Parametric X-radiation (PXR) of a relativistic electron travelling through a monocrystalline plate is examined in the Laue scattering geometry under the assumption of asymmetric and symmetric reflections of the particle field with respect to a target surfaceyesBelgorod State Universit

Blazhevich, S. V.

Noskov, A. V.

DSpace at Belgorod State University

Asymmetrie Wave Reflection in Kinematic and Dynamic Approaches to Description of Parametric X-Radiation of a Relativistic Electron in the CrystalS. V. Blazhevich and A. V. NoskovBelgorod State University, Belgorod, Russia Received July 12, 2010Abstract— Parametric X-radiation (PXR) of a relativistic electron travelling through a monocrystalline plate is examined in the Laue scattering geometry under the assumption of asymmetric and symmetric reflections of the particle field with respect to a target surface. The formulas of the kinematic and dynamic approaches to PXR characterization are analyzed and compared. The results of the experiment on relativistic electron PXR implemented at the Mainz microtron are interpreted using an expression derived for its angular distri­bution. It has been revealed for the first time that the dynamic effect of anomalous photoabsorption (the Bor- mann effect) of PXR has taken place in this experiment.INTRODUCTIONWhen a charged particle travels through a monocrystal, its Coulomb field diffracts on the par­allel atomic planes of a crystal, giving rise to para­metric X-radiation (PXR) [1—3]. Today, there are kinematic [4, 5] and dynamic [2, 3, 6] approaches to PXR characterization. It should be noted that the kinematic approach only describes the interac­tions of each atom with a primary (refracted) wave in a crystal and, in contrast to the dynamic approach, ignores its interaction with the total wave field produced by scattering from all the other atoms of a crystal; i.e., the multiwave scattering effect (in particular, interaction between elemen­tary waves and a refracted one) is not taken into consideration.In the last few years, the dynamic theory has gained significant ground in describing coherent radiation of relativistic electrons in crystals [7—11]. A striking example confirming the validity and advisability of the dynamic approach to PXR is the experimental obser­vation of PXR reflection along the relativistic electron velocity [12], which is not predicted by means of the kinematic theory.It is well known that the reflection asymmetry of a field with respect to the surface of a crystalline plate markedly affects the spectral-angular density of radiation. According to the kinematic theory, only the ratio between the charged particle path in the crystalline plate and the emitted photon path isdependent on asymmetry. The dynamic theory of relativistic electron PXR also predicts the substan­tial influence of asymmetric reflections on radiation process as such, leading to a change in the PXR spectrum. Thus, a topical problem solved in this study is to determine the limits of applicability of the kinematic approach to relativistic electron PXR in crystals.In subsequent sections, the formulas for PXR derived on the basis of kinematic and dynamic theories are analyzed and compared under conditions corre­sponding to thin and thick crystals with the aim of finding the criteria restricting an appropriate applica­tion of the kinematic formula.SPECTRAL-ANGULAR DISTRIBUTION OF PARAMETRIC X-RADIATION IN THE DYNAMIC THEORYThe expressions describing the spectral-angular density of PXR from a thick absorbing crystal has been derived in [13] via the dynamic diffraction the­ory [14]:(0-d N\PXRp ( s) g 2doxKl 4n (e2_  pW2 aPXR> (la)-XoASYMMETRIC WAVE REFLECTION IN KINEMATIC AND DYNAMIC APPROACHESE >  1Forward PXRFig. 1. Asymmetric radiation reflections from a crystalline plate at s > 1 and s < 1. The parameter (s = 1 corresponds to symmetric reflection).R <s)PXR 1 -i/P + s21 + exp(-2Z>(s)p(s)A(1)) -  2exp(-Z>(s)p(s)A(1))cosf  f  (s) + sa (s) + - ——a/?