In the present work, the parametric X-radiation (PXR) and the parametric radiation at a small angle relative to the velocity of a relativistic electron (FPXR) crossing a single crystal plate in Bragg scattering geometry are investigated within the framework of the dynamic X-ray diffraction theory. The expressions for spectral-angular density of the named radiations are derived in the general case of asymmetric reflection. The ratio of the contributions of these radiation mechanisms as a function of reflection asymmetry is considered in this work

Blazhevich, S.

Noskov, A.

English

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DOI 10.1393/ncc/i2011-10970-2Colloquia: Channeling 2010IL NUOVO CIMENTO Vol. 34 C, N. 4 Luglio-Agosto 2011On the ratio of the relativistic electron PXR in Bragg and forwarddirection within Bragg scattering geometryS. Blazhevich(1) and A. Noskov(2)(1) Belgorod State University - Belgorod, Russia(2) Belgorod University of Consumer’s Cooperation - Belgorod, Russia(ricevuto il 22 Dicembre 2010; pubblicato online il 21 Settembre 2011)Summary. — In the present work, the parametric X-radiation (PXR) and theparametric radiation at a small angle relative to the velocity of a relativistic electron(FPXR) crossing a single crystal plate in Bragg scattering geometry are investigatedwithin the framework of the dynamic X-ray diffraction theory. The expressions forspectral-angular density of the named radiations are derived in the general case ofasymmetric reflection. The ratio of the contributions of these radiation mechanismsas a function of reflection asymmetry is considered in this work.PACS 41.60.-m – Radiation by moving charges.PACS 41.75.Ht – Relativistic electron and positron beams.PACS 42.25.Fx – Diffraction and scattering.1. – IntroductionWhen a fast charged particle crosses a single crystal its Coulomb field pseudo pho-tons scatter on the system of parallel atomic planes generating parametric X-radiation(PXR) [1-3]. The theory of PXR of relativistic particle in a crystal forecasts the radia-tion not only in Bragg scattering direction, but also in the direction of radiating particlevelocity (FPXR) [4-6], which represents the appearance of dynamical diffraction in PXR.It is known that some attempts of the FPXR experimental research were made [7-11],but the first report about its experimental discovery has appeared recently in ref. [11],where the FPXR was registered in Laue scattering geometry. In the experiment [11]the relativistic electron X-radiation was registered in the condition of FPXR generation,nevertheless the sought-for peak became weakly apparent on the background of the ra-diation generated by scattered electrons in the structural material of the experimentalsetup. Thus the theoretical investigations of FPXR properties and the optimization ofviewing conditions for experimental observation of this dynamical effect remain topicalup to date.The detailed theoretical description of dynamical effect of FPXR and its accompa-nying transition radiation (TR), as a background, was made for the case of symmetricalc© Societa` Italiana di Fisica 243244 S. BLAZHEVICH and A. NOSKOVFig. 1. – Radiation process geometry.reflection in the works [12-15]. Under condition of asymmetrical reflection of the rela-tivistic electron Coulomb field the PXR, TR and DTR were theoretically investigatedin refs. [16-18] and FPXR in Bragg geometry in ref. [19]. The above-mentioned worksshowed that the spectral-angular density of these radiation mechanisms depends essen-tially on the reflection asymmetry and revealed the effects associated with it. The ratioof PXR and FPXR yield in Laue scattering geometry was considered in [20], where itwas shown that the change of the