photons resemble a flux of photons distributed over a frequency spectrum. Racah (1937) also did a calculation using the 1 93 0 version of pe rturbation theory. His r esults did not include exchange of iden- tical particles in the final state.
Experimental evidence for pair production by an electron scattering off a fixed electromagnetic field has been consid e red by a n umber of authors and a summary of some of these attempts has been discussed by Johnson (1965 ).
In 1 935 Yukawa first proposed the meson theory of nuclear fo r c es .
Since then, a large number of theoretical as well as experimental in- vestigations have b een carried out to i n crease ou r k n owled ge a bout the nature of the nucleon- nucleon interaction.
One way to study the basic problem is to investigate nucleon- nucleon e lectron-positron p a ir production. Although this process is more complicated than elastic nucleon- nucleon scattering , it y ields
/ more information than can be obtained from the study of the e lastic case. In addition i t i s worthwhile to study the process of pair pro- duction partly for its own sake , i.e. , as inte r esting phenomena in itself which must be compared with our theoretical knowledge of ele- mentary processes, and partly b ecause it can be present aa part of the "background" when other processes are studi ed. Only a thoro u gh understanding will permit us to eliminate this background. Also high- energy electron-positron pairs probe the small-distance b e havior of 3 quantum electrodynamics (Bjorken and Drell, 1 959 ). In particular, it offers the possibility of exploring the photon propagator at small distances .
Petiau (1951) and Havas (1 952) studied the c r eation of pairs by collision of particles . They treated the interactions in te r ms of virtual exchange particles with mass and estab lished the matrix ele ments for the process to lowest order in perturbation theory. Their papers have been reviewed thoroughly by Parker (1968). Parker calculated a differential cross-section fo r proton- proton electron positron pair production. He considered only electromagnetic inter actions or photon exchange. Wheeler (1968) made a calculation for neutron - proton electron -posit ron pau production u sing on! y neutral one pion exchange between nucleons . This excluded the possibility of the effect of charge exchange.
In the problem of this thesis, namel y proton-neutron electron positron pair production, the approach taken is that the interaction occurs through one-pion exchange between nucleons. We are aware that the interaction cannot be solely due to one-pion exchange.
Exchange of virtual bosons (such as P,w, fJ., etc.) between two nucleons should be considered as well. S in ce the pion is the lightest strongly interacting boson, we expect it to be the main c ontributor to nucleon nucleon interaction. Furthermore, the nucleons are treated as structureless Dirac particles. Form factors were not int r oduced to 4 take into account the fact that the pion is off the mass shell. An other
reason for u sing the one-pion e x change mode l and for not using form factors i s to keep a complicated calculation reasonably simple; and for
e n e rgies less than a couple hundred MeV, we do n ot exp ect an appre
ciable error for a first cal c ulation.
In this thesis, however, we consider one-pion exchange as a
first approximation of nucleon-nucleon interaction and neglect other
contributions . There is reason to believe that b e cause the electro- magnetic c oupling is w eak the calculation may normally b e t reated in lowest order of perturbation theory. The expansion coefficient due to the combined effects of strong and electromagnetic coupling is approxi mately equal to 0 . 01. (Figure l)
One interesting aspect of this approach is that a virtual charged pion can emit a real electron-positron pair (See F i gure l b) , a process which can not be explained in the potential model (Simon, 1950).
In our cal c ulation, because the potential for the nucleon- pion w ill be pseudoscalar and all calculations are done in a fully relativistic manner, it would be a laborious task to square the matrix e lement represented by Figure 1 and to work out by hand the traces r esulting from the spin summations for unpolarized particles . Hearn (1 967) has developed a c omputer code for such problems. This approach was not used because the final answer will contain an exceedin gly large number of terms. The difficulty was avoided by eliminating scalar 5
e N N y~ + lT ,._
N N N N
(a) (b)
Figure l. Typical Feynman diagrams for nucleon-nucleon electron positron pair production. Solid lines are nucleons, dashed lines me sons, and wavy lines are photons .
products of Dirac gamma matrices by hand , (Chisholm, 1963). This r educed the matr ix to terms involving scalar products of momentum four-vectors and traces of products of gamma matrices.
In Chapter U, a brief discussion of the solution of the wave equation by the Green's function method i s presented. Feynman graph rules are applied to proton-neutron electron-position pair production and the matrix element is displayed. The spin sum averages are then taken of the absolute value square of the matrix elem ent. This reduces the matrix to products of traces involving scalar produc t s of gamma matrices. An example of eliminating scalar products of gamma n1atrices by hand is then demonstrated. 6
Because of the tedious task of constructing the differential cross section due to the lengthy algebra involved in working out traces the problem was programmed to the computer by the author.
In Chapter III, a specific case is first considered in which data points are generated by computer for different energies. These points are substituted into a computer subroutine for the differential cross section and the results are displayed by curves.
An att.empt was then made to machine integrate the independent variables by using a random number generator to search for data points.
Each data point was given a phase s pace weight which was then changed to include the dynamic weight of the invariant matrix element. Those weights whose data points fall within a certain specified region of a distribution variable are then summed. Distribution curves are dis- played for the neutral theory (non- charge exchange) of the process.
The program used to calculate the traces was written by the a uthor. The phase space subroutines from the computer code PHASE was written at the University of California at Berkeley, California, by Alex Firestone. Du e to the complicated nature of the calculation all aspects of the programs were checked by hand calculations and compared with the value found by the computer. 7
CHAPTER II
FORMULATION OF THE PROBLEM
When calculating scattering cress sections the equation to be solved is the t ime dependent wave equation that desc ribe s the develop- ment of the system with time.
We begin vA th a brief discussion o f the relation of the wave e quation to its solution. The Green's function treatment will be used and the scattering rratrix or S matrix will be demonstrated. A more rigorous and complete discussion can be found in Schweber (1961).
Finally the Feynman graph rules will be applied to electr on- positron pair production in a neutron-proton collision t o c alcula t e a scattering matrix for the differential cross section.
Green ' s Function Treatment of
t he Dirac Equation
The Dirac equation for a particle of mass m in an external field Jl. = A ·l' with x -= {;:, t), fJ.
