Parametric X-ray radiation from relativistic electrons moving in a crystal is theoretically investigated in Bragg geometry. It is shown that the effect of anomalous photoabsorption can manifest itself within this geometry of the scattering of the pseudophoton field of a fast particleye

Gary, C. K.

Kaplin, V. V.

Nasonov, N. N.

Piestrup, M. A.

Uglov, S. R.

Physics of Atomic Nuclei

DSpace at Belgorod State University

Effect of Anomalous Photoabsorption on Parametric X-ray Radiationfrom Relativistic ElectronsC. K. Gary1), V. V. Kaplin2), N. N. Nasonov*, M. A. P iestru p 1), and S. R. Uglov2)Belgorod State University, Studericheskaya ul. 12, Belgorod, 306007 RussiaReceived November 4, 2000Abstract—Parametric x-ray radiation from relativistic electrons moving in a crystal is theoretically investigated in Bragg geometry. It is shown that the effect of anomalous photoabsorption can manifest itself within this geometry of the scattering of the pseudophoton field of a fast particle. In this case, the angular distribution of the radiation changes significantly, while the total radiation yield can increase by a factor of 31. INTRODUCTIONThe effect of anomalously low absorption (Borr- mann effect [1]) can occur in the coherent scatter­ing of x rays in a crystal. Theoretical investigations of a similar phenomenon—parametric x-ray radia­tion, which consists in the coherent scattering of the pseudophoton field of a fast particle moving in a crystal— revealed that there is no Borrmann effect in parametric x-ray radiation [2, 3]. It should be noted that the results presented in [2, 3] were obtained for parametric x-ray radiation in Laue geometry without taking into account the contribution to the formation of the radiation yield from transition radiation emitted by a fast particle at the entrance surface of a crys­tal. A more detailed analysis showed that anomalous photoabsorption can yet occur in parametric x-ray radiation owing to the Bragg diffraction of the afore­mentioned transition radiation from a fast electron at the entrance surface of the crystal [4].Of considerably greater interest is nonetheless the possible manifestation of the Borrmann effect in the scattering of the pseudophoton field of a fast parti­cle, since the yield of this process is proportional to the target thickness, in contrast to the yield of the diffracted transition radiation.The objective of this study is to analyze in detail parametric x-ray radiation generated by relativistic electrons in a semi-infinite absorbing crystal. In contrast to what was done in [2, 3], we consider here Bragg geometry. Our basic result is the prediction of the Borrmann effect in parametric x-ray radiation under the conditions of the present analysis.^Adelphi Technology, Palo Alto, USA.2)N uclear Physics Institute, Tomsk Polytechnic University,pr. Lenina 2a, Tomsk 50, 634050 Russia, e-mail: nnn@ bsu .edu .ru2. GENERAL RELATIONSLet us investigate the structure of the electromag­netic field excited by a relativistic electron moving from a vacuum and entering a crystal that is charac­terized by the periodically varying dielectric permit­tivity £ (u, r) =  1 +  Xo (w) +  E g  X g M eig'r, where g is a set of reciprocal-lattice vectors of the crystal. We assum e that the crystal thickness exceeds the pho­toabsorption length; this makes it possible to consider the crystal as a semi-infinite one.In order to find the Fourier transform of the excited electric field, Ewk =  f  dtd3r xexp (—ik  ■ r  +  i u t ) E (r, t), we will make use of the conventional Maxwell equations{h2 -u o 2) Ewk- k ( k - E wk) - W2x°E wk ( 1) /- w 2 ^ 2  X -g ^k+ g  = (w -  k • v ) ,9where v is the velocity of the radiating electron.Since the susceptibilities satisfy the condition xo,<C 1 in the x-ray region, Eq. (1) can be solved within the well-known two-wave approximation of the dynamical theory of diffraction [5]. By consider­ing that the components Ewk and Ewk+g- are virtually transverse in the x-ray range of frequencies for rela­tivistic particles [6], we