Role of the Reference Frame in Angular Photon Distribution at Electron-Positron Annihilation

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1 Journal of Modern Physis,, 5, Published Online April in SiRes. Role of the Referene Frame in Angular Photon Distribution at Eletron-Positron Annihilation Andrey N. olobuev, Eugene S. Petrov, Eugene L. Ovhinnikov Department of Physis, Samara State University, Samara, Russia Reeived February ; revised 3 Marh ; aepted April Copyright by authors and Sientifi Researh Publishing In. This work is liensed under the Creative Commons Attribution International Liense (CC BY). Abstrat The reasons of angular photon distribution ourrene at eletron-positron annihilation are onsidered. It is shown that angular photon distribution is onsequene of Doppler s effet in the referene frame of the eletron and positron mass enter. In the referene frame bound with moving eletron the angular photon distribution is absent. But it is replaed by the Doppler s shift of photons frequenies. The reeived results are applied to the analysis of a positron-emission tomograph work. Keywords Annihilation, Eletron, Positron, Photon, Doppler s Effet, Angular Photon Distribution, Positron-Emission Tomograph. Introdution The analysis of angular distribution of flying out photons at annihilation a positron and eletron has great importane for designing positron-emission tomographs (PET). PET is the advaned diagnosti devie used for searh new growth at the earliest stages of their ourrene. Unfortunately the mehanism of annihilative proess of the eletron and positron is unknown. P. Dira has been offered model of this proess. Aording to Dira [] [] for the annihilation it is possible to present as transformation of the eletron from a state with positive energy to the state with negative energy. Aording to the Dira theory vauum holes, the positron is holed in the field of vauum. Interation of the eletron and positron i.e. the annihilation is a filling vauum hole by the eletron. Thus energy as two quantums of eletromagneti radiation is alloated. How to ite this paper: olobuev, A.N., et al. () Role of the Referene Frame in Angular Photon Distribution at Eletron-Positron Annihilation. Journal of Modern Physis, 5,

2 A. N. olobuev et al.. The Angular Photon Distribution at Eletron-Positron Annihilation Quantum-eletrodynamial alulations of the annihilative proess have been arried out enough for a long time. They were repeatedly heked and reheked, inluding authors of the artile. As a result of these alulations two formulas for the differential effetive setion of eletromagneti radiation quantums sattering in a solid angle dω have been found. The first formula on time has been found by Heitler []. This formula looks like: e ( k ) + p + p sin θ p sin θ d Ω. () ( π) k p ( k ) (( ) ) p os θ k p os θ The formula is written in designations [3] where there is its detailed dedution. The so-alled rational system of units whih speed of light and Plank s onstant are equal to unit is used. In this system the units of energy, impulse and weight have the same dimension. In the formula () e there is eletron harge (or positron with an opposite sign), k the photon energy, p the eletron impulse, θ the angle between impulses of the eletron and one of the radiated photons. Formula () is found under ondition of summation on all diretions of photons polarization. At the dedution () the referene frame onneted to the enter of mass interating the eletron and positron is used whih the impulses of eletron and positron are equal on the module and are opposite on the diretion p p p. Impulses of photons also are equal on the module and opposite on the diretion k k [] [3]. We shall note that in this referene frame of the ondition of both photons supervision are idential. The seond formula has been offered a little later by Feynman []: e ω ω ω + + ) dω. () 6π m p+ ( E+ + m) ω ω The formula () is written down in designations []. As well as in the previous variant () the rational system of units is used. In the formula () e and e there are individual vetors of photons polarization radiated at annihilation, ω and ω the frequenies of the radiated photons, m the mass of eletron (or positron), p + the module of the positron impulse, E + its energy. The formula () is similar to Klein-Nishina formula for Compton effet [] [5]. The main differene there are before the third and fourth addends in the square brakets the signs are hanged on opposite. The major distintive ondition of Formula () dedution is use of other referene frame in omparison with Formula () dedution. Formula () was dedued in the referene frame in whih eletron is at rest, and positron moves. This referene frame as a whole is equivalent to the referene frame onneted with PET. Therefore we shall name this referene frame laboratory. The eletrons in the objet researhed in PET basially are in the bound state. Positrons are result β positron radioative deay of elements. Therefore eletrons in laboratory referene frame it is possible to assume motionless (if to exlude haoti thermal movement of moleules). Both formulas () and () were dedued with the help of standard diagram tehnique of Feynman and diagrams of the perturbane theory of seond order. However results of the dedutions essentially differ. First, the formula () assumes rather omplex angular distribution of annihilative photons. And this distribution is onneted only to the eletron impulse. The angle θ is present only at the omplex with impulse p. In Formula () the angular distribution of photons is absent. Seond, Formula () assumes the opportunity of photons various energy at annihilation that is forbidden by Formula () dedution owing to k k. Therefore, first of all, there is a question what nature of angular distribution of the annihilative photons in ()? Whether this distribution with annihilative proess i.e. transformation substane-energy is onneted or that is defined by other effets? Whether the given angular distribution of photons will be kept at transition to other referene frame, for example, bound with PET? 35