(S) J t ( s )- \ 2 (lb)+  S+ p (*)2a (1)2Here, notation conventions are analogous to those used in [13]:^{S} (co) = r|W(«>) + 1 -  s2v'U)'b{s) -  ■2sin20B (V 2 X* Cts) VCO y' <7<>) 1C O S\ | / 0pW =  _ £AW = s _ + i_ k ie  I W KCww ’2s 2s a/ ^ + s ^a 1 (c m \■ sC O S \|/^COS\|/n, , 1 y"Ö s)(i) _ K(i) _ Xg^Xo XoSince the inequality 2sin29B/ F 2(2)C (s) S> 1 isvalid in the X-ray frequency range, quantity r|'Vl (co) is the fast function of frequency <o. Hence, it is conve­nient to take r|(s) (co) as a spectral variable in subse­quent analysis of the PXR spectrum. Expression (lb) involves important parameter s that determines the degree of asymmetry of field reflections in the crystalwith respect to the plate surface. The wave vectors of incident and diffracted photons are directed at equal or unequal angles to the plate surface when a reflection process is, respectively, symmetric or asymmetric (Fig. 1). In the former case, s = 1 and 8 = n/2. In the latter case, s ^ 1 and 8 ^ n/2.Let us write the asymmetry parameter assin (8 + 0B) sin(S -  0B) ’(3)where 0B is the angle between the electron velocity and a set of crystallographic planes.Note that the angle 8 -  0B of electron incidence on the plate surface increases with decreasing parameter s and vice versa (Fig. 1). As is known, asymmetry parameter (3) is the ratio between the electron pathf  - ^L =length in the plate photon LL and the maxi­path length in the crystalsin(S -  0B)Lf =-mum(  '  -^ after its formation near the inputsin(S + 0B)J surface of a target.Parameter v(s) characterizes the degree of wave reflection from a set of parallel atomic planes in the crys­tal, which is determined by the constructive (v(s) ~ 1) or destructive (v(s) ~ 0) interference of waves reflected from atoms of different atomic planes. Absorption parameter p(i) can be represented as the ratio of the X-ray wave extinction and absorption lengths in theE =  1 £ <  1Forward PXR Forward PXRJOURNAL OF SURFACE INVESTIGATION. X-RAY SYNCHROTRON AND NEUTRON TECHNIQUES Vol. 5 No. 4 2011BLAZHEVICH, NOSKOVcrystal: = ■(ÙX, ■«and /  , —®Xo-, respectivelyIt should be emphasized that the primary wave energy is completely transferred to a secondary wave propa­gating in the Bragg direction at the depth equal to the extinction length.The solution to the equationM %GW +2-M ■Æ + 8 =  0 (4)4 k  = 4 1 -« \ 2sin + s (5)v7W _ + sTo observe pronounced dynamic effects, let us con­sider a crystalline plate such that the electron path length /,/sin (8 -  0B) within its thickness is many times greater than the X-ray wave extinction length ^ in the crystal; i.e., bis> > 1. Integratingt(s) —■Hext(0 X'xC«expression (la) over frequency function r]1' 1 (cd), we obtain the formula for the PXR angular density:dN,PXR _dQc j(s)2a28tï sin 0B(02 + y - 2 _ x ^)2+coX  | 4 x r ^ î 1 ( s ) M -(6)Since the spectral peak of PXR is very narrow at his) > 1, integration in (6) can be implemented with the use of the well-known approximation:sin2 ( Tx) ^—- —» n ï ’ù (x ). After integration, angular distri-xbution (6) takes the formr f C "  e2coBX: 2CW> w2* PXRdQ 47rsin2a 2 L (7)-Xos in (8 -  0B)determines the central frequency ou, of the spectrum of PXR photons emitted at a fixed observation angle.Below, dynamic formula (lb) is compared with the well-known kinematic formula [4, 5]. In addition, we find criteria restricting the appropriate employment of the kinematic formula for PXR.COMPARATIVE ANALYSIS OF KINETIC AND DYNAMIC FORMULAS:A THIN CRYSTALIn the case of a thin target (//''p1'1 <§ l j , absorptioncoefficient p1'1 can be ignored. Hence, expression ( lb) describing the PXR spectrum takes the formLet us consider the well-know expressions describ­ing the PXR angular density in the kinematic approach [5, 15]:</«K" eJ _r(8)>* p x r  _ ;  