This article reports on the transition radiation (TR) of a relativistic electron crossing a crystalline plate, which in the Bragg’s scattering geometry is considered within the framework of the X-ray diffraction dynamical theoryyesBelgorod State Universit

Blazhevich, S. V.

Noskov, A. V.

DSpace at Belgorod State University

On the dynamical effects in the characteristics of transition radiation produced by a relativistic electron in a single crystal plateS. Blazhevich *, A. N oskovBelgorod Stale University, Laboratory o f  Radiation Physics, 14 Studencheskaya Sir., 308007 Belgorod, Russian FederationAbstractThis article reports on the transition radiation (TR) of a relativistic electron crossing a crystalline plate, which in the Bragg’s scattering geometry is considered within the framework of the X-ray diffraction dynamical theory. The expressions for TR and diffracted transition radiation (DTR) spectral-angular distribution received allow to discuss the manifestation of the dynamical approach. It is shown, that the spectral-angular characteristics of TR and D TR essentially depend on the orientation of the crystal plate surface when the angle between the electron velocity and the atomic planes is kept constant. This phenomenon is explained by the modification of the frequency range of total external reflection of the radiation under a changing target surface orientation.Keywords: Relativistic electron; Transition radiation; Single crystal; Angle distribution1. IntroductionW hen crossing the boundary  between two different m edium s a fast charged particle induces transition rad ia­tion [1]. I f  the particle crosses a plate, the interference between waves radiated on in- and out-surfaces of the plate appears. The T R  waves, produced on input surface target, are scattered by a system o f atom ic planes in the crystal (see [2—5]) which bring about a change in the conditions o f interference of these waves [6].In  the given w ork a detailed theoretical analysis o f the T R  waves in Bragg’s scattering geometry is carried out. The spectral-angular distribution for T R  pho ton  is achieved by applying the dynam ical diffraction theory [7]. It shows, tha t the interference o f the waves from  the in- and out-surfaces in Bragg’s scattering geometry essentially depends on the orientation o f the crystal plate surface under a fixed angle between the electron velocity and theCorresponding author.E-mail address: blazh@bsu.edu.ru (S. Blazhevich).crystallographic plane. In  other words it m eans that the characteristics o f the T R  o f the relativistic electron in a sin­gle crystal plate essentially depend on the cut angle o f the plate surfaces. In  the current w ork this effect is explained as the result o f changing the anom alous dispersion fre­quency range (range o f to tal reflection). This effect is anal­ogous to the know n effect for free .\'-waves in a crystal [7]. It can be explained as a result o f changing the to tal reflection frequency range, in which the vectors, perpendicular to the entrance surface in the space o f the reciprocal lattice, do not cross the disperse surface, tha t corresponds to  the exponential dam ping o f waves, passing in the crystal in a forw ard direction. In  this frequency range the wave vectors take on a complex value, which correspond to  the exponen­tial weakening o f the wave intensity on passing through the crystal.In  the fram e o f the proposed model the conditions o f the T R  interference peak reduction are shown in this work, which allows to  reveal the optim um  conditions for the experimental observation o f the param etric X-ray rad ia­tion at a small angle to  the radiating electron velocity.S. Blazhevich, A. Noskov IOn the other hand, the conditions o f the interference peak increasing and narrow ing o f the T R  spectral w idth are shown, too.2. The general expressions for spectral-angular distribution TR in dynamic approximationLet us consider the radiation  o f the fast charged particle moving w