can straightforwardly reduce Eq. (1) to the simple set of equations( k 2 - u 2 -  u 2x o )  E X o  -  L 02X - g O L \ E X g  (2)( ( k +  g)2 -  w2 -  w2%o) E Xg = u 2X g a \E Xo,GARY et al.lu e  u  Xga \ CAO -VFig. 1. Geometry of scattering.where the as-yet-undefined quantities are given by 2 2=  'y '  £\oE\o, E^k+g- =  'y '  e\gE \g ,  A = 1  A= 1[k || x e*] [k x eio]eio — eig — ----- --------, e 2o —e 2gk  y ’ k[k +  g x eio](3)27r2 Ap2 \vx \ ( £ - £ i ) ( £ - & ) H C - C o ) ,2 pE\o = —r -1-—  ( £ -  ^“ Xo E \ su 2Xg®\ \  2p 1 *LÜrpv _AO —iuje 1 e A0 • v27r2 2p l^ ail A  — £ M e - & )where6,2 =  -  I A  ± A -  — Xo ) - P i  I , (8)Co =  — (w -  k||vxlk +  sla i  =  l, a 2 = k ( k +  g) / k  |k +  g | , k =  k|| +  exkx , ex ■ ky =  0.Equations (2) describe the field in the target. The corresponding equations for this field in a vacuum (beyond the target),(k2 -  u 2) E vxo =  | ^ e Ao • v6 (w -  k • v ) , (4){ k l - u 2) E l g = 0,follow from (2) in the limit xo = Xg = 0- Here, we have kg =  k +  g. Below, we assum e that the reflecting crystallographic plane is parallel to the crystal sur­face. The reciprocal-lattice vector g is then parallel to the normal to the crystal surface (see Fig. 1).The radiation field is determined as a solution to the corresponding equation in (4). The result isE vXg = a Ak|| $ (kgx -  p ) , p = ^/w 2 -  £ |.  (5)In order to determine an unknown coefficient aAky,it is necessary to find solutions to the remainingequations in (2) and (4) and to employ conventional boundary conditions for the electromagnetic field at the target surface. Considering that, in the x-ray frequency range, the photon wave vector in a vacuum differs insignificantly from that in a crystal, we intro­duce the variable £ through the relationk g x = P  + £, (6)By using relation (6), we can represent solutions to Eqs. (2) and (4) in the formEx g = bx k 5 ( Z - &  (7)+  pvx) +  A,In the case of the Bragg geometry of scattering (and we consider precisely this case), the quantity {* is determined by the relations,2 N 26 f o r ( A - - XoJ < I 32X  6  for ( a  -  y  Xo) > f % ,  a  -  y X o  > o6  for A - - x o  > P l , A - - Xo < 0 .  \  V )  V (9)By using Eqs. (5) and (7) and the boundary conditionsJ d a E x o ~ E l 0) = J d £ ( E Xg- E» Xg) = 0, (10)we find that the coefficient aAky is given by the expres­sionO-AknILüe LÜ Xga X e A0 • v ' 2TT2 4p2 \vx \ 6 - ÿ x oCo -  C* i( i i ;,(£o -  6 ) (Co -  6 ) C o - A / which completely describes the properties of the radi­ation field.3. CONTRIBUTION O F PARAMETRIC X RAYS: EFFEC T O F ANOM ALOUS PH O TO A BSO RPTIO NIn order to find the spectral and angular distri­bution of the radiation in question, we make use of the general expression (11). In order to determine the radiation amplitude A \ ,  we apply the stationary- phase method to compute the Fourier integralE r.d = A \ ( 12)A \  = - 2 m u n xa \un,EFFECT OF ANOMALOUS PHOTOABSORPTIONwhere n =  ny +  exn x is a unit vector in the direction of radiation, • ny =  0.For the purposes of the ensuing analysis, it is convenient to define the angular variables 6 and through the relations (see Fig. 1)ei 1 - ^ 7  2 -  ) +  ^  ei  • =  0,n =  e 2e 2 • e  = 0i  -  - e2 2ei  • t 2 = cos fand the dielectric susceptibilities xo and asXoCJn . // 2 +  %  ,u j z(13)(14)n<UJXg — X-g  2 + lXe ,UJZ ë^0  . "where uo is the plasmon frequency and w2 =Woe_s,2-u2 (F  (g) / Z) ,  u, F  (g), and Z  being, respec­tively, the root-m ean-square amplitude of thermal vibrations of the atoms, the atomic form factor, and the number of electrons in an atom; we consider here a reflection for which the lattice structure factor is equal to unity.From Eqs. (11)—(14), it follows that the spectral and angular distribution of the radiation being studied can be represented asd N \  e dujd29 ~  7T2 | t a ±  /a  -A l x * + A ™< XR =  S l « A |T\ ±  / aUJZ7 '-2 +  ^  +  Q2UJZ7 - 2  +  ^  +  Q 2 —  ^ |  I «A  I ( t a  ±  / a )  UJZ UJZ(15)a d t r  =7 -2  +  ^ |  +  Q2 UJZ7 2 +  Q2 'where- 1  +  2^2w2 |ckaI \2u jnx g/ a  =  \ l T l ~  1 “  2 ^ a  ( t \  -  K \ )6x =UJrXo) I I= - F  |cka| Xo(16)2w2 |cka I 'Qi =  0j_ — ipj_, 0*2 = 20 0|| +  ,Q2 =  Q2 +  ^ 2) a l =  1) a 2 =  cost/?.In Eq. (15), the quantities ^ XR and ^4°TR rep­resent the contributions of, respectively, parametric x rays and diffracted transition radiation (DTR) [7]. The plus (minus) sign corresponds to the case of £* = 6  (£* =  6 )[seeE q . (9)].Let us first consider the contribution of parametric x rays to the total radiation yield. According to (15), this contribution is given byUJd N Px x *dujd267r2 " 4 / 7 - 2 +  ^  +  Q2UJZ(17)7 2 H 1  +  ^ 2 ------ !  