(iV- m-e*(x) )\)J (x) = 0 ( 1) describe s the c hange in.. the spinor wave fun ction ljl with timet, Ifljl (1) is the wave fun ction at r i at time t , the wave function at time t > t 1 2 1 8 can always be written as
~ 2 for r , t . 2 2 It can readily been shown that, in general, G can be defined by the solution of i(\1 -m- e *{2)) G(2, 1) =6(2, 1) (3) 2 where the subscript 2 on-'9'z means that the operator acts on the variables 2 of G (2, 1 ). To derive an expression for G we write (3) in the following form: (i'9z-mlG(2,l) = 6(2,1) + eA-(2)G(2,l). (4) the total Green's function G may be written as an integral equation by noting from (3) that G (Z, 1), the free particle Green's func tion, 0 satisfies (5) 9 Therefore we can write 4 G(2,l)=G (2,l)+efG (2,3)*(3)G(3,l)d x. (6) 0 0 3 A Neumann series solution for G is 4 G(2,l)=G (2,l)+ef: (2,3)*(3)G (3,l)d x o J .... o o 3 + .. . (7) where the integral is extended over all space and time . G(2, l) is called the total amplitude for arrival at ...r , t , 2 2 startmg. f rom ~r , t • The trans it ion amplitude for finding a particle 1 1 3~ 3., s = rhJX(2)G(2,l)tjl(l)d+ r d r . (8) 21 J 1 2 This defines the S matrix when We can understand the result from Equation (7) this way. Imagine that a particle travels as a free particle from point to point but is scattered by the potential A; thus the total amplitude for arrival at 2 from 1 can be considered as the sum of the amplitudes for various 10 alternative routes. It may go directly from 1 to 2 (amplitude G (2,1), 0 giving zero order term in (7)), or it may go from 1 to 3 (amplitude G 0 (3, 1 )~. become scattered there by the potential and then go from 3 to 2 (amplitude G (2, 3)). This may o ccur for any point 3 so that summing 0 over these alternatives gives the first order term in (7). Again, the particle may be scattered twice by the potential. It goes from I to 3 (G (3,1)), becomes scattered there then proceeds 0 to some other point, 4, in space time (amplitude G (4,3)); it is then 0 scattered and proceeds to 2 (G (2, 4)). Summing over all possible 0 places and times for 3, 4 gives the second order contribution to the t otal amplitude. One can in this way write down any of the terms of the expansion (7). The Fourier amplitude of G in momentum space is given by 0 where p is the momentum and m is the rest mass of the scattered particle. G , as defined here, is known as the Feynman propagator. 0 The total transition rate, W, from 1 to all 2 is computed by using Fermi's Golden Rule Number Two 2 P is the phase space available for final states of the system. fs 2J ll in most practical problems is calculated to the lowest non-vanishing order in perturbation theory from the multiple scattering series in equation (7). The terms in the scattering series can be represented by space-time Feynman graphs. This method will now be applied to the problem in this thesis. Application of Feynman Graph Rules to Pair Production in Neutron- Proton Collisions We will conside r the creation of an electron-position pair in a neutron-proton collision. Feynman graph rules (Feynman, l949a, 1949b, 19 50; Dyson, 1949) will be used to calculate the invariant matrix ele- ment H, where His related to the differential corss section by ( 1 0) The derivation and notation of Equation ( 1 0) is pre sen ted in Appendix A and B. For definition of the symbols u!!ed in the calcula- tion see Table l . 12 Table l. Glossary of symbols four-momentum of initial proton four-momentum of final proton n. four-momentum of initial neutron t four-momentum of final neutron P_ four-momentum of electron four-momentum of positron m mass of electron M mass of proton and neutron U(p)= the Dirac four-component spinor for a particle of momentum p V(p)= the Dirac four-component spinor fo r the antiparticle of momentum p e absolute charge of an electron fLO mass of neutral meson mass of c harged meson The invariant matrix element is computed by drawing all Feynman graphs for the process in question. The Feynman approach allows matrix elements for a given interaction to be formulated from products of appropriate factors , each of which corres ponds to a parti- cular segment of the Feynman graph. The interaction probability can be constructed mathematically by simply linking appropriate terms for each step of the process in accordance with certain rules. It 13 then becomes an easy matter to obtain a formal expression for the cross section. Those factors needed for the problem in this thesis, using the formalism of Bjorken and Drell (1964), are as follows: (a) For e ach external fermion line entering the graph a factor U(p) or V(p) denotes whether the line is in the initial or final state; likewise, for each fermion line leaving the graph there is a factor V(p) or U(p). A Dirac spinor for a particle, characterized by a physical four-momentum p, is denoted by U(p) while the antiparticle is denoted by V(p). U(p) and V(p) are the adjoint spinors of the particle and antiparticle, respectively. (See Appendix A) (b) For each internal fermion line with momentum p and rest mass m, the factor i(j!=l-m) 2 2 p-m is used, where~ is the inner product of a Dirac r. matrix with a four-vector momentum p , i.e., Y pu='fl'l fl. (c) For each internal meson line of spin zero with momentum q and rest mass fl. , the factor i -2--2 q -fl. is used ; 14 (d) For each internal photon line with momentum k, the associated factor is The inte rac tions o! the process determine the terms for the vertices of the Feynman graph. The three interac tions assumed in this thesis give the following te rms : Spinor Electrodynamics . This ki nd of vertex is s hown in Figure 2(a). The rule i s a factor -iel' at each vertex. fl. Electrodynamics of a Spin--zero Boson. The vertex is denoted in Figure 2(b) with the rule that a factor -ie (p+p')fl., where p and p' are the momenta included in the charged line . :Pseudoscalar Meson-Nucleon Scattering. From charge independent theory of nucleon-nucleon interaction, the vertex in F igure 2(c) has a factor g·\ a t each meson-nucleon vertex with relative coupling strengths of-vz g for charged mesons, +g for neutral to protons, and -g for neutral mesons to n e utrons. In order to write down the matrix element, we do have to pay attention to the order in which the individual proces ses occur along the fermion line (which may not necessarily be the sequence in time). Due to time reversal in variance of the equation of motion, the end of the graph from which one starts is a matter of convenience. It has, however, become customary to s tart with the final state and c ontinue down a long the fermion lines. 