An exhaustive description of the parametric X-ray (PXR) generation by relativistic electrons in a crystal excited by an acoustic wave is proposed. A principal question as to whether it is possible to enhance the PXR yield under these conditions is consideredyesBelgorod State Universit

Ladnykh, M. S.

Mkrtchyan, A. G.

Mkrtchyan, A. R.

Nasonov, N. N.

Zhukova, P. N.

DSpace at Belgorod State University

Effect of Acoustic Wave on the Parametric X-ray Generation by Relativistic Electrons in a CrystalP. N. Zhukova, M. S. Ladnykh, A. G. Mkrtchyan, A. R. Mkrtchyan, and N. N. Nasonov*Belgorod State University, Belgorod, 308007 Russia Institute of Applied Problems in Physics, National Academy of Sciences of Armenia,Abstract—An exhaustive description of the parametric X-ray (PXR) generation by relativistic electrons in a crystal excited by an acoustic wave is proposed. A principal question as to whether it is possible to enhance the PXR yield under these conditions is considered.Increasing interest in the analysis of high-energy electromagnetic processes in oriented crystals under the action of external factors (e.g., acoustic waves) is related to the possibility of controlling these processes. In particular, the effect of acoustic waves was studied with respect to the coherent bremsstrahlung radiation [ 1, 2], coherent electron—positron pair production in a crystal [3, 4], diffraction of free photons [5, 6], X-ray transition radiation [7], and parametric X-ray (PXR) generation [8—13].Our previous experimental investigations [12, 13] showed an increase in the PXR yield with the acoustic wave power, but this phenom enon has not been given proper theoretical explanation so far. Polikarpov and Skadorov [10, 11] theoretically studied fine dynamic features of the resonant action of acoustic waves on the PXR characteristics, which are manifested only within narrow spectral and angular intervals and, hence, can­not be used to explain the observed PXR yield enhancem ent [12, 13].In this Letter, we propose a simple kinematic approach that provides an adequate description of the PXR generation by electrons with energies on the order of up to tens megaelectronvolts. The obtained formulas describe PXR emission in a broad range of angles and frequencies and, in particular, provide a simple explanation of the experimentally observed effect [12, 13].Let us consider the PXR generation by a relativistic electron that moves in a crystal with velocity V. The Maxwell equation for the Fourier image of the electro­magnetic field excited in the medium is as follows:(,k2 -  co2)Efflk -  k(kEffik) + ^ k ' G ( k' -  k )E fflk,= ^ V e x p ( - / k 'r 0)5(c« -k 'V ),2n (1)G =  e x p ( /(k '-k ) )r„2tz m ^ k ' - k )  R  ;where the last term  in the left-hand part of the equa­tion describes the induced current of electrons in the medium within the high-frequency approximation ((£>> I, I  being the average ionization potential of atoms in the target), which allows the atomic frequen­cies to be ignored in comparison to co in the energy- containing denominators of the formula for atomic polarizability [14] and the asymptotic formula (1) to be used; Z  is the num ber of electrons in the atom, R  is the radius of electron screening of the atomic nucleus; and (i is a function that describes the individual con­tributions of all target atoms with coordinates rhFor the subsequent analysis, it is convenient to rep­resent function (i as a sum of the average (refraction) and fluctuation (scattering) components as follows:G = G  + G, G = (G)ri = co26 ( k ' - k ) ,  (2)where co/; is the plasma frequency. Substituting expres­sion (2) into Eq. (1) and solving the obtained equationby iterations with respect to powers of G (which yields a well-known formula for the Coulomb field of a fast electron in a medium for the zero-order approxima­tion and an expression for the radiation field in the next approximation), we arrive at the followingshv \ / e\  q>/2 + 9’ «P/2 - yg hd NJcoJQ2 2 = e co \:d lk. d ik 1+ 2coVenk, k] + 2coVênk, x < è(k ,)f;* (k2))<exp(/(k| - k 2)r0))x ( V -  —  -  nf n V -  ^ Y ) f v  -  —  -  nf « V -  — Y)v m K  v f i 'iK - '/ v  n ip  V ri i e 7 /tron on a perturbed crystallographic plane determined by vector g is as follows:co-dN, «2 2 4e co co»dldadQ.