reflection asymmetry cardinally changes the ratio.In the present work the analytical expressions for spectral-angular density of FPXRare derived in the framework of two-wave approximation of dynamic diffraction the-ory [21] in general case of asymmetrical field reflection. Based on the obtained expressionsthe ratio of PXR and FPXR yields is investigated as a function of reflection asymmetryin Bragg scattering geometry.2. – Dynamical expressions for electric fields in crystal targetLet us consider the radiation of a fast charged particle crossing a single crystal platewith a constant velocity V (see in fig. 1). We will use the same table of symbols as inthe work [17].While solving the problem, let us consider an equation for a Fourier image of anelectromagnetic field(1) E(k, ω) =∫dtd3rE(r, t) exp[iωt− ikr].Since the field of a relativistic particle could, to a good accuracy, be taken as be-ing transverse, the incident E0(k, ω) and diffracted Eg(k, ω) electromagnetic waves aredetermined by two amplitudes with different values of transverse polarization:E0(k, ω) = E(1)0 (k, ω)e(1)0 + E(2)0 (k, ω)e(2)0 ,(2)Eg(k, ω) = E(1)g (k, ω)e(1)1 + E(2)g (k, ω)e(2)1 ,where the unit vectors of polarization e(1)0 and e(2)0 are perpendicular to vector k, andON THE RATIO OF THE RELATIVISTIC ELECTRON PXR IN BRAGG AND ETC. 245vectors e(1)1 and e(2)1 are perpendicular to vector kg = k + g. Vectors e(2)0 , e(2)1 aresituated on the plane of vectors k and kg (π-polarization) and e(1)0 , e(1)1 are perpendicularto this plane (σ-polarization); g is vector of the reciprocal lattice, defining a set ofreflecting atomic planes. The system of equation for the Fourier transform images ofelectromagnetic field in two-wave approximation of dynamic theory of diffraction has thefollowing view [22]:(3)⎧⎨⎩(ω2(1 + χ0)− k2)E(s)0 + ω2χ−gC(s,τ)E(s)g = 8π2ieωθV P (s)δ(ω − kV),ω2χgC(s,τ)E(s)0 + (ω2(1 + χ0)− k2g)E(s)g = 0,where χ0 = χ′0 + iχ′′0 is the average dielectric susceptibility, χg, χ−g are the coefficientsof the Fourier expansion of the dielectric susceptibility of a crystal over the reciprocallattice vectors g:(4) χ(ω, r) =∑gχg(ω)eigr =∑g(χ′g(ω) + iχ′′g(ω))eigr.The values N˜ (s,τ) and P (s) are defined in the system (3) asC(s,τ) = e(s)0 e(s)1 = (−1)τC(s), N˜ (1) = 1, N˜ (2) = | cos 2θB |,(5)P (s) = e(s)0 (μ/μ), P(1) = sinϕ, P (2) = cosϕ,μ = k − ωV/V 2 is the virtual photon momentum vector component perpendicular tothe particle velocity vector V (μ = ωθ/V , where θ  1 is the angle between k and V),θB is the angle between electron velocity and a set of atomic planes in the crystal (Braggangle), ϕ is the azimuth angle, counted off from the plane formed by vectors V and g, thevalue of the reciprocal lattice vector is shown by the expression g = 2ωB sin θB/V , ωBis Bragg’s frequency. The angle between vector ωVV 2 + g and diffracted wave vector kg isdefined as θ′. The equation system (3) under s = 1 describes the fields of σ-polarization,and under s = 2 the fields of π-polarization. When 2θB < π2 , then τ = 2, otherwiseτ = 1.Let us solve the dispersion equation for X-waves in crystal following from system (3):(6) (ω2(1 + χ0)− k2)(ω2(1 + χ0)− k2g)− ω4χ−gχgC(s)2= 0,using standard methods of dynamic theory [21]k = ω√1 + χ0 + λ0, kg = ω√1 + χ0 + λg,(7)λ(1,2)0 = ωγ04γg(−β ±√β2 + 4χgχ−gC(s)2 γgγ0),(8)λ(1,2)g =ω4(β ±√β2 + 4χgχ−gC(s)2 γgγ0).where β = α − χ0(1 − γgγ0 ), α = 1ω2 (k2g − k2), γ0 = cosψ0, γg = cosψg, ψ0 is the angle246 S. BLAZHEVICH and A. NOSKOVbetween the incident wave vector k and the vector normal to the plate surface n, ψg isthe angle between wave vector kg and the vector n (see fig. 1).As |λ0|  ω and |λg|  ω, we can show that θ ≈ θ′ (see in fig. 1), and hereinafterwill use θ in all the occasions.The solution