The formulae for the cross section of bremsstrahlung by relativistic electrons and positrons taking into account the contribution of the second Born approximation are obtained. The dependence of the radiation cross section in the field of atomic plane on the sign of charge of the particle is considered

Shul’ga, N.F.

Syshchenko, V.V.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

THE SECOND BORN APPROXIMATION IN THEORY OF BREMSSTRAHLUNG OF RELATIVISTIC ELECTRONS AND POSITRONS IN CRYSTALN.F. Shul’ga1, V.V. Syshchenko21National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukrainee-mail: shulga@kipt.kharkov.ua 2Belgorod State University, Belgorod, Russian Federatione-mail: syshch@bsu.edu.ru The formulae for the cross section of bremsstrahlung by relativistic electrons and positrons taking into account the contribution of the second Born approximation are obtained. The dependence of the radiation cross section in the field of atomic plane on the sign of charge of the particle is considered.PACS: 12.20.-m, 41.60.-mIn this paper we consider the second Born correction to  the  process  of  bremsstrahlung  of  high  energy electrons and positrons in an external field. The account of the second Born approximation leads to dependence of  the  radiation  cross  section  on  the  charge  sign  of radiating particle. It is demonstrated that contribution of the second Born approximation can be substantial for the case of coherent interaction of radiating particle with atoms of a crystal. 1. DIFFERENTIAL CROSS SECTION OF THE RADIATION PROCESSThe cross section of the bremsstrahlung of electrons and positrons in an external field is determined by the relation [1]kdpdMed 33242')'(')2(4ωεεδω ε εpiσ −−= ,      (1)where  ),( pε  and  )','( pε  are  the  energy  and  the momentum of the initial and final particles,  ω  and  k are the frequency and the wave vector of the radiated wave, )'( ωεεδ −−  is the delta-function that determines the energy conservation under radiation. According to the rules of diagram technique [1] the squared matrix element in (1) can be written with the account of the contribution of the second Born approximation in  the form ∫ −−= 33*212212)2(Re2piqdUUMMUUMM qgqgg ,  (2)where  gU  is  the  Fourier  component  of  the  potential energy  of  the electron  (positron)  in  an external  field, ),0( gg=µ  is  the  4-momentum  transferred  to  the external  field  (it  is  assumed that  the  external  field  is stationary),  µµµµ kppg −−= ' ,  1M  and  2M  are  the matrix  elements  which determine contributions  of  the first and the second Born approximations (see Fig. 1):ueggeebuMgg −−=τεγε σγ'2ˆˆ2ˆˆˆ' 001  ,        (3)+−+−−−+=qqqg vqevqvqpgmpeuMσεγτεγσεγγσ2ˆ1ˆ''2'ˆ12ˆ12ˆˆˆ'0'0002uepgmpvqgq ++++ ˆ'2ˆ'ˆ''2'ˆ10'0τγτεγ,                   (4)where µe  is the photon polarization vector, v and v' are the  initial  and  final  velocities  of  the  electron, µµµ qgq −=' . The values  b,  gσ  and  gτ  in  1M  and 2M  are determined by the relations ggbτσ11−= ,       pggg 22||−=σ ,  '22|| pggngg++= ⊥τ ,                              (5)where '/' ppn=  is the unit vector along the momentum 'p  direction, and ⊥n are the components of this vector orthogonal to the p.Fig. 1. Feynman diagrams corresponding to the first  and the second Born approximations in the process of  bremsstrahlung in an external fieldPROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 131-134. 131The  matrix  element  of  the  radiation  process depends on the momentum transferred to the external field g in an explicit form. The cross section itself can be  also  expressed  directly  through  the  transferred momentum (and also through the angle ϑ  between the vectors  k and  p).  Such  presentation  is  especially convenient  in  the  range  of  small  values  of  the transferred momentum mg < <⊥ , because it is possible to  make  an  expansion  in  the  matrix  element  by  the powers of  ⊥g  in this case. Transformation to the new variables  is  described  in  [2,3].  