(5)Fig. 1. Geometry ol‘PXR emission: (S) rellecting crystal­lographic plane set by the reciprocal lattice vector g; ((p) preset PXR emission angle; (O’) orientation angle deter­mined using a goniometer; (9||) component of the two- dimensional observation angle 0 (n = e(l 02/2) + 0, e0 = 0), which is lying in the scattering plane; (£||, ijj.) components of the wave vector of the acoustic wave.expression for the spectral and angular distribution of the emitted radiation photons:„ (V -g ,/c o e )2-(n V -n g ,,/c o s):(g; + 2coVeng/,)‘x S(co( 1 -  J i n \ )  -  g^V),where gp = g -  p \  and co2 =  co2e x p (-# V /2 )( l  + S2* 2) 1.For the subsequent analysis, we introduce the vari­ables On and 0^, which describe the angular distribu­tion of X-ray photons, and the angle 0' that describes the angular dependence o f the PXR yield. The geom­etry o f PXR emission is depicted in Fig. 1. In the new variables, formula (5) eventually appears as follows:d N  e2(alco------4—  « — 4dtdwdCl  Kg2 Z^(ga)0 2 + (0|| + 20' -  2 p ^ j cos'cp( p 2 + 0 21 + (o ||+2O’- 2 / 7 ^ y ) 2(6)( f( 3 )  X Ô CC3l3 1 +IOn + O1- PSvgeosc o t^2JJx 8(co(l -  VenV) -  k,V)S(co( 1 -  VenV) -  k2V).Here, the angle brackets imply averaging over the posi­tions of atoms in the target and the radius r0 that deter­mines the electron trajectory within the crystal lattice cell; n is the unit vector in the direction of radiationpropagation; e(co) =  1 — co2/co2 is the ordinary dielec­tric permittivity in the X-ray spectral range, which is given by Eqs. (1) and (2) and condition co > I.The averaging is performed using the following for­mula:r, = R,+ u, + asin(^R ;), (4)where R, is the equilibrium coordinate of an atom (evi­dently, this sequence is periodic), u, is the thermal atomic displacement, and a and £ are the acoustic wave amplitude and wave vector, respectively. The result o f averaging is conventionally presented as a sum of the incoherent and coherent components. The coherent component that is of interest in this consid­eration represents a sun of independent terms corre­sponding to various vectors of the reciprocal lattice of the crystalline target. The term  that describes the coherent diffraction of the Coulomb field of an elec-where £||Cos^ — ^jsin^ is the com ponent o f thewave vector of the acoustic wave, which is parallel to the electron velocity, coB is the Bragg frequency, p2 =y 2 4- co2/co2, and y is the Lorentz factor o f electron. Note that the simple kinematic formula (6) is valid in a broad range o f electron energies my < my* =/nco,j/cop < 100 MeV 115].Now let us consider the obtained expression and use it for a qualitative interpretation of the experimen­tal results 112, 131 (unfortunately, the available data are insufficient for a quantitative description). An analysis o f Eq. (6) shows that the aforementioned PXR yield enhancement is related to the bending oscilla­tions o f the reflecting plane (only the component of the acoustic wave amplitude that is perpendicular to this plane enters into Eq. (6)). Naturally, it is the com ­ponent \  parallel to the reflecting plane that leads to the angular modulation o f emitted radiation, since the component does not lead to bending o f the reflect­ing plane.According to Eq. (6), the periodic bending o f the system o f parallel atomic planes determined by vectorF(Q), a.u. 0.3 rP(Qd), a.u.0.20.11: :-4 -2 2 4 6x =  20’/pFig. 2. Orientation dependence of the angular distribution of the PXR photon density (/) for unperturbed crystal (y = ga = 2.9, z = 2^|i/gp = 0) and (2) in the presence of acoustic wave (y = ga = 2.9, z = 2&i/gp = 0.6).w =  0D/pFig. 3. Dependence of the PXR yield on the collimator size w for ( 1) unperturbed crystal (y = ga = 0, z = 24||/gp = 0) and (2) in the presence of acoustic wave (y = ga = 2, z = 2 |^| /gp = 0-5). The inset shows behavior of the PXR yield on a greater scale.g can be considered as replacement of one atomic plane by a series of lower-density planes rotated rela­tive to each other. Then, the final angular distribution of the emitted PXR photons consists of a set of reflec­tions with the same structure and various amplitudes, which are shifted relative to each other. Apparently, the angular density of radiation can be significantly changed under the action of the acoustic wave, but the total PXR yield integrated over angles and frequencies coincides with the yield from the unperturbed crystal (this state­ment follows from the condition yV ^(ga) — 1).pIn order to illustrate the laws under consideration, let us analyze the orientation dependence of a strictly collimated PXR beam (Gy, 9± < p). In this case, Eq. (6) yieldsdNgdtdQ.