ICKA | ("TA ±  / a )U)z w2-2It can easily be seen that the dependence of the distribution in (17) on the emitted-photon en­ergy uj is associated primarily with the fast variable t\  (u)  from (16), which shows that, at fixed values of the orientation angle O' and of the observation angle 6\\, the spectrum of parametric x rays isconcentrated in the vicinity of the frequency uj'b =( l  +  (o' +  #||) cot ( f / 2 ) ^ ,  where ujb= q/ 2 xsin (<f/2)  is the Bragg frequency, the relative fre­quency of the spectrum being very small: A uj/ uj ~  2uj2/ g 2 ~  lO“ 4.For the ensuing analysis, it is convenient to sepa­rate the real and the imaginary part of the function f \ \fx  =f'x -  *sgn ( t x  -  K \ )  f l ,  for r l  > 1- s g n  (tx -  KA) /a  +  i fx  for tI  < 1 ,fx  = l Ÿ  +  (r A -  Ka )2 + T 2x -  1,f x = + 40X ^  -  ka )2 -  ^A +  1 .(18)We note that, because of the smallness of the absorption factor (6\  <C 1 ), the function is small in the region of normal dispersion ( r 2 >  1 ) and thatGARY et al.7p x rUJd N Px XR n idujd29 7T2 ( j - 2  +  7 -2 +  Q2) ( t a )R PXR(px ~  T X -  sgn (rA -  K A ) / a )  +  ( / a )-1where7*  =  w b / w q , Pa =7- 2 +  7*“ 2 +UJ2 I « a I K(19)(20)angle, and the orientation angle. Since the spectral width of the peak of parametric x-ray radiation is A  uj ~  1 eV at a fixed value of the parameter p x ,  the approximationR PXR7T- j j y à  [ p  x  -  t x -  f xJ A(21)Fig. 2. Normalized angular distribution of param etric x rays with allowance for anom alous photoabsorption at q\ =  0.8 and® =  7 *fi forKA = ( / )  0, (2) 0.7, a n d (3) 0.95.the function f'x is small in the region of anomalous dispersion ( t x  <  1). With allowance for Eqs. (9) and(18), expression (17) can be recast into the simple form„2is sufficient for describing experiments that employ conventional x-ray detectors with an energy resolu­tion of about 100 eV.Considering that the absorption factor 5x is small, we find from ( 19) and (21 ) thatUJd N Px xRdud2 9{1 ~ 1/ p D 2(22)wXo ( I - I / p I Y  + 2 ( l - n x ) / p xQ2 -  a;r £  ( uj  -  UJB ) .The above result demonstrates the physical essence of parametric x-ray radiation as the coherent Bragg scattering of the screened Coulomb field of a fast electron by a set of atomic crystal planes. Indeed, the factor appearing in front of the function i?PXR in(19) describes the spectral and angular distribution of the Coulomb field of a particle in a medium whose mean dielectric permittivity is e ( u )  = 1 — uJq/ u 2 (in the vicinity of the Bragg frequency, uj & wB), while the quantity -Rrx r  can be treated as the coefficient of reflection of this field by a crystal.It can easily be shown that the denominator of the function -Rrx r  (ta  (u )) has a resonance character in the region of frequencies uj that correspond to the condition ta  (u ) > 1 ; that is, parametric x rays are formed in the region of normal dispersion. The position and the height of the peak of parametric x- ray radiation are determined by the single generalized parameter px (20), which is greatly dependent on the radiating-particle energy, the photon-observation(7 -2 +  7 * 2 +  £"22)This expression differs from the traditional formula in the kinematical theory of parametric x rays [8, 9] only by a factor that involves the parameters px and k\ .It can easily be seen that only in the region of sufficiently high radiating-particle energies can dynamical-scattering effects