Diffracted transition X-radiation (DTR) of relativistic electron crossing a single crystal plate of limited thickness is considered within the framework of x-ray diffraction dynamical theory in the geometry of BraggyesBelgorod State Universit

Blazhevich, S. V.

Noskov, A. V.

DSpace at Belgorod State University

A WAY TO INCREASE THE SPECTRAL-ANGULAR DENSITY OF DIFFRACTED TRANSITION RADIATION OF THE RELATIVISTIC ELECTRON IN SINGLECRYSTALS.V. Blazhevich*, A.V. Noskov **Belgorod S tate University, 14 S tudencheskaya Str., 308007, B elgorod  Russia.* *  B elgorod  University C onsum er Cooperatives, 116A Sadovaya, Str., 308023, B elgorod  Russia.E-m ail: blazh(albsu. edu.ru  Received June 9, 2006Diffracted transition X-radiation (DTR) o f relativistic electron crossing a single crystal plate o f limited thickness is considered within the framework of x-ray diffraction dynamical theory in the geometry of Bragg. The analytical expression for spectral-angular characteristics o f DTR is obtained with tacking to account the orientation o f external surfaces relative to the diffracting atomic planes in the crystal. It is shown that under a fixed angle o f electron incidence on the system o f diffracting planes in the crystal the spectral- angular characteristics DTR significantly depend on the external surface orientation.K EY  W ORDS: relativistic electron, transition radiation, dynamical diffraction, angle distribution, single crystalThe transition radiation originates [1] w hen a fast charged particle crosses a boundary betw een two different mediums. I f  one o f  the m edium s is m onocrystal the originated radiation can be diffracted on a system o f  parallel atomic planes in  the crystal [2-4]. This radiation is nam ed the diffracted transition radiation (DTR). Up to now  D TR in dynam ical approach w as considered only in the case w hen crystals entrance surface is parallel to the system  o f  the atomic planes. In the current w ork the expression for spectral-angular distribution o f  D TR is obtained on the base o f  the dynam ical diffraction theory [5] w ith  taking to account the angle between the entrance surface and the set o f  atomic planes in  the crystal, as a parameter. It is shown that spectral-angular density o f  D TR substantially depends on this parameter. So by carving the crystal in  different ways one can change the spectral-angular characteristics o f  DTR. The conditions o f  observation o f  D TR am plitude increase and decrease is shown in the work. The limit process to the case o f  the crystal entrance surface parallel to the atomic planes is specified.T H E  B A SIC  F O R M U L A SL et’s consider the radiation o f  a fast charged particle crossing a m onocrystal L thick w ith constantvelocity V  (Fig. 1). To describe this process we w ill use the equations for Fourier direct image o f  electrom agnetic fieldE(k,ft>)= j"t//t/3rE(r,/)exp(7ffi»/-7'kr). (1)Since the Coulomb field o f  an ultrarelativistic particle in  the close approxim ation is transverse, the incident E 0(k,a>) anddiffracted E j(k , co) electrom agnetic waves, can be defined by two com ponents w ith different values o f  transverse polarizationE 0(*,© ) = 4 1)(k,ffl)e [)1) + 4 2)(k,ffl)e[)2), (2)E j (k,  co) = E ^ ] (k , ft>)e j1’ + £ ’1(2)(k,ffl)eJ2).The unit vectors cJ,1 ’ . e}2’ . c j11 and ej2) are chosen in thefollowing way. Vectors e j/’ and e{,21 are perpendicular to v ec to rk  , and vectors e ^ ’and e j2) are perpendicular to vector k  + g . The vectors e[)2), e{2) are situated on the plane o f  vector k  n k  + g ( it -polarization), and e^11 n e j11 are perpendicular to this plane (<x- polarization) ; It is obvious that e j/1 = ej1’ . The reciprocal lattice vector g defines a set o f  reflecting atomic planes. B y the use o f  two-w ave approxim ation o f  the dynam ic theory o f  diffraction [5], we can w rite the w ell-know n equation set for Fouriertransform  im ages o f  the incident wave and diffracted w ave intensities [6]Fig.l. The radiation process geometry.