15 {a) \ \ P' \ \ {b) -ie(p+p')~ 17 / /p / / / { c ) Figure 2. Vertices for {a) spinor electrodynamics , {b) electrodynarr.i c s of spin- zero boson, and {c) meson-nucleon scattcrin;~. The solid, broken, and wavy lines depict the spinor, meson, and photon respectively. 16 The lowest order Feynman graphs for pair production by neutron-proton scattering are given in Figure 3. Utilizing the pre- scription given above the matrix element given by each graph is as follows: U(p )1'-V(p )U(pf)Y [~ + M]Y U(p.) l ( 11) - + f.L a 5 1 ( (1 2 ) U(p )Yf.LV(p )U(nf)Y [~ + M]Y U(p.) ( ( 13) - + 5 C f.L I ( (14) 17 n. n. l l (a) (b) n. n. l l (d) (c) n . l (e) Figure 3. Feynman graphs for process considered in this paper. 18 (15) From conservation of energy-momentum we have (16) (1 7) (18) (1 9 ) and (20) ( 21) (22) (23) The numerators of the matrix elements, Equa tions (11) through (15). can be simplified by using the Dirac operator 19 ~- m) U(p) = 0, its adjoint U(p)(-t-m) = 0, and For example Equation (11) has the factor U(pf)'Y [1>" + M]'Y5U(p.) fJ. a 1 = U(pf)'Y 'Y ~ -M]U(p.) 5 fJ. a 1 = U(pf)'Y5'Y (-p,+~>, - ....f- M] U(p. ) fJ. 1 1 l where Equation (11) can now be written as 20 Similarly for Equations (12) through (15), H = BU(nf)'Y U(n.)U(p )r""V(p )U(p£)1'_~ 1' U(p. ) (25) -0 5 1 - + ~ 1 .... 1 H = C U(pf)1' U(n.)U(p )r""V(p )U(nf)1' ..P. 1' U(p. ) (26 ) c 5 1 - + 5 2 .... 1 H = D U(nf)1' U(p. )U(p )r""V(p )U(pf)"Y .P, 1' U(n.) ( 27) d 5 1 - + fl. 3 5 1 denoting and 21 The total matrix element His (29) The absolute square is 2 +H H++H H Hd++H H+ c a cbH++IHI c + c ce +HH+HH+HH+HHd+H.+ + + + I 12 (30) ea eb ec e e Consider the square of Equation (24) 22 (31) Equation (31) has factors of the form 0 0 where Q = 'Y Q'Y , and in particular 23 therefor e, (32) In general, one does not obse r ve the polarization of the final part icles, and one does not know the initial polarization. In absence of such informa tion, one assigns equal~ priori probabilities to the different initial polarization states. This means that t h e actual cross section observed will be a sum over final spin states and an average over initial states. The spin sums can be reduced to traces if we use energy projection operators and closure: 1 \ IH 12- --fJ 2 T r 'Y ( -- n i +- M) 'Y ~nf--- + M } 4 sptnsf._. a - 4 5 2M 2M 2.4 v ( i'-_ + Tr -yli ---~ -m }1' --m) ( 2.m 2.m Tr1'f.L~ -m)1'vli'- +m)Tr1'..P. (-p,.-M)P; 1' (i>:;f+M). (33) + - .... 1 1 1 v Using the trace theore ms o f A ppen d ix C , it can be shown that the spin s ums of non - diagonal terms in E quation (30) are equal, i. e. , L H H+= LH H+, spms a b sptn b a for example. This leaves fifteen independent terms in (30). Re- clueing these terms to traces (3 0) can be written as 2 5 2 + 1Bj Trf + 2 BC + Tr(-1>..- M)~f+ M ).P. 'Y H>. + M)'Y .P. (.,...f- M) 1 1 P· 1 v 2 li 2Tr'Y (~-m}*\f>-_+m) +------~~~------2 4 256 m M 26 + + DE Tr(.p,-- M)~f+ M)Tr(-t>:f+ M)'Y .P.. (-tt:-M) 1 fl. 3 1 (34) Again utilizing the theorems in Appendix C, the scalar products of 'Y matrices can be eliminated. As an exan~ple, consider the first term in Equation (34), 2 IAITr("'"i -M)~f+M) .t:xpanding the above and using theorems 7, 9 and 10 of Appendix C , a) 2 fl. v - M Tr'Y .p. 'Y ~ Tr'Y..P,l..P, 'Y b) + - fl. 1 v 27 2 _j.J, v -m Tr T 'Y Tr"Y .t'i~1 1' v~ c) 2 2 Con sidering a) and applying theorems 5, 6 , and 12 gives Tr'Y~-yf'-1>- Tr'Y -p,f'Y -Pc "t>":P: - + v fl. 1 1 1 Applying the needed theorems to b), c), and d) yields 2 2 64 m M ( P · P ) 1 1 The scalar products were summed without use of the com pu ter for the rest of the terms in Equation (34) in a similar manner. ThesP terms were left in the form of scalar products of fou r-vectors and traces of products of matraces. The algebraic expression for Lj HjZ resulted in an exceedingly complicated array of terms. In order to 28 interpret our results, computer subroutines were written to evaluate scalar products of four- vectors and traces of products of 'Y matrices 2 for a given data point. This allowed the evaluation of L IHl . 29 CHAPTER Ill DIFFERENTIAL CROSS SECTION The differential cross section for the scattering of particles with pair production was given on page 11 by equation (1 0). Adopting it we first consider the scattering of a neutron initially at rest (n~, (;) l to the final state (n~, tf) by a proton initially in the state (p~, ;i) to a o-+ final state (pf, pf). The collision causes the creation of an electron- a+ o + positron pair in the final state describe by (p_,p_) and (p+,p+) respectively. Symmetric Coplanar Case We shall consider a specific case in the symmetrical geometry shown in Figure 4 and graph the differential c ross section versus initial proton energies for 9=10° to 60° inclusive. In order to determine how the energy and momentum in the final state were distributed the following set of equations were solved. (1) 0 0 0 pi = M = 2p f + 2p_ (2) (p.)o2/~/2 = p. + M 2 (3) l l 30 ... p . Figure 4. Symmetrical geometry for cross section. (4) o 2 I? z 2 (p_) = p_ 1 +m (5) Equations (l) and (2) are the conservation of mom e ntum and energy. The energy and momentum of the neutron and proton in the final state are assumed to be equal, and the energy and momentum of the e l ectron and positron are also equal. A Newton- Raphson (Moursund and Duris, 1967) iteration procedure was used to solve the above set of equations, (1) through (5), for a given initial energy. The equations were solved 31 by the following computer program. SUBROUTINE ITER REAL M1,M2 COMMON/S67 / ALPHA, THETA, EBU, PBO, EB1, PB1, El, Pl, R Fl (X, Y)= E*X+B* Y-C F2(X, Y)=SQR T(X**2+M1 **2)+SQR T(Y**2+M2**2)- P F2X(X)=X /SQR T(X**2+M1 **2) F2Y(Y)=Y /SQR T(Y**2*M2**2) D(X, Y)= E *Y /SQR T(Y**2+M2**2)-B*X /SQR T(X**2+M1 **2) Ml=939. M2=. 511 E=2. *COS(O. 01745 3*ALPHA ) B=2. *COS(O. 01745 3*THETA) ET=EB0+939. 0 C=SQR T(EB0**2-Ml **2) P=ET/2. X=3 . *C/(4. *E) Y=C/(4. *B) WRITE ( 6, 25) DO 50 K=1,15 T=F2(X, Y)*B-F1(X, Y )*F2Y(Y) G=X+T/D(X, Y) T=Fl (X , Y) *F2X(X )-F2(X, Y) *E H=Y+T/D(X, Y ) WRITE(6, 5) K,G, H IF(ABS(X-G). LT .. 