The  differential  cross section in this case takes the form:gdydydmMed 3222441')2(−=ωωδεεpiσ ,    (6)where '2/2 ε εωδ m= . The variable y  is connected to ϑ  by the relation( ) ayfm +=2/ε ϑ ,      11 ≤≤− y ,             (7)where−−=⊥⊥εδδ 24 2||22 ggmga ,     +−−= ⊥⊥ 222|| 21mgggfδεδδ .From the fact that the value a in the radical in (7) must be positive one can conclude thatεδ 2/2|| ⊥+≥ gg .                                  (8)Note that Eq. (7) determines the possible values of the radiation angle ϑ  under given values of ||g  and ⊥g.Eq. (4) can be simplified that makes the procedure of  summing  over  polarization  of  interacting  particles more  easy  than  for  original  Eq. (4).  Neglecting  the terms  of  the  order  of  22 / εm ,  we obtain  after  some calculations the following expression for 2M :    uQqqegeQuMqqg ++−= ⊥⊥ 2'012 2''2ˆˆ''σε σεγτε εω,(9)where 1Q  is the spinor structure of 1M , ggeggebeQτεγε σγ'2ˆˆ2ˆˆˆ 001 −−= ,andqqqgqgqeqegqqgeQστε εγγττεσσε '00'222 '4ˆˆ'ˆ'4ˆˆ'ˆ4ˆˆˆ ⊥⊥⊥⊥ +−−= .After summing over polarizations of final particles and  averaging  over  polarization  of  initial  particle  we obtain with accuracy to terms of order of  22 / εm  the following equations for the values  21M  and  *21MM  in (1):−+= ⊥2222||21 2''2|| bmggMεεεε, (10)[−−⋅−=⊥⊥ bmppMqqMMgqq)2'(2'2||2' 221'*21τεωσε σ−−+⋅+− ⊥ bgpgggg '8'2''''2εετεεσεεετεε , (11)Substituting these equations into (1), we obtain after the  integration  over  y and  expansion  on  mg /⊥  the following  expression  for  the  cross  section  of  the radiation  with  account  of  the  second  Born approximation: ⊥⊥⊥= dggdgggdmed ||2||2222 '2 ωωδεεpiσ + 2gUF×−+−−++||2||||3 1'2141')2(1gggF δε εωδδεωεpi∫ ++−−×−⊥⊥⊥qgqg UUiqiqgqqgqdU)0)(0()(Re||||||3,(12)where−−+=||||212'21ggF δδε εω.Let us consider some particular cases of (12). If the condition  effq ||< <δ  is  satisfied,  where  effq ||  are  the characteristic values of the longitudinal  component of the  momentum  q in  (12),  we  can  neglect  the dependence  of  gU  and  qgU −  on  ||g  in  (12).  After integration over ||g  we obtain that 222)2('431)()(piε εωσ ⊥⊥⊥ ⋅+=gdgdwgd − 2gU +×effeffqgO||||1 ,                 (13)whereωωεεpidmgegdw 222 '32)( ⊥⊥ = .                       (14)For  εω < <  Eq. (13) corresponds to the product of the radiation probability  ωddw /  and the cross section of elastic scattering of the particle in the external field eldσ  with account of contribution of the second Born approximation,×= ⊥⊥ 224)(piσgdgd el−−× ∫ ⊥⊥ '2||332)0(')2(Re1 qqgg UUiqqqqdUUpiε.For  the Coulomb field of  the nucleus with charge || eZ  the last equation transforms to the form132 −Ω=⊥ ϑpiϑεpiσ2||1)(24242 ZeeedeZgd el ,where the scattering angle  pg /⊥≈ϑ .  The last  result coincides with the corresponding result of the paper [4] obtained  by  different  method.  For  arbitrary  external field the formula for eldσ  was obtained in [5,6]. Note  that  radiation  of  electrons  in  Coulomb field exceeds slightly the radiation of positrons. That is due to the fact that electron attracted by the nucleus moves in the region with larger gradient of the potential than the positron. So in the range of frequencies  εω ~  the theorem about  factorization  of  the  radiation  cross  section, according to which )()( ⊥⊥≈ gdgdwd elσσ ,                (15)is  justified  with  an  accuracy  to  the  correction  which determines  the  contribution  of  the  second  Born approximation.2. THE CROSS SECTION FOR RADIATION OF RELATIVISTIC ELECTRONS AND POSITRONS IN THE FIELD OF ATOMIC PLANE IN A CRYSTALWe can see that dependence of the radiation cross section  on  the  particle  charge  sign  in  the  case  of radiation of high energy electrons and positrons in the field of single atom is rather small. Different situation arises  for  coherent  interaction  of  relativistic  particles with  atoms of  crystal  lattice.  