dQ 47rsin(\ 2  , - 2  ) + y ■Xo(8)r ( e ) = -®bXo(1 -  exp cobXo-Es sin (8 -  0B)\ \J)It is seen that asymmetry coefficient s is contained only in the geometric factor T(s) of the kinematic expression. In the case of a thin nonabsorbing crystal, the geometric factor is an electron path in the crystal­line target:Lr ( e )  = - (9)sin (8 -  0B)Therefore, the aforementioned kinematic expression can be written asd N $ k n e \ obX: 2CW> )2* PXRdQ 47rsinL(10)A 2 , - 2  0 + Y -  Xosin(S -  0B)It should be emphasized that the dynamic formula differs from the kinematic one in that asymmetry parameter s (angle 8) is contained not only in expres­sion (9) defining an electron path in the plate, but also in the multiplicand describing the PXR angular distri­bution. Hence, such a difference can serve as a base to compare the dynamic and kinematic formulas derived for a thin crystal.In subsequent analysis, it is convenient to use expressions (7) and (10) rewritten as follows:JArWdin“jvPXRdQe2®BX, L47rsin 0B s in (S -0 B)2 02-F dinj-rdin (s)F -  vw -Xo\ 2 ’(11a)(lib ),M2+ sXo7Y Xo+ 1ASYMMETRIC WAVE REFLECTION IN KINEMATIC AND DYNAMIC APPROACHESF, arb. un its9M \Fig. 2. Comparison between the PXR angular density curves constructed from dynamic and kinematic formulas. The kine­matic formula provides an error under the given conditions.F, arb. units9TlXolFig. 3. PXR angular density curves calculated at different degrees of asymmetry. It is seen that the error of the kine­matic formula increases with growing asymmetry.,^ (Y)kin a iV PXRdCle coB g(s)p(s) -rkinsin 0B s in (8 -0 B)(12a)Q27-rkin (s)F = vv -Xo\2' (12b)Xo+ ■Y Xo+ 1Functions (lib ) and (12b) define PXR angular densities and confirm the fact that only the dynamic approach allows for the influence of asymmetry on the angular component of the PXR distribution.Symmetric ReflectionLet us consider the features of symmetric reflection at 8 = 1. At all observation angles 0, the approximate equality Fiin « Fkia holds true if an emitting particle has ^ - —  (oip is the plasma fre-co„the low energy y <quency.)XoIn the case of weak reflections (e.g., v(5) » 0.3), Fdin« Fkin if the particle energy is very high(y > (£>/a>p). In the presence of strong reflections (e.g., v(s) » 0.9), the angular density dependences con­structed from dynamic and kinematic formulas are somewhat different (Fig. 2). In this case, kinematic formulas provide an error in the absolute yield of pho­tons. Hence, when the absolute yield of PXR is mea­sured with a sufficient accuracy, the results must be compared to dynamic formula (7).Asymmetric ReflectionIn the presence of reflection asymmetry caused by either a decrease (e > 1) or an increase (s < 1) in the angle ( 8 - 0 B) of electron incidence on the target (Fig. 1), the kinematic formula error increases, as is illustrated by the curves in Fig. 3. In the case of strong asymmetry (s > 1), i.e., at the small angle (8 -  0B) of electron incidence on the target, the ratio of (1 lb) and (12b) is[ din/ .8. (13)kinirrespective of whether the Lorentz factor is greater or less than co/co,,. Therefore, strong asymmetry implies that the PXR angular density is s times as large as the density predicted by the kinematic formula (Fig. 4).Thus, even if a thin nonabsorbing crystal is investi­gated, it is necessary to allow for the dynamic effects of PXR.COMPARATIVE ANALYSIS OF KINETIC AND DYNAMIC FORMULAS:A THICK ABSORBING CRYSTALIn comparative analysis it is convenient to use expressions (1) and (8) represented asdN,(^ din PXR _ 2 T)(s)2dQ 4rc sin 0Tpdin"abs 5 (14a)7-rdin (iS1)a^bs =VW-Xo\ 2Xo+ -Xo+ 1f  1 4 ! ^  (co), (14b) 2n JBLAZHEVICH, NOSKOVdW _A P X R  - 1 -+ 8f  fLcos2 LCT(i) + E,{s) -  ^  + 8wa " + £(s ) M2 + 8 + L2 n{s)2e^xt H-JJ (14c)d N ^ ?  _ e2P {s)1dQ.  F  kin. 2 .  abs »4Tisin 0R(15a)7-rkin (V) a^bs =VU-XoVp W(l -  exp (-Zyiio)), (15b)Xo+ -Xo+ 1Here,S K8 + 1 1 -8  £,(S)________. 