00 01. AND. ABS(Y-H). LT •• 0001) GO TO 60 X=G Y=H 50 CONTINUE 6 0 EB1 =SQR T(X**Z+Ml **2) F1 =SQR T(Y**2+M2**2) PBO=C PB1=X P1=Y WRITE(6, 15)C, EBl, El 5 FORMA T(l H 15 , 2F20. 1 0) 1 5 FORMA T(/1 H 5HPBO =F12. 4, 4X5HEBI =Fl2, 4, 4X, 4HEI =F12. 4) 25 FORMAT(/23X,3HPBl,l8X,2HPI) RETURN END 32 Ml mass of proton and neutron M2 mass of electron EBO initial energy of proton EBlo final energy of neutron and proton El final energy of electron and positron PBl final momentum of neutron and proton Pl final momentum of electron and positron PBO = initial momentum of proton ALPHA= 5° THETA= 9 An initial guess for the final momenta was to set PBO PBl 3/4 C Oli 5° and l PBO Pl 14 cos e This avoided unphysical negative solutions in the iteration procedure. The above computer subroutine was linked to a second program to evaluate the differential cross section for the process. This consists of subroutines, as mentioned before, for calculating inner products of four-vectors and traces of four-vectors contracted into gamma matrices . The subroutines were tested by performing a calculation by hand for a data point and compared with the value found by the computer. Other 33 aspects of the programs were also checked by hand calculations, to test each subroutine. An example of a particular set of out put is shown below. EBO PBO EB1 PBl E1 P1 THETA 990 313.65 950.37 146.57 14. 13 14.12 --;wr- 2 10 SIGMA(!)=. 2309XlO-l SIGMA(2)= .4943Xl0- SIGMA(!) and SIGMA(2) is the differential c ross section for noncharge exchange and charge exchange theory respectively. The differential cross section versus energy is shown in Figure 5. Each figure includes two curves to illustrate the effect of charge exchange and non charge exchange theori es. At low energies and e = 10° the ex 5 change theory is about 10 larger than the neutral theory, but the difference bec omes progressively smaller for larger e, approaching 2 10 for e = 60°. The peak in the exchange theory appears to shift toward threshold (941 MeV) for large e. Extrapolation seems to indicate that the cross section for the two theories approach one another at large energies and both tend to zero. Coplanar proton-proton electron-positron pair production differ- entia! corss section has been calculated by Parker (1968) for laboratory kinetic energies ranging for 1 0-l OOMeV. Parker 1s corresponding -25 -31 2 2 3 differential eros s section decreases from 10 -10 em /MeV-Sr. Comparing Parker's calculation, our calculation gives values of the -33 -35 2 2 3 order of 10 -10 em /MeV -Sr at lD-100 MeV, respectively. Figure 5. Theorectical curves of differential scattering cross section for the symmetric geometry of Figure 4. The .sytnbols have the following meaning: N.C. Noncharge EXchange (neutral theory) C. E. Charge Exchange (symmetric theory) 35 lo' (a) e- 10• :.. _10 C.E. xlO .,....."'" .. 10.. .,"Ql .J• > Ql ~.. ..,"...... " ...... "0 ..u .," "0 .."u 10 ...." "...... •u .-< ...... "' •.. ...• ~ proton.... energy in MeV Fiaure 5. 36 (b) e .. 2o• "'., ..,...." ~·., "'II) ..J > _ll .....::<:" xlO ~ ...... ," .,u ....., ...... _13 .,..." N.C.xlO "'.... "" Incident proton energy in MeV I 1000 II I'D ,, ... -Figure 5.cont1nued 37 to' (c) e - 30• _12 "' C.E. xlO .,.." .. ttl- .,..QJ ..J > ~ --..c: ..,..., .," .,.,"0 0 ..QJ !1) ..0 10° 0 00 .,.." QJ .,.. •0 ..-< .....,.. Incident proton energy in MeV ..0 .... ,.. 11\!10 taso ...... Fi&ure 5.- continued ·- 38 10.. (d) e -4o• "' ...... " "'...... Ql .J >Ql ...... ;.:: _11 ., C.E lll.O .." .D ...." ...... 0" .,0 .," "'0 10 ..u "" ...."...... " ..u "' .-<...... ,..." Ql ...... 1.£ . "' Incident proton energy in MeV '"" <100 •a.o Figure- 5. continued 3 9 (e) a ·- so• _13 xlO "'".. '" "'..., .... .J ~ .....:>::.. .." .0.. '"" "0 '".. ...,0 .. 0 .. 10' 0 ~ ...... , ...... 0 ...... '".. "., ...~ ...... "' Incident proton -..rgy ill MaV .,,,. 1100 Figure 5. continued 40 ,(/!. (t) e .. 60" _13 C.E. xlO "' Incident proton eneri!Y in MeV '""' """ ...... Figure 5. continued 41 We are not aware of any experimental data at the present time with which we can dir ectly compare our results, but it would seem reasonable to assume that the cross section for nucleon-nucleon electron-positron pair production to be no larger than the fine structure c onstant times t hat of bremsstrahlung. Therefore, if one uses the experimental mea- s urements for the differential c ross section of coplanar events observed in nucleon-nucheon bremsstrahlung, which is a lower order process in comparison to nucleon-nucleon pair production, we have values giv e n 30 2 2 b y Baier et al. (1969 ) of t he order of lo- cm lsr for proton-proton -29 2 2 bremsstrahlung and • 5xl 0 em Is r for proton-neutron bremsstrahlung corresponding t o laboratory kinetic energies 10-100 MeV and 200 MeV, respectively. Thus, using the experimental values given by Baier e t al., one may expect that the cross section f or proton-proton and proton- - 32 2 2 neutron pair production to be n o larger than of the order of 1 0 em I 11r 31 2 2 and l0- cm /sr for kinetic energies cilO~l OO and 200 MeV, respectively. Parker ' s values appear to be too large when compared to bremsstrahlung. 31 2 2 From F igure 5 our results does not exceed the upper limit of l0- cm l:sr predicted by bremsstrahlung at 200 MeV kinetic energy. It is expect~d that Parker's values may not be correct since he c onsidered only elec- tromagnetic interaction between nucleons which of course is not the dominant interaction in this energy region. The pi-meson is believed to be the major contributor to nuclear force at large distances , although heavier boson s such as o-, tl, E , P, and w may also play a rol e for small impac t parameter collisions with 42 large moment= transfer. To obtain an estimate of the contribution of heavier bosons one might consider the Feynman propagator for a particle of spin zero and mass f.l in the scattering amplitude of two nucleons 2 H = ----=2-"--2"""" (6) q - f.l In writing (6) we have suppressed all factors corning from the vertices at which the particle is absorbed or emitted by the two nucleons. The 2 I 2 I 2 invariant momentum transfer is q = (p - p ) = (p - p ) , where the 1 1 2 2 I I two nucleons have initial and final momenta p , p and p , p respectively. 