The Reason of Photons Angular Distribution at Electron-Positron Annihilation in a Positron-Emission Tomograph

Transcription

1 Advanes in Natural Siene ol 7, No,, pp -5 DOI: 3968/66 ISSN [PRINT] ISSN [ONLINE] wwwsanadanet wwwsanadaorg The Reason of Photons Angular Distribution at Eletron-Positron Annihilation in a Positron-Emission Tomograph Andrey N olobuev [a]* ; Egor D Moin [b] [a] Professor, Department of Medial and Biologial Physis, Samara State Medial University, Samara, Russia [b] Department of Medial and Biologial Physis, Samara State Medial University, Samara, Russia *Corresponding author Reeived 8 September ; aepted November Published online 6 Deember Abstrat It is shown that photon angular distribution of eletronpositron annihilation is onsequene of Doppler s effet in the referene frame of the eletron and positron mass enter In the referene frame bound with eletron the photon angular distribution is absent But it is replaed by the Doppler s shift of photon frequenies The reeived results an be used in wor of a positron-emission tomograph Key words: Annihilation; Eletron; Positron; Photon; Doppler s effet; Angular photon distribution; Positron- Emission Tomography (PET olobuev, A N, & Moin, E D ( The Reason of Photons Angular Distribution at Eletron-Positron Annihilation in a Positron- Emission Tomograph Advanes in Natural Siene, 7(, -5 Available from: DOI: INTRODUCTION The analysis of angular distribution of flying out photons by energy hω(h - Plan s onstant, ω - photon frequeny at annihilation of a positron e + and eletron e - has great importane for high-grade use of the positron-emission tomographs (PET PET is the advaned diagnosti devie used for searh tumors at the earliest stages of their ourrene Unfortunately the mehanism of annihilative proess e -+e +hω of the eletron and positron is unnown P Dira has been offered model of this proess Aording to Dira (Dira, 978; Heitler, 956 the annihilation it is possible to present as transformation of the eletron from a state of positive energy to the state with negative energy Aording to the Dira s theory vauum holes the positron it is the hole in the field of vauum Interation of the eletron and positron ie them annihilation is a filling vauum hole by the eletron Thus energy as two quantums of eletromagneti radiation is alloated ANGULAR AND POWER DISTRIBUTIONS OF ANNIHILATIE RADIATIONS Quantum-eletrodynamial alulations of the annihilative proess have been arried out enough for a long time They were repeatedly heed and reheed, 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 (Heitler, 956 This formula loos lie: e ( + p + p sin θ p sin θ ( π p ( p os θ ( p os θ ( The formula is written in designations (Bogolubov, & Shirov, 976 where there is its detailed dedution The so-alled rational system of units whih speed of light and Plan s onstant are equal to unit h is used In this system of the units the energy, impulse and mass have the idential dimension Copyright Canadian Researh & Development Center of Sienes and Cultures

2 The Reason of Photons Angular Distribution at Eletron- Positron Annihilation in a Positron-Emission Tomograph In the Formula ( e there is eletron harge (or positron with an opposite sign, - the photon energy, p - the eletron impulse, θ - the angle between impulses of the eletron and one of the radiated photons The 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 - (Heitler, 956; Bogolubov, & Shirov, 976 We shall note that in this referene frame the ondition of both photons supervision are idential The seond formula whih represents frequeny or power distribution of the radiated quantums has been offered a little later by Feynman (9: e ω + + ( ee dω 6π m p+ ( E+ + m ω ( The Formula ( is written down in designations (Feynman, 9 As well as in the previous variant ( the rational system of units is used In the Formula ( e and e there are unit 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 (Feynman, 9; olobuev, & Tolstonogov, 3 The main differene there are before the third and fourth addends in the square braets the signs are hanged on opposite The major distintive ondition of the Formula ( dedution is use of other referene frame in omparison with the Formula ( dedution The 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 oupled with PET Therefore we shall name this referene frame as laboratory The eletrons in the substane researhed in PET basially are in the bound state Positrons are result a β - positron radioative deay of the shortly-lived radiopharmaeutial isotopes, for example C 6, O 5 8, N 3 7, F 8 9, P 3 5 Therefore eletrons in laboratory referene frame it is possible to assume motionless (if to exlude haoti thermal movement of moleules Both Formula ( and Formula( were dedued with the help of Feynman standard diagram tehnique and diagrams of the seond order of the perturbations theory However results of the dedutions essentially differ First, the Formula ( assumes rather omplex angular distribution of intensity I of annihilative radiation, sine ~dp~idω where dp there is energy flux of radiation through the area ds, intensity dp I And this ds distribution is onneted only to the eletron impulse The angle θ is present only at the omplex with impulse p In the Formula ( the distint form photons angular distribution in the obvious ind is absent Seond, the Formula ( assumes the opportunity of photons various energy from annihilation that is forbidden by the Formula ( dedution owing to - Therefore, first of all, there is a question what nature of angular distribution of the annihilative radiation intensity in (? Whether this distribution with annihilative proess ie transformation substane energy is onneted or that is defined by other effets? Whether the given angular distribution of photons will be ept at transition to other referene frame onneted, for example, to the PET? THE REASONS OF ANGULAR AND POWER DISTRIBUTION OF THE ANNIHILATIE RADIATIONS 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 (Bogolubov & Shirov, 976: ( ( ( pe ( pe ( p( p ( p ( p ( ( ee ( pe( pe ( ee ( p ( p e, 8π p (3 where and there are impulses of photons ariables in square braets: an impulse of eletron, impulses of photons, unit vetors of photons polarization are written down as -vetors The formula (3 is simple for transforming to the ind: ( ( p ( p e + ( ee dω 8π p ( ( pe( pe ( p ( p ( Let s transit in ( to spatial vetors using a rule (aba 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 taing into aount absene ontravariane omponents at polarizing -vetors e the expression ( it is possible to present as: Copyright Canadian Researh & Development Center of Sienes and Cultures