2 "  2W\” W 1 Mo =  ® b X o ,  a  = ~ [ Âv y ’ Xo+ ■Y Xo+ 8 yA+ 1J(16)where the effective coefficient ^  of photon absorp­tion depends on asymmetry parameter s of a crystaland parameter kw characterizes the effect of anoma­lously low photoabsorption (the Bormann effect) [16] arising when X-ray waves propagate through the crys­tal. Its physical mechanism consists in the formationF, arb. unitsJllolFig. 4. PXR angular density curves calculated at different degrees of asymmetry. In the case of strong asymmetry, the PXR angular density is x times greater than the density pre­dicted by the kinematic formula.of an incident standing wave scattered by X rays, the antinodes of which are equidistant from neighboring atomic planes. At these points, the electron density of the crystal is at its minimum and, consequently, the photoabsorption has the lowest level. In addition, anomalously strong absorption of one of two waves generated in the crystal and anomalously weak absorp­tion of the other are observed. Expression (14) describes the PXR branch exhibiting anomalously weak absorption. As in the case of free X-ray waves, is the approximate equality between the imaginary parts of the corresponding coefficients of the Fourier expansion of its dielectric susceptibility in terms ofreciprocal lattice vectors « lj is a necessary condi­tion of manifestation of the given PXR effect in thecrystal. In this case, coefficient ^  defined by (16) reaches its minimum. It should be remembered thatparameter k(s) depends on the chosen set of parallel diffracting atomic planes in the crystal, radiation fre­quency, and its polarization.Equation (4) has the solutions a Is)2a.M(17)which determines the central frequency co* of thespectrum of PXR photons emitted at a fixed observa­tion angle. Therefore, the value of effective absorptioncoefficient depends on observation angle 0. Since the dynamic theory allows for the influence of absorp­tion, the PXR angular density curves constructed from the dynamic formula can be deformed with respect to the density curve calculated via the kinematic theory (Fig. 5).The foregoing discussion enables us to draw the following inferences. In kinematic expressions (15), asymmetry affects the process geometry only and absorption coefficient |i0 corresponds to absorption in an amorphous medium. In dynamic expressions (14),absorption coefficient |iw depends not only on the chosen set of diffracting atomic planes of a crystal(parameter k ^ ), but also on the asymmetry coeffi­cient and the observation angle. In addition, the effec­tive photon absorption length is \f \x[s).ASYMMETRIC WAVE REFLECTION IN KINEMATIC AND DYNAMIC APPROACHESF, arb. units07bco IFig. 5. Deformations of the dependence between the PXR angular density and observation angle 0 observed at different values of the effective absorption coefficient.dN/dQ., 10 3 photon/(electron sr)Angle, mradFig. 6. Manifestation of the dynamic Bormann effect in PXR.INTERPRETATION OF THE RESULTS OF THE EXPERIMENT ON PARAMETRIC X-RAY RADIATION OF RELATIVISTIC ELECTRONS AT THE MAINZ MICROTRONThe results of the experimental investigation into the PXR spectral-angular distribution of electrons with ener­gies of 855 MeV have been reported in [ 15]. It should be noted that a high quality of the electron beam (small cross-section sizes and a low divergence) and detecting equipment of the accelerator enabled the authors of [15] to determine the angular distribution of X-ray photons in absolute units at high accuracy and angular resolution. Let us compare the results of the experiment of interestwith our calculations according to well-known kinematic expressions (15) and dynamic expressions (14) derived above (Fig. 6). It is seen that dashed curve 1 correspond­ing to the kinematic formula differs slightly from the experimental results. Solid curve 2, which was calculated