\ (a2 (\ + x 0) - k 2)E^s) + a 2 x_^C(s) e {s) =8x2ieco0VP(s> <J(©-kV), w 2X » C {s)E ^  + (®2(1 + Xo ) -  (k  + g )2) ^  = 0, (3)where , x  „ is Fourier coefficients o f  crystal dialectical susceptibility expansion in  series by  reciprocal vectors o f  the crystal lattice gx(<0, r)  = 'Ltg(№ly gr = Z (x i(® )  + i l l ( ^ y gr ’ (4)L et’s consider a crystal w ith sym m etry ( / „  = % ) .The is defined by  expression= Zo (F ( g ) ! z  X ^ g )  /  ■ )exP f -  j g '2«r (5)where Zo = Zo + 'Xo is the average dialectical susceptibility o f  the crystal m edium , F(g)- form -factor o f  the atom containing Z electrons, 5 (g ) is the structure factor o f  the elem entary crystal cell containing N lt atoms, u T - mean- square am plitude o f  therm al oscillations o f  atoms in  the crystal.The quantities C l v| and l>is ' in system (3) are defined in the following w ayC (s) = e ^ )e ^ ) , C (1) = 1, C (2) = cos 20^ ,P (s) = e (c I p ) ,  = sin <p, P (2) = cos ç ,(6)where c = k -  o N  /  V is the com ponent o f  the virtual photon perpendicular to particle velocity V  ( p  = m O I V , 0 « 1 is the angle betw een k  and V ), dB is the angle betw een the electron velocity and the set o f  crystallographic planes (Bragg angle). The azim uth angle (p is counted o ff from  the plane m ade up by  V  and g . The value o f  the crystal lattice reciprocal vector is defined a s g  = 2 coB sin 6,, I V , w here coB is Braggis frequency. The system (3) describes the field o f  cr -polarization if .v = 1 or the field o f  n  -polarization if  s=2.In this w ork the case o f  Braggis geom etry is under the consideration, w hen a diffracted field exits the entrance surface o f  the plate. The diffracted radiation is directed close to vector g + o N  I V 2 (see Fig. 1.).By solving the dispersion equation following from  system  (3)(®2 (l + x 0) ~ k 2)(m1 (1 + xo )  ~  (k  + g )2) -  a 4X - sZ s C (s) = 0, (7)in standard m ethods o f  the dynam ical theory [5] we w ill have obtained the expression for wave vectors o f  incident and diffracted waves2k ^ s) -  |k + g| -  cOtJ i + Zo +2 sVi(s) _   ^(l + g ) + ^ ( j )s > _ _(8a)(8b)here the following notations are insertedK (s) =1Ç{s)~ - s - pZ " C\  (Zgf -s i - p o>2X-2* V=ri{s)((o)+ß{S)^  + s )  , C (S) =Tjis>(ca)+ ^ (S) ^ W)(1 + g)ß {s) = -q  + g ) 2 x l 2c is)1 )4 x tZo „ sin(05 -  (5 -  0))P  - I I X x 1 C* -2 "  | ^ |  c w ’ sin(05 + (5 -  0))(02 + r ~2 - z ' o )  ,7 M ( j )  (k  + g ) 2 - ^ 2 2 s in 2 <9a ^ a B (\ + dcos (pcotdB) ^2 a>*\zL\Cw  V ^ z ' A C(9)8  is the angle betw een the entrance surface o f  the crystal target and the crystallographic plane o f  the crystal.In expression (8) index i = 1,2 defines tw o branches o f  X -radiation w aves passing through the crystal. Because theinequality 2 s in 2 6B /K 2|^ g |c w  » 1  holds true in  the range o f  X-radiation frequencies r f s) {(')) is a very fast function,therefore it is convenient to use it as a spectral variable characterizing co .The solution o f  the first equation in set (3) for the Coulomb field o f  the relativistic electron in vacuum  ( /  „ = 0 )  isas follows:r(i)vac _ 08 n 2ieV dP{s)(O - y ~ 2 - 9 2(10)The same for the Coulomb field in  crystal is as2^(s)cr _  i s V ep «® z o - r ~ 2 - e 2(ii)So, the field o f  the radiation form ed on the crystal entrance surface is1 12 • f^ (s)vac-cr _  i z V  Q p (s)CO - r - 2 - e 2 z o - r 2 ~ o 2 (12)By m ultiplying the am plitude o f  the radiation field form ed on the entrance surface (12) by  the reflection am plitude coefficient Ç / v| w hich defines the reflection o f  the field in a crystal o f  lim ited thickness, we w ill obtain the am plitude o f  DTR7(s)J R ad2 • (  DTR _  £(s)vac-crq (s) _  8?T i eV nn(s)0 -6P(CO1 1- y - 2 - e 2 z o - y - 2 - e 2Q « (13)The reflection am plitude coefficient Q <s> is given as [5]iojLC (s) Y 2cos(^0)z 2+q( s -)ia>L2cos(^0)•2+^)Q (s>=- icoL- z - ^ l z 2 + q (s) |e 2cos^ >jz  +qU'il.z 2 + q (s) \e 2cos(¥t>) (14)where the following notations are inserted q = - C (s) Z g / £ > z = - (a  ~~ (l + ^ )(/o  ~ 0 2 - y  2))/ 2 e ,n / 2 -  ( 0 /; + 8) «  v|/0 is the angle betw een the incident electron velocity and external norm al to the target surface. The am plitude coefficient (14) reflects the existence o f  two branches in  the solution (13) i.e. the existence o f  two branches o f  X -radiation w aves in the crystal.I f  we substitute (13) by  expression [6]d 2W DTRdoxK l=  co2 ( 2S=1^RadD TR(15)we w ill get the following form ula for the spectral-angular density o f  the radiation2 * \ 2co­l l  i r D T R  2a N  edcodD. n 2 e2Y ] p(Sy 1 1r 2 + 0 2 r 2 + 0 2 - z ' o8w (16)SPECTRAL-ANGULAR DISTRIBUTION OF DTR IN CASE OF A THIN CRYSTALW ith a thin crystal the absorption o f  radiation can be neglected and the reflection am plitude coefficient (Q<s] can be given by( ? « = .  "  £(17)O) _ e  exp ib(s) №(■*) _l _ e  exp - i b " <Param eter e «  sin (0 /; -  8) /  s in (0 /; + 8) defines the orientation o f  the crystal plate entrance surface. If c  > 0 the crystal is oriented according to  B ragg’s geometry. In this case the part o f  TR formed on the entrance surface is diffracted on the set o f  the atomic planes and exits the crystal through its entrance surface. U nder the fixed value o f  B ragg’s angle dB param eter 8  becom es negative and its absolute value grows w hen the electron incident angle {6B + (?) decreases (in the lim it case 8  —» - 6 B ) w hich consequently leads to the increase o f  s . On the contrary, if  the electron incident angle increases e decreases (the lim it value o f  8 is reached in the case o f  5  —» 0 n ).A  very interesting expression is given for the square m odule o f  the reflection am plitude coefficient C)lv| , describing the D TR spectrum as the function o f  the orientation o f  the crystal plate entrance surfacesin\Q, , |2 (s)\  _ _b (s)-- £:( sr-  £  +  f S i n ' b (s)(18)( s r - ewhere <j(s) = r/(s) + /3(s)( 1 + s ) / 2 ,  r fs) =a 2sin2d?g |^  a>B (1 + 0cos<pcot0B)j \ x p s) v 2\ z p s) Iv CO ■ | c "[ ^ + r 2-IoThe curves describing the spectrum o f  D TR built up on formula (18) are presented in Fig.2 for different values o f  param eter s . One can see that under the increasing o f  8 the D TR spectrum becom es narrower, w hich signifies the decreasing o f  the frequency region o f  the TR total reflection in  the crystal. The total reflection region (anomalousdispersion region) is the frequency region where the wave vector k [l's) o f  the incident wave (see (8a)) acquires a com plex value in the absence o f  absoiption ( / r v| = o ) ,  the w aves radiated on the entrance surface are com pletely reflected in  the ciystal by  the atomic planes and do not m ove worward.This region is defined in  accordance w ith the expression for K <s) (8) can be presented as- 4 ^ - P {s){\ + s ) I 2 < tj{s) < ^ - j 3 (s](l + s ) / 2 ,  ( 19a)or byIq ?M l- V 7 k | c w - / ? 0 <2 s in “ 6B V 2coB(l + 0cos<pcot0B ) _ ^  < I (i)CDFig.2. Influence of the crystal plate entrance surface orientation on the DTR spectra.