3 A. N. olobuev et al. 3. The Reason of Photons Angular Distribution For researh of the angular dependene reason of differential effetive setion () we shall onsider intermediate expression of the dedution whih is not summarized yet on diretions of the photons polarization [3]: ( ) ( ) ( ) ( ) ( ) ( )( )( ) e k k pe pe k ee pe pe ) dω, (3) 8π pk ( pk)( pk) ( pk ) ( ) ( pk)( pk) pk where k and k there are impulses of photons. ariables in square brakets: an impulse of eletron, impulses of photons, unit vetors of photons polarization are written down as -vetors. Formula (3) is simple for transforming to the kind: ( ) k e ( pe)( pe) ( k ) + ) dω. () 8π pk ( pk)( pk) ( pk)( pk) Let s transit in () to spatial vetors using a rule ( ab) a b ab where a and b there are three-dimensional vetors whih omponents hange ovariane, a and b ontravariane hanging omponents of - vetors, in our ase power omponents. Transiting to three-dimensional vetors, and also taking into aount absene ontravariane omponents at polarizing -vetors e the expression () it is possible to present as: ( k e ) ( )( ) pe pe ( k ) + ) dω 8π pk ( k ) ( pk) ( k ) ( pk) e ( pe)( pe) + ) d Ω. 8π pk ( k ) pk pk ( k ) ( k ) At the dedution (5) the ondition of photons flying in strit opposite diretions k k also is used. Taking into aount k k, and also aording to the energy onservation law сk m (for learly pk evident it is entered inside brakets speed of light с ) in the formula (5) we shall replae osθ, ( k ) where speed of eletron. In result we shall reeive: e ( )( ) pe pe + ) dω. (6) 8π pk osθ osθ ( k ) Let s transit in (6) to the laboratory referene frame offered in [], and bound with eletron. In this ase p, and it is possible to examine as speed of a positron movement. The same there onerns and to value p in fator before brakets. In the given referene frame the formula (6) beomes simpler: e ) dω. (7) 8π pk osθ (5) 355