1 2 1 2 In the nonrelativistic limit in which the recoil kinetic energies of the nucleons are neglected relative to their rest energies, q 2cl- 1~12q and we may approximate ( 6) to 2 (7) H ~---~2.._"""""2 ""r-q + f.l Fourier - transforming to coordinate space, we see that (7) corresponds to the scattering in a Yukawa potential v (8) r If we consider Vrr . 03. Vrr 13 Where we have taken r l 0- em. as a characteristic distance for strong interactions. The other bosons such as f.L, £:, P, and ware estimated to con- tribute about one percent each. From this type of calculation one may expect the contribution of the heavier bosons to the interaction potential to be less than ten percent of the pion contribution. To arrive at an estimate of the energies at which one may expect to approxi mate Equation (6) by (7) w e w rite (6), for maximum momentum transfer, as (9) Where T, is the kinetic energy of the incident nucleon in the laboratory system and M is the mass of the nucleon. Equation (9) shows that q 2--+ 1.,...12q for T <- 200 MeV. 1 We would like also to mention briefly the possible effects of pionic form factors. Ueda ( 1966) investigated proton-proton bremsstrahlung by using a relativistic field theory approach. He adopted the one-pion- exchange model and considered the cases where pionic form factors 44 were neglected and where they were taken into account. Ueda used the phenomenological expression of Amaldi and Selleri (1964) for the pionic form factor. His results showed a damping effect of approximately 30 percent for an incident laboratory energy of 200 MeV. If we used Ueda's approach, the same rather large damping effect could be expected in our cross section. Frequency Distributions In order to find distributions for the eros s section the computer code PHASE was adopted for use. PHASE is composed of two separate computer subroutines. The first is a search program which uses random numbers to generate data points. The final particles are allowed to take on any configuration with the constraint that energy and momentum are conserved across the collision. A statistical phase space weight was then calculated for each data point. The second program added those weights that fell within a given rang e of a distribution variable allowing the other variables to take on any value consistant with the conservation laws. The versatility of PHASE is that the four vector momentum for eac h final particle is generated and a weight is calculated for each generated particle configuration. This allows one to insert his own weighting modification. In our problem the modification was the in variant matrix element. This follows from 45 where w is the frequency of interaction and P is the phase space factor. In principle one would have to generate an infinite number of data points, but since this is impossible a rather small finite number of ten thousand data points were generated. This would be consistant with the number one might observe in a experimental situation. For the ten thousand points generated it was found that the matrix varied 8 by a factor of 1 o In order to smooth out the graphs for the distribu 8 tions, weights greater than 10- were not included. This excluded 3 23 data points from the total generated. From the study of scatter plots it was determined that this would not greatly bias the results. The following Figures 6-17 are only for the noncharge exchange part of the problem. Charge exchange has not been included. Further more, no attempt has been made to give an absolute value to the vertical scale; the values represented are relative. The incident kinetic energy of the initial proton in the laboratory is 200 MeV. Figure 6 shows the angular distribution of the final proton in the laboratory system. The distribution is strongly peaked in the forward direction. Figure 7 shows the angular distribution of the e l ectron in the laboratory system peaked in the forward direction about the incident proton. The angular distribution of the proton in the center of momentum depicted by Figure 8 shows a maximum around 90° while, the angular 46 distribution of the electron in the center of momentum system Figure 9, appears isotropic. The angular distribution between the pair in the laboratory, Figure 10, and the center of momentUIIJ, Figure ll, shows a peak about zero degrees. 2 The t. distribution of the exchange gamma ray for the pair, 2 Figure l 2, has a maxima near zero, while, the t. for the pion, 2 Figure 13, has a maxima around .125 (BeV) . Figure 14 gives a maxima at about an angle of 90° for the angular distribution between the final proton and neutron in the laboratory and in the center of momentum, Figure 15, the maxima is at an angle of 180°. Momentum distributions for the electron are given by Figures 16 and 1 7. For comparison phase space distributions have been included in Appendix D. For a test the above method was applied to the elastic scattering of two electrons (M61ler scattering). The results agreed very well with previous calculations (Gasiorowicz, 1966). Because of the extreme variation of the matrix, the conclusion is that any computer evaluation of c omplicated matrix elements for finding distributions, one can not ignore physical insights about possible singularities. In particular, a program designed to evaluate any (lowest order) process is feasible, but if it is to be useful it will probably have 47 to include some means of locating and treating carefully the near zeros of denominators. Because of the greater amount of computer time needed only ten thousand data points were generated. Future work should increase the number of data points by one or two orders of magnitude. Figure 6. Angular distribution of final proton in laboratory system with the dire.:_t~on of incident proton with matrix for weights < 1 0 • 49 -l;lO .)05 .o - ~ ....• ••••• Cox a Figure 7. Angular distribution of electron in laboratory with the direction of the incident proton with matrix for weights -8 < 10 • 51 . C)i .O.S+ ·- .o... >- u a "' .OJ "cr ..."' ~ .oaf. ·OIIJ .1&10 ~·-·-· Cos e t- Figure 8 . Angular distribution in final proton with the direction of the incident proton in center of momentum system with matrix for weights < 1 o-S 53 .... .•u .oJo .oaf' >- u .,~ ::l .,r:r ..~.. [>," · 01.