manifest themselves. To demonstrate this, we note that, in the region of low energies,7  < 7 * ,  (23)it follows from (20) that px »  1 , irrespective of the observation-angle value. Expression (22) then coin­cides with that in kinematical theory.In the energy region where the condition oppo­site to (23) is satisfied, the parameter px is on the order of unity for observation angles in the region ^  7 ^r\ where the bulk of the radiation is con­centrated. In this case, the distribution in (22) is extremely sensitive to variations in the parameter kx . In order to demonstrate this, we consider, for 7  »  7 *, the angular distribution of parametric x rays that follows from (22). We haved N PXRUJ-e2u jta{sin2 (ip/2)dxx A "rX oF pxr ( x , nx , qx ) ,(24)F PXR =( 1 -  qx + x 2)2 +  2qx (1 -  k\ )  (1 +  x 2)2 N 2 Qx1 -(1 + x 2)2,where x  = 7 *Q and qx = e_s,2-l<2/2 ( F  (g) / Z )  |cka|- The function F pxr calculated for a strong reflection (qx 1) is illustrated by the curves in Fig. 2, whichEFFECT OF ANOMALOUS PHOTOABSORPTIONare plotted for various values of the parameter nx . We can see that the polarization-brem sstrahlung yield grows significantly as n \  tends to unity, and this is the main result of the present study.The effect being discussed is similar to the anom a­lous photoabsorption of x rays in a crystal [1]. The latter is manifested in the Bragg diffraction of x rays under the condition n \  & 1 . In the diffraction of the pseudophoton field of a fast particle—it is precisely the case considered here—the effective absorption factor f x also decreases for —> 1 . It is impor­tant to note that, according to formula (18), which determines the dependence f x (t\ )  (for example, we have f x ~  5 \ \ / ( tx — 1) / ( t \  +  1) in the limiting case of n \  = 1 ), the above suppression of photoabsorp­tion is realized only near the region of anomalous dispersion, t\  (u) ~  1. In accordance with (21), the position of the maximum in the spectral distribution of parametric x-ray radiation is determined by the parameter p \  [the corresponding value is t\  =  tx (p \  +  l)  /2p \\ .  At low radiating-particle energies that satisfy the condition in (23), the parameter p \  is large. In this case, f x ~  5\, that is, the effect of anomalous photoabsorption does not manifest itself in kinematical parametric x-ray radiation. At high energies, 7  »  7 *, we have p \  ( l  +  x 2) q\ ~  1 for strong reflections (q\ & 1 ) in the region x  ^  1 , which is of particular interest [see formula (24)]. In this case, the effect of anomalously low photoabsorption, f x  (t a) ^  <^x, can show up if the coefficient n \  is sufficiently close to unity.4. CONTRIBUTION O F DIFFRACTED TRANSITION RADIATION: IN TERFERENCE O F DIFFRACTED TRANSITION RADIATION AND PARAMETRIC X-RAY RADIATIONReturning to the general formula (15), we con­sider the contribution of diffracted transition radia­tion, whose spectral and angular distribution can be represented in the formd N ,DTRÜJ-P 2x _  _ 0 27T2dud? 91 17  2 +  Q2 “ 2 +  7 * 2 +  n 27(25)(ta),( t a  +  S g n  ( r A -  K a )  / a )  +  (<*A +  f x )-1which is similar to that in (19). As follows from for­mula (25), diffracted transition radiation is generated when transition radiation from a relativistic electronjQ.Fig. 3. Normalized angular distributions of param etric x-ray radiation, diffracted transition radiation, and to ­tal radiation with and w ithout allowance for anomalous photoabsorption (presented in the figure are the quan­tities P ^ TR, P £ XR, and Pa =  P °'rR +  f * XR defined by the relations d N ^ / d 29 =  A XP£ TR and d N ™ / d 26 =  Aa-PaXR> where Aa =  e2<jA/47rsin (t^ /2)2: (a)  curves /  and 2 representing the contribution of param etric x rays at ka =  0 and 0.95, respectively, correspond to q\  =  0.8, 7*Xo =  1/300, and 7 / 7 * =  0.5 (there is virtually no con­tribution from diffracted transition radiation); (b ) curves / ,  2, 3, 4,  and 5 representing the contributions of, respec­tively, diffracted transition radiation, param etric x rays