In  this  case,  due  to  the coherent  effect  the  dependence  of  the  radiation  cross section on the particle charge sign can be substantially amplified in comparison with analogous dependence of the  radiation  cross  section  in  an  amorphous  medium. The  attention  to  this  fact  was  paid  in  [7]  during consideration  of  contribution  of  the  second  Born approximation into coherent radiation cross section of relativistic electrons in the field of atomic plane of the crystal. It was demonstrated that in considered case the relative contribution of the second Born approximation into coherent  radiation cross section is  determined by the parameter 222θεαaRZep = 22~θθ c                                   (16)which represents by the order of value the ratio of the squared  critical  angle  of  plane  channeling  [8]  to  the squared angle of incidence θ  of the beam to the atomic plane (here || eZ  is the charge of the nucleus of crystal lattice  atom,  R is  the  screening  radius  of  the  atomic potential, a is the average distance between atoms in the crystal  plane).  In this case the Born expansion of  the radiation  cross  section  is  valid  if  1< <pα .  The parameter pα  rapidly increases with θ  decrease. Under 1~pα  the account of effects of channeling and above-barrier  motion  of  particles  in  respect  to  the  crystal atomic plane is necessary [3,8,9].So consider the coherent radiation of electrons and positrons in the field of continuous potential of one of the atomic planes  in  a  crystal  under  incidence of  the beam under small angle  θ  to this plane. The potential energy  of  the  particle  in  continuous  potential  of  the plane is determined by Eq. (8,9)∫ ∑=−=NnnzyrrudydzLLxU1)(1)(,      (17)where )( nrru−  is the particle potential energy in field of the single atom of crystal plane located in the point nr,  yL  and  zL  are the linear dimensions of the plane and x is the coordinate, orthogonal to the atomic plane of the crystal (summation in (17) is made over all atoms of the crystal plane). Taking the atomic potential in the form of the screened Coulomb potential, andRrereZeru /||)( −= ,we find  the  expression  for  the  Fourier  component  of (17): gzyyzg uaaggU 1)()()2( 2 δδpi= ,            (18)where  ya  and  za  are the distances between atoms in the plane along the axes y and z, and22||4−+=RgeeZu g pi.Substituting the Fourier  component (18)  into (12), we  obtain  the  following  expression  for  the  radiation cross section×= 2232 '16θωωδεεpiασ xzydgdmaaNZd++−−+×− 2222)(112'21Rggg xxx θδθδε εω×++− 2212|| RgaaZeexzyεα+−−+×θδθδε εωxx gg12'212×−+−−+θδε εωθδθδεωxxx ggg1'2141'2+×− 222 421RgRxpiθ.                          (19)Here we have used the fact that in the case under consideration xgg θ≈|| . The value xg  here covers the range θδ /≥xg . Under εω < <  Eq. (19) transforms to the corresponding result of the paper [7]. Note that in the case of interaction of the particle with continuous potential of the plane the radiation cross section cannot be  presented  in  the  form  (15)  for  any  photon frequencies. This is due to the fact that elastic scattering on the continuous plane  can take place only to  some fixed angles to the plane [5,6] because of energy and momentum conservation laws in the process of elastic scattering. Eq. (19)  demonstrates  that  for  all  frequencies  the cross section of radiation by positrons turns out larger 133than  the  cross  section  of  radiation  by  electrons,  in difference  to  the  case  of  radiation  in  Coulomb field. This result can be explained by the following way. The sign of the effect is determined by competition of two factors:  (i)  the  electron  is  attracted  to  the  plane  and moves in the region with larger gradient of the potential than  the  positron,  that  leads  to  increase  of  radiation; (ii) in distinct to the positron, it spends less time in the region with large gradient of the potential, that leads to decrease of radiaton. In Coulomb field the first factor plays the determinative role, in the field of atomic plane - the second one. Eq. (19)  demonstrates  also  that  radiation  spectrum ωσω dd /  posesses  the  maximum  in  the  range  of frequencies satisfying the conditionθωωεε Rm2~)(22−.                                (20)With  the  particle  energy  growth  the  position  of  this maximum moves to the region of high frequencies. For θε /~ 2 Rm  the maximum is located in the region of frequencies for which the effect of recoil under radiation is  substantial.  The  parameter  (16)  that  determines dependence of the cross section on the particle charge sign for θε /~ 2 Rm  takes the formθα 222~amZep .So in  the  range  of  energies  under  consideration  with decrease  of  θ  the  dependence  of  coherent  radiation cross section on the charge sign of the particle becomes substantial in the whole range of frequencies of radiated photons. 3. COHERENT RADIATION ON A SET OF ATOMIC PLANES IN THE SECOND BORN APPROXIMATIONThe cross section of bremsstrahlung on the crystal is determined by relation [3]:)( incohcoh ddNd σσσ += ,where N is the whole number of atoms in crystal, cohdσ  is the coherent part of radiation cross section caused by interference  of  radiation  produced  on  different  atoms regularly  arranged  in  the  crystal,  incohdσ  is  the incoherent  part  caused  by  thermal  spread  of  atom positions in the crystal. For the case of interaction of the particle  with  the  set  of  parallel  atomic  planes  in  the crystal, we can obtain the equation for cohdσ  from (19) by change of  integration over  xdg  to  the  summation over nagxnxpi2)( = :∑∫≥∞→θδθδpi//...2...xgxx adg   ,where  ax  is the distance between atomic planes. The cross section of coherent radiation of 1 GeV positrons and  electrons  incident  under  the  angle  4104 −⋅=θ  radians to the <011> plane of the Si crystal is shown on the  Fig. 2.  We  can  see  that  the  difference  between radiation cross  sections for  positrons  and electrons in the case illustrated is of order of 10%. Fig. 2. The  cross  section  of  coherent  radiation  of  1 GeV positrons (solid line) and electrons (dashed line)  incident  under  the  angle  4104 −⋅=θ  radians  to  the <011> plane of the Si crystal. The dotted line shows the Bethe-Heitler cross sectionThis  work  is  supported  in  part  by  Russian Foundation  for  Basic  Research  (Project  № 00-02-16337).REFERENCES1. A.I. Akhiezer,  V.B. Berestetskii.  Quantum Electrodynamics. New York: “Interscience Publ.”, 1965, 868 p.2. M.L. Ter-Mikaelian.  High-Energy  Electro-dynamic  Processes  in  Condensed  Matter.  New York: “Wiley Interscience”, 1972, 457 p.3. A.I. Akhiezer,  N.F. Shul’ga.  High-Energy Electrodynamics in  Matter.  Amsterdam: “Gordon and Breach”, 1996, 388 p.4. W.A. McKinley, G. Feshbach. The Coulomb Scattering  of  Relativistic  Electrons  by  Nuclei  // Phys. Rev. 1948, v. 74, p. 1759-1763.5. V.V. Syshchenko, N.F. Shul’ga. Elastic scat-tering  of  high  energy  charged  particles  in  an external field in the second Born approximation // Ukr. Fiz. Zh. 1995, v. 40, №1-2, p. 15-21.6. A.I. Akhiezer,  N.F. Shul'ga,  V.I. Truten', A.A. Grinenko,  V.V. Syshchenko.  Dynamics  of high-energy charged particles in straight and bent crystals  //  Physics-Uspekhi. 1995,  v. 38,  №10, p. 1119-1145.7. A.I. Akhiezer,  P.I. Fomin,  N.F. Shul’ga. Coherent bremsstrahlung of electrons and positrons of ultrahigh energy in crystals //  JETP Lett. 1971, v. 13, №12, p. 506-508.8. J. Lindhard.  Influence  of  Crystal  Lattice  on Motion of Energetic Charged Particles //  K. Dan.  Vidensk. Selsk. Mat.-Fyz. Medd. 1965, v. 34, №14.9. D.S. Gemmell. Channeling and related effects in the motion of charged particles through crystal // Rev. of Mod. Phys. 1974, v. 46, №1, p. 129-228.134

The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal

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The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)