4 A. N. olobuev et al. We researh an auxiliary task. The observer who are taking plae in motionless (onneted with the Earth) referene frame, Figure, examines some partile moving with a speed whih in ertain moment of time radiates two opposite direted quantums. At the quantum frequeny is ω. The angle between speed of the partile and diretion of one quantum propagation is equal θ. In the observer diretion the partile has a omponent of speed osθ. Due to Doppler s effet the quantum moving in the observer diretion will have the inreased frequeny [6]: ω ω. (8) osθ For the quantum moving in an opposite diretion so-alled the red displaement of frequeny will be observed: ω ω. (9) + osθ ω ω Using (8) and (9) we shall find size of the omplex + + whih is inluded into the formula (): ω ω ( ω + ω ) ω ω ω ω ωω + + osθ. () Let s note that distintion in frequenies of quantums in an examined task is determined by distintion in onditions of these quantums supervision: one quantum moves to the observer another leaves from him. In Formula (7) the onsidered auxiliary task is atually realized. Thus the moving partile is meant as a positron, and the observer is on motionless eletron. Therefore substituting () in (7) we shall find: ω ω + + Ω e ) d 8π pk ω ω e ω ω + + ) dω 8π pk ω ω Let s note that at use of Formula () we have atually refused the ondition k k. If in fator before brakets in Formula () to use E+ m ω Formulas () and () beome idential. In summary we shall summarize the formula () on photons polarization. Coming bak to polarizing -vetors with the aount e also using ) [3], we shall find: e, e () ω θ osθ ω Figure. Supervision of the photons whih be emitted by the moving partile. 356

5 A. N. olobuev et al. e ω ω + + Ω ) d 8π pk ω ω ( ω ω ) e d Ω 8π pk ωω The sign on the module is written owing to standard use of the ompound matrix element module at finding of the differential effetive setion of proess []. Taking into aount that in the positron-emission tomograph the speeds of positrons are small, and also taking into aount (8) and (9) it is possible to write down: Substituting (3) in () we shall reeive: () ωω ω ω. (3) os θ ( ω) e d Ω, () 8π pk ω where it is designated ω ω ω. On Figure the basi sheme of photons registration in the positron-emission tomograph [7] is shown. The researhed objet is loated in the ring of detetors. At the annihilation of the positron and eletron whih is in a point a the two quantums with energies ω and ω in opposite diretions flying out. If the quantums flying on line A-A are registered by detetors D and D simultaneously the point of quantums emission is in the middle between detetors D and D. Detetors in the ring from the point of Doppler s effet view in the referene frame bound with eletron have a role of motionless observers. By the number of the quantums whih are flying in different diretions the proess it is spherial-symmetrially. Therefore the density of detetors in the ring should be uniform. However the quantum frequenies and onsequently also their energy depending on the detetor diretion (observer) due to Doppler s effet an differ on size ω ω ω.. Conlusions By results of the arried out analysis we an draw the following onlusions. Formulas Heitler () and Feynman () it is adequate in different referene frames to desribe sattering photons at annihilation of eletron and positron. A ω D ω D a θ Figure. The basi sheme of photons registration in the positron-emission tomography. 357

6 A. N. olobuev et al. In the laboratory referene frame bound with eletron the angular distribution of number photons is absent however due to distintion in onditions of quantums supervision there is a distintion in frequenies of the radiated quantums (owing to Doppler effet). At transition in the referene frame bound to the enter of mass of eletron and positron the distintion in frequenies of the radiated quantums is redued in angular distribution of photons whih is also onsequene of Doppler s effet. In the laboratory referene frame bound to the positron-emission tomographs radiation of the annihilative quantums by number is spherial-symmetrially however it is neessary to take into aount some distintion of the opposite radiated photon frequenies. Referenes [] Dira, P.A.M. (978) Diretion in Physis. Edited by H. Hora and J. R. Shepanski, John Wiley & Sons, New York. [] Heitler, W. (956) Quantum Theory of Radiation. In. Lit., Mosow, 3-3. [3] Bogolubov, N.N. and Shirkov, D.. (976) Introdution in the Theory of Quantum Fields. Siene, Mosow, 3-5. [] Feynman, R.P. (9) Quantum Eletrodynamis, A Leture Note. Book House LIBROKOM, Mosow, [5] olobuev, A.N., Petrov, E.S. and Ovhinnikov, E.L. () Journal of Modern Physis, 3, [6] Landay, L.D. and Lifshits, E.M. (967) Theory of Field. Siene, Mosow, 56. [7] olobuev, A.N. () Bases of Medial and Biologial Physis. Samara Publishing House, Samara,