of the combined equations (3) gives us the relativistic particle field inthe vacuum in front of the crystal as(9) E(s)vac I0 =8π2ieV θP (s)ω1γ0|γg|(−χ0 − 2ω γ0γg λg + βγ0γg)δ(λ∗g − λg),where λ∗g =ωβ2 +γgγ0λ∗0, λ∗0 = ω(γ−2+θ2−χ02), δ(λ∗0 − λ0) = |γg|γ0 δ(λ∗g − λg).The incident and diffracted field inside the crystal asE(s)cr0 =8π2ieV θP (s)ω(−ω2β − 2ω γgγ0 λ0)δ(λ0 − λ∗0)4γgγ0 (λ0 − λ(1)0 )(λ0 − λ(2)0 )(10a)+E(s)(1)δ(λ0 − λ(1)0)+ E(s)(2)δ(λ0 − λ(2)0)E(s)crg =8π2ieV θP (s)ωω2χgC(s,τ)4γ20γ2g(λg − λ(1)g )(λg − λ(2)g )δ(λ∗g − λg)(10b)+E(s)(1)δ(λg − λ(1)g)+ E(s)(2)δ(λg − λ(2)g),where E(s)(1)and E(s)(2)are the free fields, corresponding to the two solutions (8) ofdispersion equation (6), and the fields in the vacuum behind the crystal asE(s)vac II0 =8π2ieV θP (s)ω1−χ0 − 2λ0ωδ(λ0 − λ∗0) + E(s)Rad0 δ(λ0 +ωχ02),(10c)E(s)vacg = E(s)Radg δ(λg +ωχ02),(10d)where E(s)Rad0 , E(s)Radg is the coherent radiation amplitude in the electron velocity andBragg directions appropriately.3. – Spectral-angular density of the radiation in Bragg directionFor definition of amplitudes E(s)Rad0 and E(s)Radg the ordinary boundary conditionson the inlet and outlet surfaces of the crystal plate are used and by substituting derivedformulae for E(s)Radg to the well-known [22] expression for spectral-angular density ofX-radiation(11) ωd2NdωdΩ= ω2(2π)−62∑s=1∣∣∣E(s)crg ∣∣∣2 ,ON THE RATIO OF THE RELATIVISTIC ELECTRON PXR IN BRAGG AND ETC. 247the expression for PXR spectral-angular distribution is obtained:ωd2N (s)PXRdωdΩ=e2π2P (s)2θ2(θ2 + γ−2 − χ′0)2R(s)PXR,(12)R(s)PXR = R(1)(s)PXR + R(2)(s)PXR + R(INT)(s)PXRR(1)(s)PXR =(ξ(s) −√ξ(s)2 − ε)2Δ(ω, ε)sin2(b(s)(Δ−(ω, ε)− σ(s)) /2)(Δ−(ω, ε)− σ(s))2 ,(13)R(2)(s)PXR =(ξ(s) +√ξ(s)2 − ε)2Δ(ω, ε)sin2(b(s)(Δ+(ω, ε)− σ(s)) /2)(Δ+(ω, ε)− σ(s))2 ,(14)R(INT)(s)PXR =(15)ε cos((b(s)√ξ(s)2−ε)/ε)(cos(b(s)(ξ(s)/ε−σ(s)))−cos((b(s)√ξ(s)2−ε)/ε))Δ(ω, ε)((ξ(s)/ε− σ(s))2 + (ε− ξ(s)2)/ε2) ,whereξ(s) = ξ(s)(ω) = η(s)(ω) +(1 + ε)2ν(s),(16)η(s)(ω) =2 sin2 θBV 2|χ′gC(s)|(ωB(1 + θ cosϕ cot θB)ω− 1),ε =|γg|γ0, σ(s) =1|χ′gC(s)|(θ2 + γ−2 − χ′0), b(s) =ω|χ′g|C(s)2· Lγ0,Δ+(ω, ε) =ξ(s) +√ξ(s)2 − εε,Δ−(ω, ε) =ξ(s) −√ξ(s)2 − εε, Δ(ω, ε) = ξ(s)2 − ε + ε sin2(b(s)√ξ(s)2 − εε).Here the PXR spectrum is represented in the form of the sum of two spectra, R(1)(s)PXRand R(2)(s)PXR , corresponding to the two branches of dispersion equation and the summandR(INT)(s)PXR is the result of their interference.The function R(s)PXR describes the PXR spectrum. Parameter b(s) numerically equalshalf of the ratio of the relativistic electron path length L/(2γ0) and the length ofX-radiation extinction 1/ω|χ′gC(s)| in the crystal. The asymmetry parameter ε can berepresented as ε = sin(θB − δ)/ sin(θB + δ), where θB is the angle between the electronvelocity and the diffracting atomic plane. Hence we can see that parameter ε increaseswhen the angle of electron incidence (θB + δ) decreases and vice versa (see fig. 2).248 S. BLAZHEVICH and A. NOSKOVFig. 2. – Circuitry of asymmetric (ε > 1, ε < 1) reflection of the radiation from single crystalplate. The case of ε = 1 corresponds to the symmetric reflection (δ = 0).4. – Spectral-angular density of the radiation in the direction of incidentelectron velocitySubstituting derived formulae for E(s)Rad0 to (11) we will obtain the expressions de-scribing the FPXR spectral-angular density:ωd2N (s)FPXRdωdΩ=e2π2P (s)2θ2(θ2 + γ−2 − χ′0)2R(s)FPXR,(17)R(s)FPXR = R(1)(s)FPXR + R(2)(s)FPXR + R(INT)(s)FPXR ,R(1)(s)FPXR =1Δ(ω, ε)sin2(b(s)(Δ−(ω, ε)− σ(s)) /2)(Δ−(ω, ε)− σ(s))2 ,(18)R(2)(s)FPXR =1Δ(ω, ε)sin2(b(s)(Δ+(ω, ε)− σ(s)) /2)(Δ+(ω, ε)− σ(s))2 ,(19)R(INT)(s)FPXR =(20)cos((b(s)√ξ(s)2−ε)/ε)(cos(b(s)(ξ(s)/ε−σ(s)))−cos((b(s)√ξ(s)2−ε)/ε))Δ(ω, ε)((ξ(s)/ε− σ(s))2 + (ε− ξ(s)2)/ε2) .5. – Ratio of the PXR and FPXR yieldsThe contributions of the first (13), (18) and the second (14), (19) branches of PXRand FPXR spectra will be considerable under the conditionsΔ−(ω, ε)− σ(s) = 0,(21)Δ+(ω, ε)− σ(s) = 0.