3 Andrey N olobuev; Egor D Moin ( Advanes in Natural Siene, 7(, -5 8π 8π e p e p p ( ( ( p ( ( ( ( pe ( pe ( ( p ( pe ( pe p ( + + ( e e ( e e dω (5 At the dedution (5 the ondition of photons flying in strit opposite diretions - also is used Taing into aount, and also aording to the energy onservation law m (for learly evident it is entered inside braets the speed of light in the p Formula (5 we shall replae ( speed of eletron In result we shall reeive:, where - e ( pe ( pe + ( ee 8π p ( os θ os θ (6 Let s transit in Formula (6 to the laboratory referene frame 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 braets (p positron impulse In the given referene frame the Formula (6 beomes simpler: 8π e p os θ Our researh is an auxiliary tas ( e e dω Figure Supervision of the Photons Whih Was Emitted by the Moving Partile The observer who is taing plae in motionless (onneted with the Earth referene frame, Figure, examines some partile moving with a speed whih in ertain moment of time radiated two quantums opposite direted 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 (7 Due to Doppler s effet the quantum moving in the observer diretion will have the inreased frequeny (Landay & Lifshits, 967: ω (8 For the quantum moving in an opposite diretion so-alled the red displaement of frequeny will be observed: ω ω (9 + Using (8 and (9 we shall find size of the omplex ω + + ω ω whih is inluded into the Formula (: ω + + ω ω ( ω + ω ω ω ( Let s note that distintion of frequenies of quantums in the examined tas is determined by distintion in onditions of these quantums supervision: one quantum moves to the observer another leaves with him In the formula (7 the onsidered auxiliary tas 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: 8π 8π e ω + p ω e ω + p ω + + ( e e ( e e dω ( Let s note that at use of the formula ( we have atually refused the ondition - If in fator before braets in the Formula ( to use E + mω the formula ( and Formula ( beome idential We shall note one important point arising at transition from the referene frame of the eletron and positron mass enter to the referene frame bound for eletron or laboratory referene frame If to divide the Formula (9 on the Formula (8 the result whih differs from the result reeived in monographies For example, (Itzuson & Zuber, 98; Bjoren & Drell,978 turns out At division (9 on (8 and aepting we reeive: ω ( + In (Itzuson & Zuber, 98; Bjoren & Drell, 978 the following ratio is offered: 3 Copyright Canadian Researh & Development Center of Sienes and Cultures

