Features of polarization bremsstrahlung radiation (PB) by a relativistic charge on medium electrons bound in atoms are discussed. PB is considered as a dispersion of virtual photons of an electromagnetic field of a fast charge on

atomic bound electrons. In this case atomic electron can get dispersed at a recoil energy in only certain portions, as in the nuclear M¨ossbauer process. Because of this a spectrum of dispersed photons is degenerated in a series of narrow peaks

Grishin, V. K.

Nikitin, D. P.

English

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DOI 10.1393/ncc/i2011-10947-1Colloquia: Channeling 2010IL NUOVO CIMENTO Vol. 34 C, N. 4 Luglio-Agosto 2011Polarization bremsstrahlung by fast charge on atomic boundelectrons—Analog of nuclear Mo¨ssbauer’s effectV. K. Grishin(∗) and D. P. NikitinSkobel’tsyn Institute of Nuclear Physics, Lomonosov Moscow State University - Moscow, Russia(ricevuto il 22 Dicembre 2010; pubblicato online il 29 Agosto 2011)Summary. — Features of polarization bremsstrahlung radiation (PB) by a rela-tivistic charge on medium electrons bound in atoms are discussed. PB is consideredas a dispersion of virtual photons of an electromagnetic field of a fast charge onatomic bound electrons. In this case atomic electron can get dispersed at a recoilenergy in only certain portions, as in the nuclear Mo¨ssbauer process. Because ofthis a spectrum of dispersed photons is degenerated in a series of narrow peaks.PACS 41.60.-m – Radiation by moving charges.PACS 03.50.-z – Classical field theories.1. – IntroductionRecently PB by fast electrons on medium electrons has been actively discussed sincePB is rather sensitive to the features of electronic configuration. Consequently it can beused, e.g., for diagnostics of a substance structure. One of the ways of PB characteris-tics research is an approach, which represents PB as a process of dispersion of virtualphotons, “contained” in an electromagnetic field of a fast charge, on atomic electrons ina medium [1-3]. But usually it is proposed that in the process of dispersion the electronsare actually free. Nevertheless, as it follows from the laws of conservation of energy andmomentum, the recoil energy transmitted to electrons during interaction with a fast elec-tron field appears as an enough appreciable portion only in the high-frequency area ofphotons. Therefore, in this case (as opposite to Compton’s photon-electron interaction)the dispersion shows interesting features, because probability of transferring a share ofpulse and energy (lost by virtual photons during dispersion) to atomic electrons sharplydepends on the quantum state of the atom. So, a portion of the energy transmitted tothe atomic electron appreciably differs from the energy of bound electron’s transition toa new quantum state; the momentum can be transferred during dispersion to the wholeatom, i.e. actually without transfers of energy due to ruggedness of atom. Probability of(∗) E-mail: vkgris@mail.ruc© Societa` Italiana di Fisica 335336 V. K. GRISHIN and D. P. NIKITINa similar dispersion is strongly suppressed, but it becomes strong when the specified por-tions of energy coincide. To sum up, the spectrum of dispersed photons (i.e. irradiatedphotons) results in the peak structure of PB on bound electrons.Thus, a radiation spectrum is displaced in a X-ray range as it follows from kinematicratios and also it is divided into two groups, which range poorly and strongly depend onthe energy of a fast charge.Specified PB features on bound electrons are described below on an example ofhydrogen-like one-electronic atom.2. – Analytical description: a stream of virtual wavesSo, let us present electromagnetic field of a fast charge (a relativistic electron) asa stream of virtual photons of various frequencies, addressing to the following scheme.Fast electron at a speed v, with a charge e and relativistic factor γe = 1/√1− v2/c2distributed in the medium along some directions z, is scattered by a medium atom. Asa result of interaction of a projectile with one or several atomic electrons (further theprocess of scattering of a fast electron field is considered), a photon is emitted under theangle ϑ along the unit vector n′.The electric field of a fast electron with spatial density of charge ρ = eδ(r − vt) canbe submitted as a stream of plane waves(1) E(t, r ) =∫Eω,k exp[i(kr − ωt)]dω dk.However, in this case it becomes useful to describe the axial-symmetric field as a wavestream (a stream of pseudo- or virtual photons; their velocities coincide with the velocityof fast electrons) with indication of coordinate dependence in transverse planes [4]:(2) E(t, r⊥, z) =∫Eω(r⊥) exp [i(kzz − ωt)] dω dkz,where r = (r⊥, z); r⊥ is the radius-vector transverse to the trajectory plane.Using the ratio for the electric field of a fast electron (see for example [4, §18.5]), wecan write for the Fourier components:(3) Eω(r⊥, z) =(e⊥eζK1(ζ)πε(ω)vr⊥+ ezeζK0(ζ)πε(ω)vγ⊗r⊥)exp[ikzz],where ζ = ωr⊥/vγ⊗.Here e⊥, ez are the transverse and longitudinal unit vectors, accordingly, ε(ω) = 1−χis the dielectric permeability of the medium crossed by the electron, χ = ω20/ω2, ω0 isthe plasma frequency of the medium (ω0  30–40 eV for various substances), 1/γ2⊗ =1/γ2e + χβ2, β = v/c, K0,1 are the Macdonald functions. The wave stream is distributedalong the axis Z with velocity v and wave vector kz = ω/v, i.e. k = ωv/v2.In the description (3) the sizes of “a central stain” of the virtual stream are preciselytraced and are defined by the ratio ζ ≤ 1, as at the big values of argument K0,1 → 0.The radius of the spot being(4) Rs  βγ⊗c/ω → βγ⊗λ/2π,POLARIZATIONAL BREMSSTRAHLUNG BY FAST CHARGE ON ATOMIC BOUND ETC. 337Fig. 1. – Scheme of the virtual photon (wave vector k) dispersion on the electron. k′ is the wavevector (real) of the dispersed photon, pe is the recoil momentum transmitted to the electron.where λ is the wavelength. The same ratio defines the width of virtual frequenciesspectrum and its change in the process of radiation separation from electron trajectory.It is typical that spectra for longitudinal and transverse parts of the fields differ by thefrequency dependence as well as by the intensity one. As 1/γ⊗  1 even for moderatelyrelativistic electrons, the contribution of longitudinal fields in (3) as a whole is small, seefor example [5].3. – Scattering of virtual photons by electronScattering of a virtual photon with initial momentum k = ωv/v2 and a real photonwith momentum k′ = ω′n′/c, where the velocity v of the fast electron is proposed to beconstant during the dispersion (here h¯ = 1; ω and ω′ are the frequencies of a photonbefore and after scattering, accordingly, c is the velocity of light in vacuum), is describedby the ratio (see fig. 1)(5) k = k′ + pe,or(6)ωvv2=ω′n′c+ pe,where pe is the momentum transmitted to the electron.As ω − ω′  ω (that is quite natural in the range of moderate energies of photons asthe latter are more “simple particles” in comparison with electrons), we obtain(7) ω =|vpe cosϑ|1− β cosϑ ,where β = v/c.Further using the relation (k′)2 = k2+p2e−2(pek) we can derive the ratio for electrontransmitted energy Erec = p2e/2me:(8) Erec =ω22meβ2c2(1− 2β cosϑ + β2).338 V. K. GRISHIN and D. P. NIKITINAt γ2e  1 from the ratio (7) it follows that the maximal (ϑ → 0) and minimal(ϑ → π) values of photon energy are equal according to (see also [6])(9) ωmax  2|(v pe)|γ2e ,(10) ωmin  pec2 .Thus the dispersion of virtual photons appreciably differs in frequency-angular de-pendence on the classical Compton dispersion of real photons. First of all, we specifyrelativistic effects that was in particular marked on an example of the parametrical radi-ation in [6] (see also [7]). We remind that energy transmitted to electron (recoil energy)remains appreciably smaller than energies of virtual and irradiated photons.4. – PB due to bound atomic electron. PredictionsAs underlined above, one of the traditional ways to estimate the PB intensity isbased on the classical representation of this process as a dispersion of a stream of virtualphotons of fast charge on atomic electrons. Appropriate analytical PB description follows,for example, from the procedure stated in [8]. As a result, PB spectral-angular intensityat charged particle scattering by the atom with Z electrons is equal to(11)d2WdωdΩ=e48πm2c3∣∣∣∣∣∞∑s=1[n′ Eω,s]exp[iper′s]∣∣∣∣∣2,where dΩ is the element of a solid angle, rs are the electron radius-vectors within the atomframe. Vector product [n′Eω,s] is calculated in conformity with the ratio (1) (therefore,atomic electrons have various values of transverse radii (i.e. impact parameters), r⊥,s; inthe chosen system of coordinates r2⊥,s = x2s + y2s).Summing in (10) is distributed over all atomic electrons which are “covered” by thevirtual stream. Therefore, the ratio (10), being divided into a stream intensity of virtualphotons, represents the effective section of PB process.However, this classical description does not take into account that atomic electronsare in bound states. And though, the atomic bound weakly affects the accuracy of theratios (7) and (8), the recoil energy Erec can be transferred effectively to atomic electronsonly by certain portions, appropriate for transition of particles to a new quantum state(obviously, it is valid if Erec does not exceed the energy of electrons ionization). Inintermediate state the transfer of energy results actually