ith constant velocity V  th rough a crystal o f L  thick (Fig. 1). Let us consider the equations for the Fourier direct image of electrom agnetic field. W hile solving the problem , let us take up the equations for the Fourier direct image o f the electrom agnetic field E(k ,  co) = f  d t d 3rE(r, 1)  exp(ittrf — i kr).Since the Coulom b field o f an ultrarelativistic particle can be allowed to transverse with a good degree o f accu­racy, the incident E0(k,co) and diffracted Ei(k,co) electro­magnetic waves are defined by two am plitudes with different values o f transverse polarization.E 0{k, to) = E [Q]( k , co)e[0n + E [^  (k, to)e\E \ ( k , co) = E[1\ k , co)e\11 + E\~!(k, co)e\(i)o„ID,<21/ {2) K0 ’r<2 )/ ( 2 ) (1)The unit vectors e ^ ,  e ^ \  e\1] and e(2) are chosen in the following way. Vectors and eo ] are perpendicular to vector k,  and vectors e ,1' and e(2) are perpendicular to  vec­to r k  +  g. Vectors Cq2) and e{2) are situated on the plane of vectors k  and k + g  (ir-polarization) and e ^ ,  e\1] are per­pendicular to  this plane (cr-polarization); g  is vector of the reciprocal lattice, defining a set o f reflecting atomic planes. Using the two-wave approxim ation o f the dynamic theory o f diffraction, we will apply the well-known system o f equations for the Fourier transform  images o f intensity o f incident and diffracted waves [8]:(ffl2 ( l + X o ) -=  8iz2ieco9VP[s)S ( c o -  kV) ,  co2x , C ls)E [0s) +  (®2(1 +  xo) -  {k -(2 )Mwhere xg, X-g, Xo are the factors in expansion o f the dielec­tric susceptibility in the Fourier series by reciprocal lattice vectors gr ) =  +  i4 ( « ) ) e ^ .  (3)g gWe will consider a  crystal w ith the following symmetry:(.Xg ~  X-g)‘The values O s) and  R s) are defined asC(i) =  C(1) =  1, C (2) =  cos(20B),P [s) = e ^ \ p / p ) ,  P {1) =  sin <p, P {2) =  cos (p, (4)where p — k  — coV/V2 is the virtual pho ton  m om entum  vector com ponent perpendicular to  the particle velocity vector V (p — cod/V, where 9 (9 -c  1) is the angle between k  and V), 9B is the angle between the electron velocity and a set o f atom ic planes in the crystal (Bragg angle), cp is the azim uth angle, counted off from  the plane formed by vectors V  and g, the value o f reciprocal lattice vector is shown by expression g — 2<yBsin0B/F , <yB -  Bragg’s fre­quency. U nder 5 = 1  the system (2) describes the fields of cr-polarization and under s — 2 the fields o f n-polarization.The solution o f the system (2) for the coulom b field of relativistic electron in vacuum  is like-,(j)vac 8n2ieV 9P[s)co -y 9(5a)and for the coulom b field in crystal (w ithout the PX R  con­tribution) it is likeWcr _  8n2ieV 9P[s)co xo -  I’- 2 -  02 '■A,-1/2(5b)where y =  (1 — V~)~ is the particle Lorentz factor.The field o f the radiation generated on the entrance sur­face o f the crystal is£ ( j ) v a c - c r  _  8 n ~ i e V  0 CO1 1- r Xo -  r• (6a)The one on the exit surface o f the crystal is1 1£ ,(s )c r—vac _  ^ K - j e V  Q p {s)1 1 ,. (6b)X o - ' r 2 - 0 2 - y - i - 9 21From  this it follows, that the electric field o f the T R  wave radiated ahead o f the crystal will be expressed asFig. 1. The geometry of the radiation process on the relativistic electron- monocrystal plate interaction. 6B -  Bragg’s angle; k  -  wave vector o f  the incident pseudo photon; g  -  reciprocal lattice vector; V -  electron velocity vector; 6 -  angle o f photon observation.t \1K''Rad^ ,(»vac—cr j j (s )  _|_ £(V )cr-vacTR 8n2ieVE [s) -■R ad — 9p(s)CO(1 -  U [s))(7a)(7b)vy-2 +  9- y- 2 + 9 - -  xoyThe am plitude transm ission coefficient l / s} defines the field am plitude o f T R , radiated on the entrance surface o f the crystal and left the crystal through its exit surface. The coefficient in Bragg’s geom etry taking into consid­eration such a view [7] asS. Blazhevich, A. NoskovU [s) =  ^—2 \ Jz2 +  j  z — \ / z 2 +g2cos(iPq) (xo-02-y 2--+\z + \ / z 2 +  eKSferU-®2- ’'"2- - ^ , (8)where # (i) =  - C ^ ^ / s ,  z  =  - ( a  -  (1 +  s)(x0 -  d2 -  y 