at ka =  0, param etric x rays at ka =  0.95, total radiation at ka =  0, and total radiation at ka =  0.95 correspond to 7 / 7 * =  20, all other param eters being identical to those in Fig. 3a.that traverses the entrance surface of the crystal ta r­get undergoes Bragg reflection from the crystal [7].It can easily be shown that the coefficient of reflec­tion -R°t r  attains a maximum value of about unity in the anomalous-dispersion region r 2 (u) < 1 .Absorption reduces the yield of diffracted transi­tion radiation; however, it does not play a crucial role, as it does in parametric x-ray radiation from a relativistic particle moving in a semi-infinite medium. The formula for the angular distribution of diffractedGARY et al.transition radiation— it follows from (25) without tak ­ing into account absorption— has the forme\  (26)d2el3ir2g2- Q 217 - 2  +  Q 2  7 - 2  +  7 - 2  +  Q 2This formula and the corresponding formula fol­lowing from (22) for the angular distribution of parametric x rays make it possible to compare the contributions of the radiation mechanisms under investigation. First of all, we note that our analysis of the interference term in the general formula (15) has revealed that the interference between parametric x-ray radiation and diffracted transition radiation is insignificant.The expression that follows from (22) and which describes the angular distribution of parametric x rays has the formdiVAPXRd2ee u g I^aI qx Q2* 9 2 7*2X o (7 - 2 +  7  *-2 +  Q2) ‘(27)(1 -  1 /Pa)2 +  2(1 -  n\)  / p \where px = (7  2 +  7 * 2 +  Q2) /<?a7 * 2-In order to compare the dependences in (26) and (27), it is convenient to choose the quantity 7 Q for an angular variable. In this case, the ratio 7 / 7 * be­comes the main parameter in both distributions. The distributions calculated on the basis of (26) and (27) are displayed in Fig. 3, along with the computed dis­tribution of the total radiation. The curves presented in this figure demonstrate that parametric x-ray radi­ation is dominant at low radiating-particle energies 7  <  7 * such that the effect of anomalous photoab­sorption is not manifested here. At the same time,we can see that, at energies so high that 7  »  7 *, the contribution of diffracted transition radiation domi­nates the observation-angle region Q ^  7 “ 1, while the peak of parametric x-ray radiation is formed near the observation-angle value of Q w 7 “ 1. From these curves, we also see that the yield of parametric x-ray radiation increases considerably under the conditions where the effect of anomalous photoabsorption m an­ifests itself and that the peak in the angular distri­bution of parametric x-ray radiation is shifted to the region of small observation angles.ACKNOW LEDGM ENTSThis work was supported in part by the Russian Foundation for Basic Research (project no. 00-02- 16388) and by the program Russian Universities— Basic Research (grant no. 991702).R EFEREN CES1. G. Borrmann, Z. Phys. 42, 1157(1941).2. V. A. Bazylev and N. K. Zhevago, Radiation from Fast Particles in Matter and in External Fields (Nauka, Moscow, 1987).3. A. Caticha, Phys. Rev. B 45 , 9541 (1992).4. N. Nasonov, Phys. Lett. A 260 , 391 (1999).5. Z. G. Pinsker, Dynamical Scattering of X  Rays in Perfect Crystals  (Nauka, Moscow, 1974).6. A. M. Afanas’ev and M. A. Aginyan, Zh. Eksp. Teor. Fiz. 74 , 570 (1978) [Sov. Phys. JETP 47 , 300 (1978)].7. A. Caticha, Phys. Rev. A 40 , 4322 (1989).8. M. L. Ter-Mikaelian, High-Energy Electromagnetic Processes in Condensed Media  (Akad. Nauk Arm. SSR, Yerevan, 1969; Wiley, New York, 1972).9. I. D. Feranchuk and A. V. Ivashin, J. Phys. (Paris) 46 , 1981 (1985).Translated by A. Isaakyan

Effect of anomalous photoabsorption on parametric X-ray radiation from relativistic electrons

C. K. Gary

V. V. Kaplin

N. N. Nasonov

M. A. Piestrup

S. R. Uglov

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Effect of anomalous photoabsorption on parametric X-ray radiation from relativistic electrons

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