& .0/0 ,...., Cos e • Figure 9. Angular distribution of electron with the direction of the incident proton in center of momentum system with matrix for weights< 1 o-·s 55 · ·~ .o+e ...... . •.So >- u .,,:: ::1 .,0" ...... ~. . ~ . Dill . ot~ . ~ .....J7.f ·1&1'0 Cos e Figure l 0. Angular distribution betwee~Bair in laboratory system with matrix for weights < l 0 . 57 ..uo Cos a Figure 11. Angular distribution between pair in center of mamentum system with matrix for weights < 1 o-8 59 . 10 .og ... .OT .... >- u ... ""' " ·" .ol 2 2 Figure 12. 1:1 = (p + p ) distribution of gamma ray with matrix for - + weights < l ~. 61 .l1 , il .It .. ~ >- v " "0' "... .11 ~ .09 ... ~ .o3 'I .OCt., 2 2 0 8 Figure 13. 6. = -{nf-ni) distribution of rr with matrix for weights < 10- . 63 ... ... .n .IUD ·- 2 2 A (BeV) Figure 14. Angular distribution between final proton-neutron in the laboratory system with matrix for weights< 1 o-8 . 65 ... >- .,u " .OS .,"'0' r...... ... Cos e Figure 15. Angular distribution between final proton-neutron in center of momentum system with matrix for weights < 1 a--8. 67 .w >- 0 . ~ "::> C1'"' "'... "" .2. .~ . ..• · ·~ _,.ii7S . liUO Cos e Figure 16. Momentmn distribution of electron in laboratory system with matrix for weights < l o-8• 69 Momentum in MeV Figure 17. Momentum distribution of electron in center of momentum system with matrix for weights< 1 O_s. 71 . I~ .Ol Momentll!Il in MeV 72 BIBLICXJRAPHY Amaldi, U., and F . Selleri. 1964. "Unitary Condition and Connections between Pion- Pion and Pion-Nucleon Scattering," Nuovo Cimento 31:360-376. Baier, R., H. Kuknelt and P. Urban. 1969. "A Relativistic and Gauge Invariant Model of Nucleon-Nucleon Bremastrahlung," Nuclear Physics 811:675-691. Bhabha, H. 193 5. "The Creation of Electron Pairs by Fast Charged Particles." Proc. Roy. Soc. (London) Al52:559-586. Bjorken, J. D. and S. D. Drell. 1959. "Tridents, Pairs and Quantum Electrodynamics at Small Distances," Phys. Rev. 114:1368-1374. Bjorken, J. D. and S. D. Drell. 1964. Relativistic Quantum Mechanics. New York: McGraw-Hill Book Company. 300 pp. Caianiello, E. R. and S. Fubini. 1952. "On the Algorithm of Dirac Spurs," U Nuovo Cimento IX(l2):1219-l226. Chisholm, J. S. R. 1963. "Relativistic Scalar Products of 1' Matrices," Il Nuovo Cimento XXX(1 ):4984-4986. Dyson, F. J. 1949. "The S Matrix in Quantum Electrodynamics," Phys. Rev. 75:1736-1755. Feynrnan, R. P. l949a. "The Theory of Positrons," Phys. Rev. 76:749-759. Feynman, R. P. l949b. " Space Time Approach to Quantum Electrodyna mics," Phys. Rev. 76:769-789. Feynman, R. P. 1950. "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction," Phys. Rev. 80:440-457. 73 Furry, W. H. and J. E. Carlson. 1933. "On the Production of Positive Electrons by Electrons," Phys Rev. 44:237-238. Gasiorowicz, S. 1966. Elementary Particle Physics. New York: John Wiley and Sons. 613 pp. Havas, P. 1952. "Sur La Creation de Faires de Corpuscules Dans Les Processus de Collision Entre Corpuscules de Spin-f," Le Journal de Physique et le Radium Tame 13:438-440. Hearn, A. C. 1967. "Reduce User's Manual, Subroutine for Dirac Gamma Matrix Algebra," Institute of Theoretical Physics, Stanford University Report ITP-120 (unpublished). Heitler, W. 1960. The Quantum Theory of Radiation. London England: Oxford University Press. 430 pp. Johnson, E. G. 1965. "Differential Cross Section for Trident Production," Phys. Rev. 140 (4B):Bl005-Bl008. Moursund, D. G. and C. S. Duris. 1967. Elementary Theory and Application of Numerical Analysis. New York; McGraw Hill Book Company. 297 pp. Oppenheimer, J. R. and M. S. Plesset. 1933. "On the Production of the Positive Electron," Phys. Rev. 44:53-55. Parker, B. R. 1968. "Theoretical Calculation of the Cross Section for Pair Production in a Proton- Proton Collision," Dissertation, Utah State University. Petiau, G. 1951. "Sur La Creation de Faires de Corpuscules Dans Lee Processus de Collision Entre Corpuscules de Spin+," Le Journal de Physique et le Radium Tame 13:911-919. Racah, G. 1937. "Sulla Nascita de Coppie per Urti di Particelle Elettrizzate, " Nuovo Cimento 14:93-1 01. Schweber, S. S. 1961, An Introduction to Relativistic Quantum Field Theory. Elmsford, New York: Row, Peterson and Company. 905 pp. Simon, A. 19 SO. "Bremsstrahlung in High Energy Nucleon- Nucleon Collisions," Phys. Rev. 79:573-576. 74 Ueda, Y. 1966. " Proton-Proton Bremsstrahlung," Phys. Rev. 45: l 214-1228. Wheeler, R. D. 1968. "Differential Cross Section for Electron Positron Pair Production from Neutron Proton Scattering, " Dissertation, Utah State University. 75 APPENDIXES 76 Appendix A Notation and Definitions In our units c and il are set equal to one. With thi:s particular c hoice of units, time appears to have the dimension of a length; energies, momenta, and masses the dimension of an inverse length; e lectric 2 2 e 1 c harges appear as dimensionless quantities (e = ~<;::) ~ ). Momentum four-vectors are defined (1) and the inner product is (2) ~ Three vectors are written as p . The inner product of a Dirac 'Y matrix with a momentum four- vector is denoted by (3) The 'Y matrices are defined through the relation (4) 77 with the Hermitian condition 10 += 10' ::; += - y . (5) Equation (5 ) may be written in the c ondensed form (6) with 1. 2 0 2 (1' ) = - 1 , (1' ) =1 i = 1, 2, 3. (7) gfl.v is a metric tensor used for the lowering and raising of Spitmrs U and V satisfy the Dirac equation H>-- m) U(p) = 0 (8) (-t>-+ m) V(p) = 0. (9) In terms of the adjoint apinora 0 U = U + 1' , where U + is the Hermitian conjugate, the Dirac e quation becomes U(p) ~~m) = 0 (1 0) V(p) H>- + m ) = 0. ( 11) Summing over spin states lead to energy projection operators: 78 ~ for positive energy states 2m and ..:£::- m for negative energy states. 