(22)The solution of eq. (21) or (22) defines the frequency in whose vicinity the PXR spectrumis concentrated at a certain observation angle. It follows from eqs. (21), (22) that theON THE RATIO OF THE RELATIVISTIC ELECTRON PXR IN BRAGG AND ETC. 249maxima of the and FPXR spectra is situated out of the frequency region of the radiationtotal reflection (extinctions):(23) ξ(s)(ω) =√ε +(σ(s)√ε− 1)22σ(s)>√ε,therefore formulas (13)-(15) and (17)-(20) correctly describe the PXR and FPXR spectrafor a thin crystal.Equation (21) is soluble only under the condition(24) ε <1σ(s)2or ε <ν(s)2(P (θ, γ) + 1)2.where P (θ, γ) = θ2/|χ′0|+ γ−2/|χ′0|.One can prove that eq. (22) has a solution under the condition(25) ε >1σ(s)2or ε >ν(s)2(P (θ, γ) + 1)2.When the strong reflection of X-ray waves from the atomic planes takes place, theparameter ν(s) = |χ′gC(s)||χ′0| is close to 1, in the case of weak reflection the parameterν(s) ≈ 0. Since ν(s) < 1, than inequality (25) is satisfiable in the case when ε ≥ ν(s)2and it means that only eq. (22) is soluble and the contribution to PXR and FPXR givesonly the second branch of dispersion equation solution. In the case when ε < ν(s)2bothinequalities (24) and (25) can be satisfiable as a function of the observation angle θ andthe relativistic electron energy γ. It means that for this case both branches of the excitedX-ray wave can give the contribution to PXR and FPXR yields.Let us consider the case when the inequality (25) is complete. In this case the secondbranches of PXR and FPXR give the main contribution to radiation yield and the spectraratio has the form which follows from (14) and (19):(26) R(2)(s)PXR /R(2)(s)FPXR =(ξ(s) +√ξ(s)2 − ε)2.It follows from expression (26) with taking into account (23) that in case when ε > 1or ε ≈ 1, the PXR yield exceeds considerably the FPXR yield R(2)(s)PXR  R(2)(s)FPXR. Todemonstrate this fact the curves plotted by formulas (14) and (19) are presented in fig. 3for the parameters pointed in the picture. As was expected, the PXR yield materiallyexceeds the FPXR yield.At the increase of the parameter ε, i.e. increase of the angle of particle incidence to thetarget surface δ + θB (see in fig. 2), the intensity of FPXR (at a small angle to particlevelocity) can considerably exceed the intensity of PXR in Bragg scattering directionR(2)(s)PXR  R(2)(s)FPXR, which is demonstrated by the curves in fig. 4. In the considered casethe PXR photon will move from the plate at a small angle to its surface (see in fig. 2).When we use eq. (22), whose solution is the frequency in the vicinity of which thePXR photon spectrum is concentrated and the amplitude maximum is situated, we will250 S. BLAZHEVICH and A. NOSKOVFig. 3.write expression (26) as(27) R(2)(s)PXR /R(2)(s)FPXR = (ε2/ν(s)2)(θ2/|χ′0|+ γ−2/|χ′0|+ 1)2.This ratio allows us to estimate the relative contributions of the radiations as a func-tion of the observation angle θ, the radiating particle energy defined by the Lorentz factorγ, and the parameter ν(s).It results from (37) that at observation angle decrease the FPXR yield more and moreexceeds the PXR yield for low values of the asymmetry parameter ε.We can obtain this ratio for the PXR angle density maximum (θ =√γ−2 + |χ′0|) inthe following form:(28) R(2)(s)PXR /R(2)(s)FPXR = (2ε2/ν(s)2)(γ−2/|χ′0|+ 1)2.As the value of parameter ν(s) is always less than 1, it follows from expression (38) thatfor γ2|χ′0|  1 the PXR yield will practically always exceed the FPXR yield. Otherwise,for γ2|χ′0| ≥ 1 this ratio can become close or less than 1 for a small value of the asymmetryparameter ε. In this case the feature of PXR and FPXR contribution to the totalFig. 