from our dynamic formula and characterizes asymmetric photon reflection by the set of {111} planes, agrees closely with the experiment. It is important to note that the con­ditions of the experiment considered indicate the pro­nounced manifestation of the effect of anomalously low photoabsorption (the Bormann effect) in the asymmetric case of PXR [11]. This assertion is illustrated by solid curve 3 calculated from the dynamic formula without allowance for the Bormann effect.BLAZHEVICH, NOSKOVThus, the dynamic Bormann effect of PXR can be assumed to have occurred in this experiment.CONCLUSIONSComparative analysis between the formulas of the kinematic and dynamic approaches to description of parametric X-radiation has been carried out under the conditions of symmetric and asymmetric reflections. In the case of reflection symmetry in a thin crystal (including the nonabsorbing one), the PXR kinematic formula is shown to be erroneous at the emitting par­ticle energies y > (o/(op. However, the PXR kinematic and dynamic theories provide identical results at y (o/(o/;. It has been found that the error of the kine­matic formula increases with growing asymmetry and, moreover, the PXR angular density calculated from the kinematic formula is s times less than the actual density predicted by the dynamic formula under the condition of strong asymmetry (s > 1). Thus, even if a thin nonabsorbing crystal is examined, the dynamic effects of PXR must be taken into consideration. In the case of a thick absorbing crystal, the dynamic the­ory enables us to calculate the PXR absorption coeffi­cient with allowance for reflection asymmetry and the direction of photon emission, which lead to the defor­mation of the angular distribution shape constructed from the kinematic formula.The developed theory has made it possible to inter­pret the results of the experiment on relativistic elec­tron PXR implemented at the Mainz microtron and reveal the anomalous photoabsorption effect (the Bor­mann effect) in PXR.REFERENCES1. M. L. Ter-Mikaelyan, The Influence o f the Medium on High-Energy Electromagnetic Processes (AN ArmSSR, \ferevan, 1969) [in Russian],2. G. M. Garibyan and Yan Shi, Zh. Eksp. Teor. Fiz. 61, 930 (1971) [J. Exp. Theor. Phys. 34, 495 (1971)].3. V G. Baryshevskii and I. D. Feranchuk, Zh. Eksp. Teor. Fiz. 61, 944 (1971) [J. Exp. Theor. Phys. 34, 502 (1971)].4. H. Nitta, Phys. Lett. A 158, 270 (1991).5. I. D. Feranchuk and A. V. Ivashin, J. Phys. 46, 1981 (1985).6. V. G. Baryshevsky and I. D. Feranchuk, J. Phys. 44, 913 (1983).7. A. S. Kubankin, N. N. Nasonov, V. I. Sergienko, and I. E. Vnukov, Nucl. Instrum. Methods Phys. Res. B 201, 97(2003).8. N. Nasonov and A. Noskov, Nucl. Instrum. Methods. Phys. Res. B 201, 67 (2003).9. S. Blazhevich and A. Noskov, Nucl. Instrum. Methods. Phys. Res. B 252, 69 (2006).10. S. Blazhevich and A. Noskov, Nucl. Instrum. Methods. Phys. Res. B 266, 3770 (2008).11. S. V Blazhevich and A. V. Noskov, Nucl. Instrum. Methods. Phys. Res. B 266, 3777 (2008).12. A. N. Aleinik, A. N. Baldin, E. A. Bogomazova, et al., Pis’ma Zh. Eksp. Teor. Fiz. 80, 447 (2004) [JETP Lett. 80, 393 (2004)].13. S. V Blazhevich and A. V. Noskov, Poverkhnost’, No. 4, 40 (2010) [J. Surf. Invest. 4, 303 (2010)].14. Z. G. Pinsker, Dynamical Scattering o f X-rays in Crystals (Nauka, Moscow, 1974; Springer-\ferlag, Berlin, 1978).15. K.-H. Brenzinger, C .  Herberg, B. Limburg, et al., Physica A 358, 107(1997).16. G. Borrmann, Zh. Phys 42, 157 (1941).

Asymmetric wave reflection in kinematic and dynamic approaches to description of parametric X-radiation of a relativistic electron in the crystal

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Asymmetric wave reflection in kinematic and dynamic approaches to description of parametric X-radiation of a relativistic electron in the crystal

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