2m 79 Appendix B Expression for Cross Section Cross sections can be divi ded into two parts, the invariant amplitude H in which the physics lies, and the phase space and kine- matical factors. In terms of H the expression for the differential cross section d 3...,... ~ d pf __m_2_M_4_-=-~ I /2 d n f d » d P_ __3_o_ (21T) p_ Whe r e i s a scalar invariant and ~l, ~ are the collinear velocities of the 2 two initial particles, If the following identity i s u s e d d3:p 4 fl. 2 0 2 d p 6(pf'.p - m ) S (p ). ( 2) E J 80 Wher e 0 0 ~ l for p > 0 e(p l = o 0 for p < 0 is the unit step function. The c ross section becom es , when equation (2) is substituted fo r the final neutron, d 3~ d P_ 0 P_ 0 + nf 2 2 d 3+ d3~ d p_ p+ ( 3 ) 0 0 P_ p+ By energy momentum conservation 81 (4) 3- ,~1 0 0 and writing d p = p p dp dn for a final particle to emerge into a given solid angle dn about an angle 8, we have dcr=f[ f' _llil,'tt,l .J;J ·IP.I (p . . n . )2-M (21T) l l (5) If the argument of the delta function, equation (4) is set equal to y, we have differentiating with respect to p; (6) Substituting the following relation into Equation (6) 82 0 p - b (7) we have (8) Therefore by taking the reciprocal of (8) and using the relation The cross section becomes 83 (9) 84 Appendix C Trace Theorems and 'Y Identities The following trace theorems and properties of Dirac matrices are derived from the commutation algebra of the 'Y's, -yfL-yV + -yV-yfJ.= 2gfJ.V' and are valid independently of the choice of representation for the 'Y ' s. l. 5. Tr-AB€ = Tr B:A B = Tr ft-6--A = Tr -b-fh'l 6 . -yf'. . ~-y = 4A· B fJ. 7. Tr h-B· = 4A· B 8. Tr -f1'r +-B-) = Tr-A + Tr -B- 9. Tr 1 = 4 10. Tr (odd number of slashed terms)= 0 11. Tr A B 6 D = 4 ((A· B)(C· D)- (A· C)(B· D)+ (A· D)(B· C)] 85 12. Tr ('Y S)Tr'Yfi. S' = 2 Tr (S + R)S' fl. Where R is S written in reverse and Sand S 1 are "odd strings" or strings containing an odd number of matrices. If S = 'Y 'Y . .. 'Y n then R = 'Yn .•. 'Y 'Y l. 1 2 z Where S and R are defined in l 2 above 14. 'Y SA~= 2 f-*S + R.,o,.j fl. Theorems 12 and 14 are derived by Chisholm (1963) and Caianiello and Fubini (1952) r espectively. 86 Appendix D Phase Space Distributions for N e utron-Proton Elec tron-Positron Pair Production Figure 18 . Angular distribution of final proton in laboratory with the direction of incident proton without matrix. 88 > g .I ""' "'... ~ .oa. Cos e F i gure 1 9. Angular distribution of electron in laboratory with th e direction of the incident proton without matrix. 90 -1.000 .I&JO .ens •1.000 Cos e Figure 20. Angular distribution of final proton w ith the direction of the incident proton in c enter of momentum s y stem with out matrix. 92 Cos e Figure 21. Angular distribution of electron with the direction of the incident proton in center of momentum system without matrix. 94 .... -.•. ,.. .l&fo ... .,. Cos·e Figure 22. Angular distribution between pair in laboratory system without matrix. 96 , O)U> . 1&10 ·011. >- .,() ":::1, .,0" ... . OliO rx. .0171 .,.ooo -.+•1• .r&lo ... ,. Cos e Figure 23. Angular distribution between pair in cente r of momentum s ystem without matrix. 98 >- .oao u ":;1 "c:r ".. ~ .o•• .o 1& Cos e 2 2 Figure 24. D. = (p + p ) distribution of gamma ray without matrix. - + 100 . ItO . 108 . 09 .oooo . 0011 ...... 2 6. {BeV) 2 2 0 Figure 25. ~ = -(nf- ni) distribution of rr without matrix. 102 . . 06 >- . or. .,u "::> 0'., ... [,.,. ·0+ . o:J Figure 26. Angular distribution between final proton-neutron in the laboratory system without matrix. l 04 .060 .IIH .o+& ·OK . OJ .oo ·- .,... Cos e Figure 27. Angular distribution between final prot on-neutron in c enter of momentum system without matrix. ' 106 Cos e Figure 28. Momentmn distribution of electron in laboratory system without matrix. 108 . 10 ..• .., ·06 ... t2.f.ooo ·- Momentum in· MeV Figure 29. Momentum distribution of electron in center of momentum system without matrix. 11 0 . 0~ .OJO MOmentum in MeV lll Appendix E Computer Program PHASE C 011 MO N PL AC [ ( 2 O(J l C0Mf'10N/COUNT/ '101>"102•11103 COMMON/ PH A<; PC AI GH T, WG, W5, EM, P .T • T MAX • fC M, NP, NU M,NOT RY • P CM 0 IM [N 5 I ON £ M( 1 0 l , P ( 1 0 • 3 l , T I 1 0 l • T MAX I 10 l • K NUM ( 50 l • KP l ACf (50 l • U I 'iC l 0 IM EN '51 ON 8 I SOl , 11 I SOl , H T L I 50, 2 0 l, ST Lll 0, 20 l , K A l I 5(1 l • PC M I 1 n l OIMEN5ION BIN l 'i[J .tOOl , f1 JN')O l50 •100l •WGI SOl •W '>l 101 OIMfN'iiON KSCAT I 101 .NJIIOl•NYllOl •XUl1Dl•YUllOl•XlllOl•Yl 1101 0 IM ENS ION <;CAT( l U• ;>Q,2(ll .wT<;UMH l'iOl, A5UMH 1501, I/T5UMSI I Ol, ASUMSI JOl 0 H1 EN<; tON L HI 50 l , l A I SO l •II A 15 0 l • LB I 50 l • WB I S 0 l • l C I 50 l • WC I 50 l NOI ::cO N02=0 N03=0 ZIP=t.OilO• •llJ C THIS PPOGRAM U5ES A RANDOM NUMBER ANO A WEIGHTING PROCESS TO C GENERATEEVENTS . C CUT<: TO BE MA [£ . T H( RESULTS CAN BE ADDEO USING WEIGHTS SUPPLI[f) BY C BY DATA CARDS. THE RE5ULTS ARE THEN HISTOGRAHH[O OR PLOTTED IN CHANNELS C Of 1•2•5 BIN WIDTHS. THE INFORMATION IS READ INTO PHASE USING DATA C C AR OS 0 f TH E FO L L 0 WIN G f 0 R MATS. C .. •••fYENT INFORMATION E lQ.Q, 6IIO C EOhC£NTER Of MASS [N(RGY. KNORM•FINAL NORMALIZATION. C KHIST•NUHSER N' HISTOGRAM£<;. KAO•NUHBER OF SUPfRIMPOSINGS. C K S• NUMBER OF R.. OTS. NEVT•NUMRER Of EVENTS GENERATED. C .... •MASS IN FORMAT ION 8(\0.D C EMIJl=HASSES CF OUTGOINGI'~CLBS •J=l,NP C .. •••HlSTOGI'IAH INFIRMATION lllOo3ElO.O C KNUM•NUMBER OF HIST. KPLACE•PLACEIKPLAC[l IN HJST KNUH. C K ..._ •PLACE IKPL ACEl +PLACE IKPLACf+l l ·ECT •• KAL C B• LOWER LIHIT. U• UPPER LIMIT. O• RIN WIDTH C .... •H TL •• TITLE 1 QA R C•••••SuPERIMPOSING INFORMATION•• 2ItO.E10.0.IlD·fiO.O.IlO•E1fJ.O C LH• NUf'l'lER Of FIRST HJST. LA·LB.LC• HIST TO FIE ADOFO WITH WTS WA.wR,WC. C .... •PLOTS INFORMATION 3ll0•4E10.Q C KSCAT• NUHBEP OF PLOT. NX• NY• NUMBER OF BINS IN X AND Y COOROINATES. C XL•YL~ LOWER LlMIT'S . XU• YU• UPPER liMIT<; C .... •<;TL .. TlTLE lOAB N REAOISoiOOOl ECI' oNPoKNORMoKHI SToKAOoKSoNfVT R [A 01 ~, 11'11 3 l If: M I J l , J:: 1o NP l WRI Tf 16• 8701 [CMoNPoKNORMo KHI<;T .KAO otiS• NEVT WRITE lf;o87ll IEI'IJl,J::l,NPl A NORM::~ NO RM I Fl I(HT'>Tl9'lo 74,71 7 I 0 0 1 I:: 1 • KH I<; T R EA 01 o • I 00 I l K I'() M I I l ,I( PLACE I I l • K AL I I l , R I I l • U I I l , 01 ll R [A 01 !', I 04 2 l I HT l I I • J l • J:: 1• 20 l WPlTEi f; o8SRl lr.'JUM IIIoKPLACEIII•KALI IloSIIloUIIloOII WRT If I r; • R 7 6 l I HT l I I • J l • J:: I, ?0 l CONTINUE 70 IFIKA0l99.72o73 73 DO 2 I::l,KAO R [A 01 S • 1 00 2 ll H( I l, l A I ll , WA I ll • l B I I l , WB I I l , l C I I l • WCl I I WRITEI6o8771 LHIIloLAIIloWAIIloLBIIloWBIIloLCIIloWCIII ? CONTINUE 72 IFIKSiq9,74o75 75 00 76 I::loKS R (A 01 'i ol 014 IK SCAT II l, NX II l , NY I I l, XL II I , YL I I l , XU I I l, YU I I l R [A 01 5 • 1 04 211 <;T L I I, J I • J:: l• 20 l WR!TEIF.o8781 KSCATIIloNXIIIoNYIIloXLIIloYLIIloXUilloYUIIl WRITE I f. • 8 7 6 II ST L I I • J l • J:: lo 20 I 76 CONTINUE 7 4 E CM:: S QR T I 2. •E )o!l 1 l • E C'4 l NUM ::0 Ntot::O WT'iUM::O.O E M'iUM ::0 .0 00 7 I::1oNP EMSUM::F:MSUM+E"I ll 7 CONTINUE DO 8 I::loNP T MAX I II ::I fC M• •2 +EM I ll ., 2- IE MSU M-E 111 J l I" 2 l/ I 2. O•E fM 1- fl' II l 8 CONTINUE DO ea I:: 1• ~0 IITSU'1HI II ::o . O ASU,..H!Il ::O . O 00 q 6 J=1· 100 BINfi,Jl::O.O BINSOfi,Jl::Q.O q6 CONTINUE 88 CONTINUE DO 8'3 r = 1· 10 WTSUM S I II ::O.D A SU "'<; II l :: 0. D 00 97 J = l· 70 DO Q8 K:: l• 20 SCAT( I,J,Kl :: 0.0 98 CONTI "U E '!7 CONTI NUE 8'l CONTI NUE 100 NUM::N U... +l NOT RY ::0 DO 14 I:: 1 • 20 0 P LA CE I I l :: 0 . 