4.ON THE RATIO OF THE RELATIVISTIC ELECTRON PXR IN BRAGG AND ETC. 251radiation yields considerably depends on the asymmetry of relativistic particle Coulombfield reflection (parameter ε).Let us consider the case when inequality (24) is completed. In this case both branchesgive the contribution to the radiation and their interference can be considerable. For thiscase the ratio of the first branch of the PXR and FPXR waves following from (13) and (18)has the following view:(29) R(1)(s)PXR /R(1)(s)FPXR =(ξ(s) −√ξ(s)2 − ε)2.As it follows from comparison of (26) and (29) in the last case the FPXR yield muchmore exceeds the PXR yield for small values of ε. So under decrease of ε it is the firstbranch of excited X-ray waves which makes the main contribution to the radiation andit leads to the increase of FPXR yield.6. – ConclusionBased on two-wave approximation of the dynamic scattering theory analytical ex-pressions for the spectral-angular distribution of FPXR and PXR in Bragg scatteringdirection are derived in the general case of asymmetric reflection for Bragg scatteringgeometry.The analyses of the obtained expressions have shown that at a fixed angle betweenrelativistic electron velocity and the system of diffracted atomic planes θB and length ofelectron path in the crystal plate 2b(s) the ratio of the PXR and FPXR yields dependssignificantly on the angle between reflecting atomic planes and the inlet surface of theplate, i.e. on reflection asymmetry. The decrease of the asymmetry parameter ε =sin(δ − θB)/ sin(δ + θB) (increase of the angle δ + θB of the electron incidence on thecrystal plate) results in the decrease of PXR spectral-angular density and the increase ofFPXR density which can exceed the density of PXR when the parameter ε value becomessmall enough.REFERENCES[1] Ter-Mikaelian M., High-Energy Electromagnetic Process in Condensed Media (Wiley,New York) 1972.[2] Garibian G. and Yang C., J. Exp. Theor. Phys., 61 (1971) 930.[3] Baryshevsky V. and Feranchuk I., J. Exp. Theor. Phys., 61 (1971) 944.[4] Garibian G. and Yang C., J. Exp. Theor. Phys., 63 (1972) 1198.[5] Baryshevsky V. and Feranchuk I., Phys. Lett. A, 57 (1976) 183.[6] Baryshevsky V. and Feranchuk I., J. Phys. (Paris), 44 (1983) 913.[7] Yuan Luke C., Alley P., Bamberger A. et al., Nucl. Instrum. Methods Phys. Res. A,234 (1985) 426.[8] Kalinin B., Naumenko G., Padalko D. et al., Nucl. Instrum. Methods Phys. Res. 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B, 201 (2003) 67.[15] Kubankin A., Nasonov N. and Noskov A., Proceedings of 7 International Russian-Japanese Symposium on Interaction of fast charged particles with solids, Nov. 24-30, Kyoto,Japan (2002), pp. 217-225.[16] Blazhevich S. and Noskov A., Nucl. Instrum. Methods Phys. Res. B, 252 (2006) 69.[17] Blazhevich S. and Noskov A., Nucl. Instrum. Methods Phys. Res. B, 266 (2008) 3770.[18] Blazhevich S. and Noskov A., Nucl. Instrum. Methods Phys. Res. B, 266 (2008) 3777.[19] Blazhevich S. and Noskov A., Russ. Phys. J., 50 (2006) 574.[20] Blazhevich S. and Noskov A., J. Phys.: Conf. Ser., 236 (2010) 012013.[21] Pinsker Z., Dynamic Scattering of X-rays in Crystals (Springer, Berlin) 1984.[22] Bazylev V. and Zhevago N., Emission From Fast Particles Moving in a Medium andExternal Fields (Nauka, Moscow) 1987.

On the ratio of the relativistic electron PXR in Bragg and forward direction within Bragg scattering geometry

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On the ratio of the relativistic electron PXR in Bragg and forward direction within Bragg scattering geometry

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