(x - p z )(l + ( x - p z Ÿ Ÿ(7)acterized by the reciprocal lattice vector g, while par­ticular cases are specified for calculations of the amplitude a and wave vector £ of the acoustic wave). Note also that the parameters y  = ga and z  = 2^||/gp of these curves are quite accessible. For example, with the lattice parameter I = 5 x 10~8 cm, acoustic wave­length X — 5 x 10-5 cm, and electron energy 100 MeV, the wave amplitude must be a ~ 0.51 to provide for the plane bending by an angle as small as 0.5 mrad.Another interesting issue is the dependence of the PXR intensity on the collimator size 9fl. Assuming for the simplicity that cp <1 1 in Eq. (6) and integrating this equation with respect to the frequencies and angles of observation, we eventually obtain the following for­mula:where x= 20'/gp, y  =  ga, and z  =  2 ^ / g p .  Figure 2 shows plots of the function F calculated using formula (7) for different values of parameters. These curves demonstrate the possibility of a sharp change in the angular distribution of PXR photons under the action of a control acoustic wave. In particular, it is possible to provide for a significant increase in the radiation yield as measured by a detector with a small angular aperture (0D < p), which is oriented in the direction of Bragg scattering. This possibility explains the results obtained in [12, 13].It should be noted that function F in Eq. (7) and the curves in Fig. 2 are universal (the crystal is only char-1 (A(w)  ^(0)> 12\B(w) B(0y B(w) 5(0). (8)A(w) -  \ + w 2- A p 2z ,B(w) = 7 ( 1+ w2 -4p2z") + 16p1 z ,where w = 0D/P- The curves plotted in Fig. 3 show that an increase in the PXR yield is only possible provided that the collimator size does not exceed the width of the characteristic dip in the angular distribution of the PXR photon density for an unperturbed crystal. Therefore, the presence of this dip is a necessary con­dition for manifestation of the PXR yield enhance­m ent under the action of acoustic waves [12, 13].The inset to Fig. 3 presents the P(w) curves con­structed in a broader range o f variation of the argu­ment, which show that (in agreement with the above assumptions), the PXR yield into a sufficiently wide collimator is independent of the acoustic wave param ­eters.Thus, the above analysis provides an exhaustive description of the effect of acoustic waves on the PXR generation by relativistic electrons in a crystal, which is based on simple kinematic formula (6). It is dem on­strated that the total PXR yield is independent of the wave parameters, but the yield into a limited collima­tor can be significantly changed under the acoustic wave action, which explains the experimentally observed effect [12, 13].Acknowledgments. The authors are grateful to the referee for constructive remarks.This study was supported by the Russian Founda­tion for Basic Research (project no. 09-02-97528) and the Ministry o f Education and Science of the Russian Federation (project nos. G K  02.740.11.0545 and GK P2317).REFERENCES1. A. A. Saharian, A. R. Mkrtchyan, V. V. Parazian, and L. Sh. Grigoryan, Mod. Phys. Lett. A 19, 99 (2004).2. N. F. Shulga, V. V Boiko, Pis’ma Zh. Eksp. Teor. Fiz. 84, 366 (2006) [JETP Lett. 84, 305 (2006)].3. A. R. Mkrtchyan, A. A. Saharian, L. Sh. Grigoryan, and B. V Khachatryan, Mod. Phys. Lett. A 18, 2581 (2002 ).4. V V. Parazian, J. Phys. Cond. Matter 21, 185 401 (2009).5. I. R. Entin, Zh. Eksp. Teor. Fiz. 77, 214 (1979) [Sov. Phys. JETP 50, 110 (1979)].6. A. V. Kolpakov and Yu. P. Khapachev, Kristallograflya 18, 474 (1973) [Sov. Phys. Crystallogr. 18, No. 3 (1979)].7. L. Sh. Grigoryan, A. H. Mkrtchyan, and A. A. Sahar­ian, Nucl. Instr. Meth. Phys. Res. B 145, 197 (1998).8. V. G. Baryshevskii and I. V. Polikarpov, Zh. Eksp. Teor. Fiz. 94, 109 (1988) [Sov. Phys. JETP 67 (1988)].9. A. R. Mkrtchyan, M. A. Aslanyan, A. H. Mkrtchyan, and L. Sh. Grigoryan, Phys. Lett. A 152, 293 (1991).10. I. V. Polikarpov and V. V. Skadorov, Phys. Status Solidi (b) 143, 11 (1987).11. I. V. Polikarpov and V. V. Skadorov, Izv. Akad. Nauk BelSSR, Ser. Fiz.-Mat. Nauk, No. 3, 83 (1988).12. A. R. Mkrtchyan, H. A. Aslanyan, A. H. Mkrtchyan, et al., Solid State Commun. 74, 187 (1991).13. A. R. Mkrtchyan, A. G. Mkrtchyan, A. A. Aslanyan, et al., Izv. Nats. Akad. Nauk Arm., Fiz. 47, 282 (2005).14. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Allowance for Spatial Dispersion (Nauka, Mos­cow,1979) [in Russian],15. N. Nasonov and P. Zhukova, Rad. Phys. Chem. 75, 1409 (2006).

Effect of acoustic wave on the parametric X-ray generation by relativistic electrons in a crystal

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Effect of acoustic wave on the parametric X-ray generation by relativistic electrons in a crystal

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