The possibilities of using broadband X rays for diagnostics of the atomic structure of solids are considered. It is proposed to use virtual photons of the Coulomb field of a relativistic electron beam as primary probe radiationye

Astapenko, V. A.

Gostishchev, N. A.

Kubankin, A. S.

Nasonov, N. N.

Zhukova, P. N.

DSpace at Belgorod State University

 ISSN 1062-8738, Bulletin of the Russian Academy of Sciences: Physics, 2008, Vol. 72, No. 6, pp. 863–866. © Allerton Press, Inc., 2008.Original Russian Text © V.A. Astapenko, N.A. Gostishchev, P.N. Zhukova, A.S. Kubankin, N.N. Nasonov, V.I. Sergienko, V.A. Khablo, 2008, published in Izvestiya Rossiiskoi Aka-demii Nauk. Seriya Fizicheskaya, 2008, Vol. 72, No. 6, pp. 912–915. 863 Modification of the EDXD Method for Diagnostics of Polycrystalline and Fine-Grained Media V. A. Astapenko a , N. A. Gostishchev b , P. N. Zhukova b , A. S. Kubankin b , N. N. Nasonov b , V. I. Sergienko c , and V. A. Khablo c a  Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700 Russia b  Belgorod State University, ul. Pobedy 85, Belgorod, 308015 Russia c  Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russiae-mail: nnn@bsu.edu.ru Abstract —The possibilities of using broadband X rays for diagnostics of the atomic structure of solids are con-sidered. It is proposed to use virtual photons of the Coulomb field of a relativistic electron beam as primaryprobe radiation. DOI:  10.3103/S1062873808060312 1.  The conventional approach to the diagnostics ofthe atomic structure of matter is based on measurementof the angular distribution of quasi-monochromaticX rays scattered by an object studied. Advances indevelopment of X-ray detectors made it possible to pro-pose a new approach, which is based on measurementof the spectral distribution of broadband X rays scat-tered at a fixed angle [1]. This approach, known as theenergy dispersive X-ray diffraction (EDXD) method,can be modified by replacing the primary flux of freeX-ray photons with virtual photons of the Coulombfield of relativistic electrons transmitted through a sam-ple under study. The modified EDXD method has thefollowing advantages: broadband spectrum of virtualphotons, possibility of studying small portions usingfocused electron beams, high sensitivity to interatomiccorrelations, and possibility of using an X-ray detectorin a fixed position.It is noteworthy that the radiation that is due to thescattering of the Coulomb field of a fast electron duringits collision with an atom is well known as polarizationbremsstrahlung [2–4]. Polarization bremsstrahlung hasbeen investigated in detail for the collision of fast parti-cle with an isolated atom [4]. At the same time, study ofthe collective contribution of the atoms of a medium tothe formation of the polarization bremsstrahlung yieldhas only begun [5]. The purpose of this study is to the-oretically and experimentally analyze the possibility ofusing the coherent component of the polarizationbremsstrahlung of relativistic electrons in diagnosticsof the structure of polycrystalline and fine-grainedmedia. 2.  The Maxwell equations yield the following rela-tion for the Fourier components of an electromagneticfield: (1) Here,  is the velocity of a fast electron and  is theFourier transform of the induced current density of theelectrons of the medium, defined as the mean of theconventional quantum-mechanical operator of the cur-rent density. Generally, the expression for  has theform (2) where the tensor  G lp  is generally cumbersome [6]. Inthe case under consideration (motion of a fast electronthrough a polycrystal), the main mechanism of Cou-lomb field scattering is the Bragg scattering from thelattice, which generally occurs in the frequency range ω   ≈  10  keV. This value of  ω  significantly exceeds theelectron binding energy in light atoms; therefore, theatomic electrons can be considered as free during thegeneration of radiation in the atom. Under these condi-tions, the tensor  G lp  takes a simple form [7]: (3) Here,  F (  – )  is the atomic form factor and  isthe coordinate of the  j th atom.k2 ω2–( )Eωk k k Eωk⋅( )–=  4πiωJeωkiωe2π2--------Vδ ω kV–( ).+V JeωkJeωkJeωk l, d3k 'Glp k k ',( )Eωk ' p,∫p∑=  4πiωGlp k k ',( )=  δlpe22π2m------------ ei k k '–( )r jF k ' k–( )j∑– δlpGlp k ' k–( ).