in a virtual excitation of atom.And here it becomes interesting to compare discussed problem and situation when nuclearMo¨ssbauer effect [9] is observed.The effective section (i.e. probability) of a similar process in the elementary nu-clear system-atom of hydrogen or deeply “cleared” (bared) more complex atom, i.e.one-electron ion, is described following [9, 10] by the ratio (here the most importantresonant factor is only underlined)(12) P ∝ Γs(E1s −Erec)2 + Γ2s/4,POLARIZATIONAL BREMSSTRAHLUNG BY FAST CHARGE ON ATOMIC BOUND ETC. 339Fig. 2. – Intensity Iω and PB peak locations at γ = 20 with different angles of radiation θ = π,π/4, π/8 (from left to right).where E1s is the energy difference of electronic states 1 and s, Γs is the full width ofresonance, directly connected to a constant of disintegration of the excited state (lifetimeof this state). Nevertheless, a rather important specification appears in (11) because herea recoil energy transmitted to electron appears directly, see ratio (11) in [10].Being limited to similar representations, it is necessary to admit that the basic PBfeatures here are defined by the product of a stream of virtual photons representing therelativistic electron field and the effective section (11). As the lifetime of excited statesis small enough (for hydrogen about 10−8 s), PB spectrum should degenerate actually ina set of narrow peaks corresponding to various excited states.For numerical calculations we have to be limited to the approached ratio. So, forelectrons in the basic state of the hydrogen atom, the average size of a field Eω in avirtual stream is of the order (as follows from the ratios (3) and (4))(13) 〈Eω〉  2e(1− exp[−Ω0γe/ω])πca0,where Ω0 = c/a0  4 keV, a0 is Bohr’s radius. Further it is possible to take advantageof the ratio (11).Thus, there is an additional interesting effect: due to the return of transition-excitedatomic electron, characteristic radiation (CR) can be emitted. Paradoxical situations:the frequencies of peaks PB and CR essentially differ, though it would seem both peaksare defined by the same energy transition of the atomic electron. But in reality, for PBpeak position is defined by the ratio (8) and only for CR by the energy E1s. Thus, totalradiation at the fixed angle of radiation contains two groups of peaks, namely, soft (CR)and hard (PB). Some results of the obtained estimations are presented in fig. 2.5. – ConclusionThus, consideration of polarization bremsstrahlung radiation on bound electrons en-riches in a certain extent the traditional picture of the description of the radiating inter-action of relativistic electrons with substance. Hence, the analog between considered PBprocesses and Mo¨ssbauer’s effect becomes evident.340 V. K. GRISHIN and D. P. NIKITINOf course, the analysis carried out is open for the discussion. It certainly requiresspecification and wider scope of the object of research. So one of the obvious momentsis searching in the considered scheme the conditions for coherent radiations in multi-electron atoms and other collective effects.∗ ∗ ∗The authors express sincere gratitude to Profs. S. B. Dabagov and I. E. Vnukovfor the opportunity to present our work at the conference.REFERENCES[1] Fermi E., Z. Phys., 29 (1924) 315.[2] Ter-Mikaelian M. L., High Energy Electromagnetic Processes in Condensed Media(Wiley, New York) 1972.[3] Amus’ia M., Buimistrov V., Zon B. et al., Polarization Bremsstrahlung of Particles andAtoms (Plenum Press, New York) 1992.[4] Panofsky W. K. H. and Phillips M., Classical Electricity and Magnetism (Addison-Wesley Publishing Company, Inc., Cambridge) 1962, §18.5.[5] Grishin V. K., Ishkhanov B. S., Shvedunov V. I. and Nasonov N. N., Coherentradiative diagnostic of nanostructure parameter by electron accelerators//vestnik MGU,Vol. 3, Phys. Astron. (2007) No. 6, pp. 29-33.[6] Baryshtvsky V. G., Feranchuk I. D. and Ulyeanenkov F. P., Parametric X-rayRadiation in Crystals, ISBN-10 (Springer Berlin, Heidelberg, New York) 2005.[7] Basylev V. A. and Zhevago H. K., Izluchenie bystrykh chastits v veshchestve i vovneshnikh polyakh (Nauka, Moscow) 1987.[8] Landau K. D. and Lifshits E. M., Field Theory (Nauka, Moscow) 1960, §80.[9] Wertheim G. K., Mo¨ssbauer Effect. Principles and Application (Academic Press, NewYork) 1964.[10] Akhieser A. I. and Berestetski V. B., Quantum Electrodynamic (Nauka, Moscow)1969, §27.2.

Polarization bremsstrahlung by fast charge on atomic bound

electrons—Analog of nuclear M¨ossbauer’s effect

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Polarization bremsstrahlung by fast charge on atomic bound electrons—Analog of nuclear M¨ossbauer’s effect

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