4 The Reason of Photons Angular Distribution at Eletron- Positron Annihilation in a Positron-Emission Tomograph E+ p+ m (3 Taing into aount E + m and p + m we find: e ( ω 8π p ω (7 ( The Formula ( differs from the Formula ( a little It is onneted by that the Formula ( is reeived within the framewor of the first approximation of the perturbation theory Therefore it is essentially inexat The Formula ( follows from exat formulas of Doppler s effet Thus remaining only within the framewor of the first approximation of the perturbation theory it is impossible to establish equivalene of Formula ( and Formula ( In summary we shall summarize the Formula ( on photons with polarization Coming ba to polarise -vetors with the aount e also using ( e e e, e find: 8π 8π (Bogolubov & Shirov, 976, we shall e ω + p ω e p ( ω ω ω ω + ( e e (5 The module used owing to the standard use in the module of a ompound matrix element of the finding of the proess differential effetive setion (Heitler, APPLICATION OF THE ANNIHILATIE RADIATIONS IN A POSITRON-EMISSION TOMOGRAPH Taing into aount that in a positron-emission tomography the positrons speed are insignifiant, and also taing into aount Formula (8 and Formula (9 it is possible to write down: ω ω ω os θ Substituting (6 in (5, we shall find: (6 Figure The Basi Sheme of Photons Registration in the Positron-Emission Tomography Let s find the frequenies differene of radiated photons, ie size Δωω -ω, using the Formula (8 and Formula (9: ω ω ω ω + (8 If the angle θ, ie a positron moves on the line onneting detetors γ - radiation D and D, the differene of the photons frequenies will be greatest and the Formula (8 will be transformed to the ind: ω max ω Taing into aount <, we shall find: ω ω max (9 ( The size ω an be found from the approahed equality hω m In this ase: ω max ( where 3 3,8659 m m there is Compton s length of an eletron wave (Javorsy & Detlavs, 99 In Figure the basi sheme of the photons registration in the positron-emission tomography (olobuev, is shown Copyright Canadian Researh & Development Center of Sienes and Cultures

5 Andrey N olobuev; Egor D Moin ( Advanes in Natural Siene, 7(, -5 The researhed objet is plaed in the ring of detetors At the annihilation of a positron and eletron, taing plae in a point a, the two quantums energies hω and h _ ω in opposite diretions radiated (Plan s onstant h used for learing 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 a ring from the point of Doppler s effet view in the referene frame bound with eletrons play a role of motionless observers By number of the quantums whih are radiated in different diretions proess is spherial symmetri Therefore the density of detetors in a ring should be uniform However the quantum frequenies and onsequently also their energy depending on a diretion on the detetor (observer due to the Doppler s effet an be different size Δωω -ω Measuring the frequenies or energies differene of the quantums whih radiated opposite diretions also using the maximal value of this differene during measurement Δω max it is possible to find the speeds of positrons movement under the Formula ( Taing into aount that speed of positron is proportional to the density of a tissue ρ~ through whih it moves we reeive the neessary information on density of a tissue in the tumor This additional information an be reeived during diagnostis of an organism with help of the positronemission tomograph CONCLUSION By results of the arried out analysis we an draw the following onlusions Formulas Heitler ( and Feynman ( it is adequate in different referene frames desribe sattering photons of annihilation of eletron and positron 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 owing to Doppler s effet there is a distintion in frequenies of the radiated quantums To transition in the referene frame bound to the enter of mass of eletron and positron the distintion in frequenies of the radiated quantum is redued in angular distribution of photons whih also is onsequene of Doppler s effet Investigating angular distribution of eletromagneti radiation intensity at annihilation of a positron and eletron in the referene frame of their mass enter our researh not annihilation, and other physial phenomenon Doppler s effet whih aompanies with annihilation radiation Hene the first not disappearing amendment of the perturbation theory reeived on the basis of a holes Dira s hypothesis does not result in onfirmation or denying of this hypothesis even if experiments onfirm angular distribution of the annihilative radiation intensity Measuring the frequenies or energy differene of the quantum whih have flung out opposite diretions it is possible to find the speeds of positrons movement, see Formula ( Taing into aount that speed of a positron is proportional to density of substane through whih it moves it is possible to reeive the information on density of researhed substane REFERENCES Bjoren, J D, & Drell, S D (978 Relativisti quantum theory (p38 Mosow: Siene Bogolubov, N N, & Shirov, D (976 Introdution in the theory of quantum fields (pp3-5 Mosow: Siene Dira, P A M (978 Diretion in physis In H Hora & J R Shepansi (Ed New Yor, NY: John Wiley & Sons Feynman, R P (9 Quantum eletrodynamis, a leture note (pp35-37 Mosow: Boo House LIBROKOM Heitler, W (956 Quantum theory of radiation (pp3-3 Mosow: Lit Itzuson, C, & Zuber, J-B (98 Quantum field theory (p8 Mosow: World Javorsy, B M, & Detlavs, A A (99 Handboo on physis (p576 Mosow: Siene Landay, L D, & Lifshits, E M (967 Theory of field (p56 Mosow: Siene olobuev, A N ( Bases of medial and biologial physis (p636 Mosow: Samara Publishing House olobuev, A N, & Tolstonogov, A P (3 Malus law for X-ray radiation Journal of Surfae Investigation, X-Ray, Synhrotron and Neutron Tehniques, Mosow, 7(, Copyright Canadian Researh & Development Center of Sienes and Cultures