–≡k ' k r j 864 BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS        Vol. 72        No. 6        2008 ASTAPENKO et al. Separating  G (  – )  into the mean and random com-ponents,  G  =  + ,  =  〈 G (  – )  =  δ (  – ) ,where  ω p  is the plasma frequency of the medium, weobtain the equation (4) which is convenient to solve by iterations (thisapproach is admissible in view of the small size of themicrocrystallites forming a polycrystal). Here,  ε ( ω ) = 1 –/ ω 2  is the conventional permittivity in the X-rayregion. In the zero-order approximation with respect to, Eq. (4) yields an expression describing the Cou-lomb field, which is screened due to the polarization ofthe electrons of the medium. In the first-order approxi-mation with respect to , we obtain the desired radia-tion field (generated as a result of scattering of Cou-lomb virtual photons): (5) The radiation amplitude  and the spectral angular distribution of the radiation intensity are derived from (5)by the standard methods (6) where  n   is the unit vector in the radiation direction andthe angular brackets  〈   〉  mean averaging over the coor-dinates of target atoms. 3.  As a result of averaging of the squared modulus ofthe amplitude over the atomic coordinates in a microc-rystallite and over the random orientations of microc-rystallites, we obtain from (6) the final formula for thecoherent component of the radiation intensity, (7) where   =  V cos θ ,  S ( )  is the unit-cell structure fac-tor,  is the Debye–Waller factor, g is the reciprocallattice vector, and na is the atomic density. The contri-bution of the incoherent component of radiation is sup-pressed and does not exceed several percent (7). Thefunction (ω, θ) in (7) has a cumbersome form and isomitted here. In the case of relativistic electron ener-gies, the spectrum of scattered photons contains sharppeaks with amplitudes(8)and relative width(9)These peaks arise at the frequencies(10)which are the Bragg frequencies for a fixed observationangle θ and a given reciprocal lattice vector g (the con-tribution to the radiation yield in the vicinity of ωg isformed mainly by the microcrystallites that are in theposition of specular reflection of virtual photons of theCoulomb field of fast electrons). Here, γ is the electronLorentz factor.Formulas (7)–(10) indicate the possibility of simul-taneous determination of many reciprocal vectors of thepolycrystal lattice from measurement of the spectrumof the scattered Coulomb field of a fast electron. Thispossibility was experimentally verified in [5]. Figure 1shows the measured and calculated curves of the spec-tral distribution of the polarization bremsstrahlung(collimated at an angle of 90°) of 7-MeV electronspassing through a copper film. Note good agreementbetween the measurement results and theoretical pre-k ' kG G˜ G k ' k 〉r j ωp2 k ' kk2 ω2ε ω( )–( )Eωk+ d3k 'G˜ k ' k–( ) Eωk ' kkEωk 'ω2ε ω( )-----------------–⎝ ⎠⎛ ⎞∫=  iωe2π2-------- V kωε ω( )---------------–⎝ ⎠⎛ ⎞ δ ω kV–( ),ωp2G˜G˜EωkS iωe2π2--------1k2 ω2ε–-------------------d3k 'k '2 ω2ε–-------------------- G˜ k ' k–( ) V k 'ωε------– kkV kk '/ωε–ω2ε------------------------------–⎝ ⎠⎛ ⎞ δ ω k ''V–( ).∫–=AnωdNdtdωdΩ-------------------- An2〈 〉 ,=An iωed3kk2 ω2ε ω( )–---------------------------- G˜ k ω εn–( ) V kωε------– n nV nkωε------–⎝ ⎠⎛ ⎞–⎝ ⎠⎛ ⎞ δ ω kV–( ),∫–=ωdNdtdωdΩ-------------------- ωdNgdtdωdΩ-------------------- ,g∑=ωdNgdtdωdΩ-------------------- AgΦg ω θ,( ),=Agπe6m2-------na2g3---- S g( ) 2F2 g( )e g2uT2–,=nV geg2uT2–ΦgΦg max2γ θ/2( )tan1 γ 2ωp2 /ω2+---------------------------------≈∆ωω-------1 γ 2ωp2 /ω2+2γ θ/2( )tan---------------------------------.≈ω ωggV2 1 εV θcos–( )--------------------------------------------g2 θ/2( )sin----------------------- ,≈= =BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS      Vol. 72      No. 6      2008MODIFICATION OF THE EDXD METHOD 865dictions, not only in the positions of coherent peaks butalso in their amplitudes.Note also the very strong influence of density (fol-lowing from (8)), which leads to saturation of the radi-ation yield at sufficiently high energies of probe elec-trons: γ > γ∗ = ωg/ωp (generally, ωg ≈ 5–10 keV and ωp ≈20–50 eV; therefore, γ∗ ≈ 100–500). These estimatesshow that electron beams with energies not higher than100 MeV are most appropriate for the diagnosticmethod under consideration.4. When a fast electron moves through a fine-grained medium, the mechanism of Bragg scattering ofthe Coulomb field is not implemented due to the smallgrain size. Under the conditions considered here, themain factor is the coherent scattering from grains,which is most effective in the range of wavelengthsexceeding the grain size. In this case, the dipole approx-imation is more appropriate to calculate the tensor Glpin (3). Within this approximation, expression (2) givesthe formula(11)where α0(ω) is the conventional dipole polarizability ofan atom, ωn0 = En – E0 is the