0 14 CONTINUE CALL EVENT IITSUM ::W TSUM+W(H T 00 25 r::t.so WGl I l :: 11 GH T 2~ CONTI NUE DO 26 I :: 1• 1D WS!Il ::1 .0 26 CONTI NUE CALl c;us IF! NUl'- 10 l 9 . q , 8 7 q IIRI T[ (1; , lfJ03 l DO l 1 I:: 1• 25 J::I•S-8 - -"" 00 1? K=1•A l ::J +K IFIPLACEIU 113•12•13 1~ N::J+1 N Ill= J+ 8 WRITE 16•100'11 N, IPL~CE IMI,"'=N,NNI G 0 TO 11 1? CONTINUE 11 CONTINUE 87 IFIKHTSTI'J'J,69d 0 10 DO 33 I=1•KHIST KX=KALI II DC 500 J=1•KX KH=KPLACE II I+J-1 IF( PLACE I KH II 39 • qq d'l 3q IFIPLACEIKHI-Ul II l~'ld4.4lt 3'1 I Fl PLACE I KH 1- !ll II 144.35• 35 35 N=l PL AC:El KH 1- Bl III/DC II +1. IF1Nilt4•44•38 36 IFIN-100136•36•44 31i BIN II ,N I= BINI I• Nl +WGI II WT'i UH HI I I = WT S lJol H ( I I +W G I I I A SU HH II I= AS UH HI I I + 1 • 0 B IN SQ I I , N I= B I 1'1> Q I I , Ill I +W G I I I • •? q q CONTI NlJE 500 CONTINUE 3 3 CONTI IIIli E 69 IFIKSIF;f;,!;!;,52 52 DO 53 I=1·KS J=NXI II IF l PL ACE I Jl I 54 • 5 J, 5'1 54 I Fl PL ACEI Jl -XlJI II I S~.o;s, 53 55 IFlPLAC£1JI-XUI1153•56•56 56 M X= 2 0 • • l P LA CE IJ I- Xl II II /l lC lJ l I I - XL li I I + I • IF! HX 153,53,59 S'l IFIMX-20157o57o53 C,7 J::NYI II IF! 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AI "'I loNI l 19 5 ol qs.t q11 I 'l5 A N'i 0= 0. 0 GO TO 197 J GO TO 105 00 lO'l If( N?-50 1 I 10 • 16• 16 110 [Nl:o[NI+l.O EN2=EN?+l.O Nl=Nl+l N7=N2+l GO TO Ill 10 3 WPI Tf I f>olOHd [Nl=l.O EN2=2f>.O EN3=5l.O N l= l N2=26 N3=5l ll'i EIH:oB III+ IEN\ -\ .01• 0 1 II E11;>:oB II I+IEN2-1.0hDI II E111 = B II I+ IE N3-l.Ol•DI II A Nl =A NO RI1•B IN II, Nl l IW TS LIMH I I I A N2 =A NORM •B IN II , N2 l IW TS U MH I I l AN3=4NORihBIN IIoN>l IWT <;UMH III A NS Ill = S llR T I AI N SIll I , N 1 ll I BIN I I , N 1 I A NS ll2 -:: S GR T( BI N SIll I , N ;>I l I B IN II , N 7 I ANSG3 = SGRTifliNSGIIoN3ll/BINIIoN31 WRITE (6o1037l t: llloE112o EI'U WRITEifiol03BI .ANloANSGloAN2oANSQ;>, AN3oANSQ3 If IUH-EI1 31 ll2oll?ol13 11? 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IH1120H TOTAL CM ENERGY = oFI2.4/20H NO OF PARTICLES = 1ol2/l 1001 FOR,.,AH 107H MASS T MAX E KIN 1 PCM PX PY PZI 1002 FORMAT! 2H o7F1S.lll 100 3 FORHAT!1H /17H NO OF TRIALS= oi31 1 Ofl 4 F OR 1'1 A T( 1 H I 1 1 H Wf I GH T = • fl 2 • 4 I '1'1 RETUR"' ENO SUB POUT INE SUB CO "!M 0111 A I 20 0 I C OMHON/ PH ASPC IW GHT, WG oWS oEMo p, T, T MAX, ECHo NPo NUMoNOT RY, PCM OIMENSION WGISUio£MilClloPI10dloTI10ioTMAXI10ioPCHIIOI OIME"'SION PIL llOloW<;IlOI REAL M lo H 2 111=939. H 2= • 51 I Altl=o. o A 121=0.0 A 13 l= SQ RT I I EC W 2. I • • 2-E MI 1 I .. 2 I A 1'1 l= FC M/ 2. AISI=O.O Al61=n.o Al71=-4131 AIBI=AI~I J= DO 20 J=t.N c D IS SC ALAR PRODUfT OF P AND C. D =C ( 4 I• " ( 4 • J I -( ( 1 I • P ( I • J I - C I 2 I • P I ? • J I -C ( 3 I • P I 3 • J I c FORH CH ENERGY. E =o /W c A 15 CON'iTANT FOR T'HNSFORM J NG MOMENTA A :::1 PI q, Jl +E II l c S TORE ENERGY. c NOW TR ANSFORM MOMENT A. DO lD 1::1 , 3 I 0 Q I I • J 1-:: PI I, J I -A • C I I I c LO OP S ACK F OR N(XT VECTOR . 20 CONTINUE c DON E . RETURN END FUNCTION OTIIoJI C OH 1'00 N AI ? 00 I I fl I. NE .J I GO T 0 3 D T:: AI II • • 2+ AI I+ 1 l • • 2 + A! I+? l • • 2 R ElWIN 3 DT::: A!Il•A!Jl+AII+ll•A!.J+ll+A!I+ 2 1• AIJ+21 RETUR N ~ FUNCTION O OT II • J l C OH 110 N A I 2 00 l IFII.NE.Jl G O T O 3 00 T:::AI I+ 31 .. ?- AI 1 1 .. 2- AI 1+11 .. 2- H I+? l ••2 RETURN 00 T:: AI I+ 31 • A U + 3 1- AI II • A I J l - A I I + II • A I.J • ll- A I I + 2 1 • A IJ + ?' I RETUR N ...£!10 FU NC Tl ON T R I lo J, K, Ll C OH HO N AI ? 00 I TR ::: q .• ID OT I I ,J I • CO Tl K •l I - OO T I I , K l• CO Tl J , L I +00 T IT • l l • 00 T C J , K I I !IE TUR"J EN D FUNCTION TR6fiJ .Y2.Y 3 .!4oi5oi£;1 COMMON A I 2 00 I T R£; ::Q (IT I I I • I 2 h T R I I 3 • I 4 • I 5 • I 6 1- 0 0 l I I l • I 3 I • T Rl I ? , I 4, I S , l 6 I +0 0 T IT I , 1 I 4 I• TR I I 7. • I3 oi 5 • I 6 1- 00 l I I I • I 5 I • T R I I 2 • I 3 , I 4, I 6 I +DOT li I , J 6 I • l R I I ;> , J 2 3ol4.YSI RETUPN EN Q.._ FUN CT I 0 N TR 8 I Il, I 7 , I 3 ol 4 , I 5, I 6 .I 7 ol 8 I C 0'1 MO 'l A I 7 00 I T Rfl =0 OT I I 1, I 2 I• TR6! I3, I 4 , I 5, I 6, I7, I 8 I -0 OT II 1 • I 3 I •lR 6 I I ? • 14, I S , I 6 1 • I 7, I 8 I • 00 T I I I • I q I • T R 6 I I 2 , I 3 • IS • 16 • I 1 • I 8 I- 00 T I I I • I 5 I • T R6 I I 2 • I 3 • 2 . 14, I 6, I 7 .r 8 I +0 0 T I I I • J 6 I •lR6 I I 2 .r 3 • 14, I 5 • I7, I 8 I-DOT II 1 , I7 I • l R6 l I 3 2oi3oi4oiSoi6oiBI+OOTII!oi81•TR6II :? oi3oi4oiSoi6oi71 R ETUR!\1 [NO FUNCTION RAND 10 I DATA R/782451/ RANO::RG[NIRJ R £TURN £NO FUNCTION PGENIIARGI C NUMBER GENERA Too PROGRAM OR FUNCTION DATA IX/0/ IF! URG.£0. IX I GO TO 3 IX::JARG I Y= I X 3 IY=IY•Z62147 I Fl I V. L T • 01 I v= I Y + J 4 3 5 9 7 J8 36 7 + I R G£ N-:: I Y R G£ N= RG EN •O. 2910 38 Jf- 1 0 RETURN E NO 136 VITA William A. Peterson Candidate for the Degree of Dr. of Philosophy Dissertation: A Relativistic One Pion Exchange Model of Proton Neutron Electron- Positron Pair Production Major Field: Physics Biographical Information: Personal Data: Born at Idaho Falls, Idaho, December 1 2, 1 929, son of Will A. and Jnella Peterson; married Emma L . Johnson February 14, 1969- divorced; one child- Patric i a . Education: Attended elementary school near Rire, Idaho; graduated from Ririe High School in 1948; received the Bachelor of Science degree from Utah State University with a major in physics and a minor in mathematics in 1965; did graduate work in physics at Utah State University, 1965-72; com pleted requirements for the Doctor of Philosophy degree , specializing in Theoretical Physics, at Utah State Univer sity in l 973. Professional Experience: 1965-67, graduate teaching assistant for the Department of Physics, Utah State University; 1967-70 National Science Foundation Fellowship through the Department of Physics Utah State University; 1971- 72 Research Scientist with Academic Research at Weber State College Ogden, Utah.