2) ) /  2s, s =  sin(0B — <5)/ sin(0B +  <5), <Ao =  ^ /2  — (#b +  <5) is the angle between the incident wave vector and the external norm al to  the target, <5 is the angle between the entrance surface of the target and the crystal plane, a =J_ _|_ g'y _  2^  _  4 sin2 eR ^toB(l+fl cosy cot 6B) _  ^The T R  spectral-angular distribution can be expressed as [8]4 2 rr/TRdcodQ=  CO'i 2 n r 6 J 2 \ E \(  \  IK .CJ)Rad (9)By substituting (8) w ith (7b) and then (7b) with (9), we will draw  the expression for the spectral-angular d istribu­tion o f T R  in the crystal:COd jV dcodQ Ji202 J > (sY1 1r  V ^ 2 +  ^ 2 +1 +  (2K[s)) /  ( ( ( c is) - JS:(")N) e ^ ^ ö ( CtI)+Ä'tI)^ /itI)+vtI))( 10)p )  +  K i AwhereZn . X c (s)\y!s \A i +  £)2 t f c < ‘)2\ \ 1/24 c i 's)2 V 4 t o  JR [s^ ) + IS>(1 +  6) 2 C s) =  >i{s) +XÖ ß ( S ] = 1 1( f t 2 1 Vi z ; i c w ’ lx ; i cw (( ß [s) -  i p ^ ) ( l  + s)(s) a 2sin~0B ^wB(l +  0cos<pcot0B) ^2 \ x ' \ c {s) V2\X' \ C ^  V CO( 11)Since in X-ray frequency region the condition 2 sin2 0B/  V2\x'g\Cls) >  1 is held true, the quantity  i f s\co) depends strongly on frequency, therefore it is very convenient as a spectral variable for analysis o f T R  spectrum.The expression (10) can be used in the investigation of dynam ic effects m anifestation in transition  radiation.3. The dependence o f spectral-angular distribution o f the transition radiation on radiator orientationParam eter s — sin(0B — <5)/sin(0B +  8) defines the orien­ta tion  o f the crystal plate surfaces. U nder a fixed value of 0B on the decrease o f the electron incidence angle 0B +  <5 the param eter 8 becomes negative and its absolute value increases, which leads to  the increasing o f param eter s. On the opposite, on the increase o f angle 0B +  <5 param eter s decreases (in the extreme case <5 —> 0B). W hen s > 0 the crystal is oriented according to  Bragg’s geometry and the part of T R  form ed on the entrance surface o f the crystal and then diffracted on a system o f parallel atom ic planes in the crystal leaves the crystal through its entrance surface.Let us consider the influence o f crystal orientation  on the spectral-angular properties o f T R  in the asym ptotic case o f thin crystal, when the absorption o f the radiation is negligible (xq =  0). In  this case the expression (10) becomes( 12)b{s)=coL\x'g \Cls)/ 2  sin(0B +  8) is param eter dependent on crystal thickness; is the factor describing the interfer­ence o f the radiations form ed on bo th  of the crystal plate boundaries.It is necessary to  note, tha t in the case, when the surfaces o f the plate are parallel to  the crystal plane (<5 =  0 or s — 1) expression (12) transform s to  the expression for T R  in the dynam ic approxim ation, obtained in [9]. Expression (12) is valid for all possible values o f quantity  t f s) and significantly differs from  the form ula for T R  on an am orphous plate of the same thickness L.  Such a difference is stipulated by the diffraction o f the radiation  on atom ic planes and represents a dynam ic effect. This difference is significant only in the vicinity o f Bragg’s frequency \tfs) + + s)/2\ <  sl/2.Out of Bragg’s frequency vicinity the condition |t f s) + [3{s)(l + s)/2\ >  s1/2 is valid and expression (12) takes theS. Blazhevich, A. Noskovview o f the well know n form ula for T R  form ed on an am orphous dielectric plate:2 ArT Rd-wdcodQ2 f l  e2ÿ pn-(sT 1 1j=l 1 — COSV)'-2 +  02 y-2 + 02 -x 'o;coL , ,2 sin(0B +  <5)(13)As we are interested in the dynam ic effects, let us con­sider the frequencies close to  Bragg’s frequency <yB. F rom  expression (13), in particular, it follows, that the destruc­tive interference o f the T R  waves, em itted on the entrance and exit surfaces o f the crystal plate, leads to  a full suppres­sion of radiation in the frequency range close to  Bragg’s frequency under condition [10]:coL2 sin(0B +  <5) + 7 2 -  Zo) = (14)where n is a natural num ber. In  such conditions in the vicinity o f Bragg’s frequency the