difference of atomic ener-gies in the |n〉 and |0〉 states, fn0 is the oscillator strengthof the |0〉 → |n〉 transition, and Z is the number of elec-trons in an atom.Further calculations are similar to those reportedabove. In particular, general formula (6) remains valid,in which one has to make the replacement ε(ω) → εd(ω) =1 + 4πnaα0(ω) ≡ 1 + χ(ω). When necessary averagingsare performed in general formula (6), one has to takeinto account the grain size finiteness. The calculationsgive the formula(12)where the function FN(ω) (omitted here because of itscomplexity) describes the coherent component of thepolarization bremsstrahlung from grains (in this study,we considered grains in the form of a cube with Natoms on a side and a distance a between atoms). Thefunction ω4|χ(ω)|2 can be considered as known, becausethe dielectric susceptibilities of many materials havebeen experimentally determined in a wide frequencyrange. Obviously, this function becomes zero at ω → 0and tends to a constant ( ) with an increase in ω. Atthe same time, the analysis performed showed that4πiωGlp k k ',( ) = δlp ω22π2--------α ω k ', k–( ) ei k ' k–( )r j,j∑–α ω k ', k–( ) α0 f k ' k–( ),=α0e2m----f n0ω2n0 ω2–----------------------, f k ' k–( )n∑ F k ' k–( )Z----------------------,= =ωdNdtdωdΩ--------------------e24 2π( )4na--------------------- ω4 χ ω( ) 2FN ω( ),=ωp4FN(ω) is a decreasing function of ω; at the same time,this function increases with an increase in N and shiftsto small values of ω (this effect is quite natural becausethe coherent response of a scattering object shifts tolower frequencies with an increase in the object sizes).The most important property of distribution (12) is thesharp dependence of the shape of this distribution on N,i.e., on the grain size. Figure 2 shows as an illustrationdN/dω dΩ, 10–6 keV–1 sr–1ω, keV6543210 2 3 4 5 6 7 8Fig. 1. Spectra of collimated polarization bremsstrahlungfrom a copper foil, obtained by absolute measurements (cir-cles) and calculated from formula (7) (solid line) with thefollowing parameters: electron energy 7 MeV, emissionangle θ = 90°, and foil thickness 15 µm; the electron veloc-ity vector makes an angle of 45° with the foil surface.(ωa)4FN(ωa)ωa200 0.2 0.4 0.6 0.815105123Fig. 2. Model spectrum of collimated polarizationbremsstrahlung from a fine-grained medium. The quantityω4a4FN(ωa) was calculated at N = (1) 6, (2) 8, and (3) 10and fixed parameters ρ = γ–1 = 0.04, R/a = 0/1, and θ = 90°;the mutual grain screening was disregarded. 866BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS      Vol. 72      No. 6      2008ASTAPENKO et al.the dependences ω4FN(ω) for different N, which disre-gard mutual screening of grains. The plotted curvesdemonstrate a clear dependence of the spectrum on thegrain size.5. The analysis performed here shows good pros-pects of using the polarization bremsstrahlung of rela-tivistic electrons in solution of the following diagnosticproblems: (i) establishment of the structure of theatomic lattice of microcrystallites in polycrystallinesamples and (ii) determination of the grain sizes in fine-grained media.ACKNOWLEDGMENTSThis study was supported by the program “Develop-ment of the Higher School Scientific Potential” of theMinistry of Education and Science of the Russian Fed-eration (project RNP.2.1.1.3263) and the Russian Foun-dation for Basic Research, project nos. 07-02-12226and 06-02-16942.REFERENCES1. Waseda, Y., The Structure of Non-Crystalline Materials,New York: McGraw-Hill, 1980.2. Buimistrov, V.M., Ukr. Fiz. Zh., 1972, vol. 17, p. 640.3. Polyarizatsionnoe tormoznoe izluchenie chastits i ato-mov (Polarization Bremsstrahlung of Particles andAtoms) Tsytovich, V.N. and Oiringelya, I.M., Eds., Mos-cow: Nauka, 1987.4. Korol’, A.V., Lyalin, A.G., and Solov’ev, A.V., Pol-yarizatsionnoe tormoznoe izluchenie (PolarizationBremsstrahlung), St. Petersburg: Izd-vo SPbGPU, 2004.5. Astapenko, V.A., Kubankin, A.S., Nasonov, N.N., et al.,Pis’ma Zh. Eksp. Teor. Fiz., 2006, vol. 84, no. 6, p. 341[JETP Lett. (Engl. Transl.), vol. 84, no. 6, p. 281].6. Afanas’ev, A.M. and Kagan, Yu., Acta Crystallogr., Sect.A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr., 1968,vol. 24, p. 163.7. Garibian, G.M. and Yang, S., Zh. Eksp. Teor. Fiz., 1972,vol. 61, p. 944 [Sov. Phys. JETP (Engl. Transl.), vol. 34,p. 502].

Modification of the EDXD method for diagnostics of polycrystalline and fine-grained media

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Modification of the EDXD method for diagnostics of polycrystalline and fine-grained media

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