radiation, leaving the crystal in a forw ard direction, will represent a peak w ithout any ordinary T R  background in im m ediate proxim ity to  it. This peak is form ed by the uncom pensated part o f TR  from  the exit surface o f the m onocrystal plate. The reali­zation o f the destructive interference condition (14) for different values o f param eter s can be accomplished, for example, by the choice o f the corresponding values o f plate thickness L:L4 ■ n ■ n ■ sin(0B +  (5) co ■ (02 +  y -2 -  Xo)4 ■ K ■ n ■ sin ( 0B +  arctan f r f  ta n  0bco • (0- +  y-2 -  x'o)(15)In  Fig. 2 the curves, describing the T R  spectrum  of cj-polarization, calculated under the anti-resonance condi-Fig. 2. The influence o f crystal plate entrance surface orientation (parameter e) on the spectral distribution T(i]) o f the relativistic electron TR in a forward direction (on formula ( 12)).tion (14), are presented. The curves are calculated by fo r­m ula (12) for different values o f param eter s and the fixed value o f param eter In the frequency region of anom alous dispersion o f the radiation, emitted on the entrance crystal surface (the region o f to tal internal reflec­tion) in a forw ard direction the peak o f the radiation, con­sisting practically o f the radiation  emitted on the exit surface only, will be observed. One can define the anom a­lous dispersion region from  expression (11) as.cl/2 . ß [s)( l  +  s)/2  <  <  £1/2 -  ß [s)( l  +  s )/2  (16)and the w idth o f this region can be defined as 2s1/2.It is seen from  the figure that under the increasing of the spectral d istribution shifts to  smaller values o f param ­eter t f s}, the to tal internal reflection region widens and the interference peak decreases and under large values of s practically vanishes. U nder the decrease o f param eter s the to tal reflection region converges and the interference peak increases, that leads to  the rise and grow th o f the interference peak and to  the narrow ing o f its spectral width.4. Spectral-angular distribution of diffracted transition radiationIn  the preceding section the T R  o f the relativistic elec­tron  crossing the crystal plate was considered in a forward direction. The influence o f a diffracted system of parallel atom ic planes on the radiation in the crystal was shown. In  this chapter we will consider the diffracted transition radiation (D T R  ), propagating through the entrance surface o f the plate in a backw ard direction. Let us handle the influence of plate surfaces orientation on the spectral- angular properties o f the D TR .The radiation  field, emitted by the relativistic electron on the entrance surface (6a) after the diffraction on the sys­tem  o f parallel atom  planes in the crystal, ispMDTR _ j^ [s)ViiC cr )R a d  —  ^ 08n2ieV0P{S)CO - 2 _ 0 2 Xo -  7 -2 _ 02Q[is)(17)where Q(s) is the am plitude coefficient o f the radiation  field reflection of the crystal, given by [7]:C[s) yQ (s )  _  ------------ t L  I  e  2 c o s ( 0 q  ) . g2cos(iPq)/-------------\  i (ùLr _|_ - ^ / ^ 2  _|_ q (s )  J Q 2cos(iAo) y /z2+q(s) (18)In  the case o f a thin crystal, when absorption is neglected, this coefficient can be given byS. Blazhevich, A. NoskovQ[,w -16'»expexpi b{s)- i  b{s)■Asr(19)To define the spectral-angular density o f T R  let’s substi­tu te (17) to  the form ulaj2  rr /D T R2(27i)-6 g | £ ;d ö d ßä= ithen we will draw  the nextd27VDTRRad (20)d ö d ß02 27T■02 ^ P (j=l1Q[s)\- (21)0’-2 + ö2 y-2 +  02-x 'o.It turns out, tha t the square m odule o f coefficient (19) can be presented as well-behaved function in the followingway:sin’ b\Qis) I2 tisr esin(22)is defined in (11).In  Fig. 3 the curves describing D T R  by form ula (22) are presented for different values o f param eter s characterizing the crystal plate orientation. It is seen, tha t by increasing param eter s the intensity o f D T R  decreases.By substituting (22) for (21) we can obtain the general expression for the spectral-angular d istribution o f D T R2 a rDTRd 2Vd ö d ß  7T^ 02ï > 'TT-1 1siny- 2 +  e2 y- 2 +  ö2 -  / 0/Ç[s) — £ +  £ sin’ [ b(23)It should be noted, that in the case, when the entrance surface o f the crystal is parallel to  a system o f crystal planes, expression (23) turns into the expression for D T R , obtained in [11].F o r studying the dependence o f the D T R  angle density on param eter s, let us integrate the expression (23) over the frequency function ^ s>(co)dML 2 atD TR 2 a rD TR COd Q âcoâQd co' à 1 N ' C[s)dcodQ 2 sin 2 0,-d t .  (24)Fig. 3. The influence of crystal plate surface orientation on the spectral distribution o f D TR  (on formula (22)).In  order to  plot the curves o f the radiation angle density distribution (in dependence on angle 0) let us rewrite the expression (24) for cr-polarization in the followingway:dTVD TR  e l (OndQ Ti“ 2 sin2 0B ®bwhereF 0( r 2 +  0-l)2( r 2 +  0: + S "sin^(1) — £ +  £ sin2 f fe(1)dÇ, (25)0± =  0sin cp.The curves, plotted for different values o f param eter s, are presented in Fig. 4. Here one can see that the angle dis­tribution o f the D T R  density is strongly dependent on £. It is im portant, w ith relation to the problem  o f creating X -radiation quasim onochrom atic sources on the base of relativistic electron-single crystal interaction.61(mrad)Fig. 4. The angular distribution o f the D TR  density F(6) integrated on photon energy co calculated for different values o f parameter e (calculated on formula (25)).S. Blazhevich, A. Noskov5. ConclusionsThus in this w ork the detailed theoretical analysis o f TR  and D T R  o f the relativistic electron in a m onocrystal plate is carried out in Bragg’s geometry. Using the dynam ic theory o f diffraction the expressions for spectral-angular distributions o f T R  and D T R  are deduced in a form , that allows to study the dependence o f spectral-angular charac­teristics on the relative orientation o f the crystal lattice and the entrance surface o f the plate as a single crystal radiator. The studies showed tha t the spectral-angular properties of T R  and D T R  vary significantly w ith this orientation. This effect is explained by the m odification o f the frequency total reflection range under a changing surface orientation. It can be used bo th  for the intensity enhancem ent o f the X-radia- tion source on the base o f the R T R  m echanism  and for the suppression o f the transition  radiation as a background in the cases when T R  hinders the experimental studying of other radiation m echanisms o f electron-crystal interaction.AcknowledgementsThis w ork is supported by the R ussian F oundation  for Basic Research (Project 05-02-16512), the Federal Agencyo f Education o f R ussia (Belgorod University, R E  N o 1.1.05) and an In ternal G ran t o f Belgorod State University 2005.References[1] V. Ginzburg, I. Frank, J. Phys. (USSR) 9 (1945) 353.[2] A. Caticha, Phys. Rev. A 40 (1989) 4322.[3] V. Baryshevsky, Nucl. Instr. and Meth. A 122 (1997) 13.[4] X. Artru, P. Rullhusen, Nucl. Instr. and Meth. B 145 (1998) 1.[5] N. Nasonov, Phys. Lett. A  246 (1998) 148.[6] N. Imanishi, N. Nasonov, K. Yajima, Nucl. Instr. and Meth. B 173 (2001) 227.[7] Z. Pinsker, Dynamical Scattering of X-rays in Crystals, Springer, Berlin, 1984.[8] V. Bazylev, N. Zhevago, Emission From  Fast Particles Moving in a Medium and External Fields, Moscow, USSR, 1987 (in Russian).[9] N. Nasonov, A. Noskov, Nucl. Instr. and Meth. B 201 (2003) 67.[10] G.M. Garibian, C. Yang, X-ray Transition Radiation, Erevan, USSR, 1983 (in Russian).[11] N .N. Nasonov, V.V. Kaplin, S.R. Uglov, V.N. Zabaev, M.A. Piestrup, C.K. Gary, Nucl. Instr. and Meth. B 227 (2005) 41.

On the dynamical effects in the characteristics of transition radiation produced by a relativistic electron in a single crystal plate

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On the dynamical effects in the characteristics of transition radiation produced by a relativistic electron in a single crystal plate

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