- A (19b)w here = -ZoThe w idth o f  the area is A /;l v| = 2 -Je . It is im portant to  note that the m agnitude o f  this area depends on the orientation o f  the ciystal. From  (19) one can see that the presence o f  the total reflection area is a dynam ical effect, in  cinem atic approach, w hen = 0 . this area turns into the point o f  exactBragg resonance for the pseudo photon o f  the relativistic electron Coulomb fieldC0B c o l \  + 6 c o s i p c o X 6 B  + - ^ — —  ( $ 2 + T  24 sin " 6 ,The m iddle point o f  the m axim um  is defined from  (18) bymax ------1— - i d 2 + r ~2 ~ Z ' ( ) f \  +  s ) l 24  F  (21)(22)In accordance w ith (21) the total reflection area shifts to  the left w ith  increasing param eter a (see Fig. 2) and the m idband frequency o max grows (20). This effect is not connected w ith dynam ic scattering because ®max does not depend on param eter ■/„. W ith decreasing a the am plitude o f  D TR spectrum  is on a considerable increase, w hich isconnected w ith the decreasing cross direction dim ensions o f  the reflected photon beam.The cause o f  the form ation o f  subordinate m axim a (see Fig.2) in D TR spectrum  is the interference betw een the second wave field penetrating the ciystal and the first wave field originateng on the input ciystal surface passing backw ard inside the ciystal.I f  we substitute (18) for (16) we w ill get the general expression for the spectral-angular distribution o f  DTR.Sill(O-d 2N m Kdcodfl j f - * 2 Z  p(sVs=l ■e 2 y  2 + d 2 -  x'ob (s)( s r - a: (s) - a  + a s i n 2 b{s)■ \  '(23)- aIt is necessaiy to  note that in the particular case o f  entrance ciystal surface parallel to a ciystallographic plane ( 8  = 0 or <: = 1 ) the expression (23) turns into the expression for D TR obtained in  [7].ANGULAR DENSITY OF DTRIn order to study the dependence o f  the angular density o f  D TR on param eter 8 le t’s integrate (20) w ith thefrequency function ^ {s\co)r D TRfd N LdQ.Cd2N Dm dcodD..  , 2 ArD T R  r J r  r ( 1rd  N  Ldco= — ;— —---------- 9 d ç  .dcodD. 2 sin 2 6 R (24)To build the curves describing the angular density for cr -polarization le t’s write (24) asF  = -(;r 2+0l )Z -2r D T R  2  y*  dN _ e Uf 1/^ (1) 4 COç,dQ. k 2 2 s in 2 (9R co4y B(Sil l ~?2+4CO"b il)- e- a  + a sin" b il)?>(■-  a(25)where 6  = 6  sm cp.The curves o f  the angular density o f  D TR is calculated by formula (25) presented in Fig.3. The curves are built for different value o f  param eter a and fixed values o f  other param eters shown in the figure. Fig.3 shows that w ith the increasing o f  the incidence angle o f  the electron on target (with decreasing 8 ) the angular density o f  radiation rises steeply, w hich is im portant from the point o f  view  o f creating alternative quasim onochrom atic sources o f  X-radiation.FCONCLUSIONThe detailed theoretical analysis o f  relativistic electron DTR in ciystal is carried out in  B ragg’s Fig.3. Influence of orientation of the scattering geom etiy. O n the base o f  the dynam icalcrystal plate on angular density of DTR. theory o f  x-ray diffraction the analytical expressionfor spectral-angular characteristics o f  DTR is obtained by taking into account the orientation o f  external surfaces relative to the diffracting atom ic planes in  thecrystal. It is shown that under a fixed angle o f  electron incidence on the set o f  reflecting atomic planes the spectral- angular characteristics o f  DTR depend substantially on the orientation o f  the entrance surface o f  the crystal. The abrupt increase o f  the angular density o f  DTR w ith the increasing o f  param eter 8 we explain by  the decreasing o f  cross dim ensions o f  the reflected photon beam  under increasing the angle o f  electron incidence on the entrance crystal surface. A nd for all that the total reflection area w ill narrow  and its m iddle frequency shifts dow nw ard 0] shifts upward). This effect can be used for intensity enhancem ent o f  x-radiation source built on a basis o f  D TR mechanism.This w ork is supported by  the Russian Foundation for Basic Research (project 05-02-16512) and an Internal G rant o f  B elgorod State University 2005.REFERENCES1. V. Ginzburg, I. Fran Emission from uniformly moving electron appeared when crossing a boundary between two different media // J. Phys. (USSR). - 1945. - V  9. - P. 351- 356.2. A. Caticha Transition- diffracted radiation and Cherenkov emission o f X- rays // Phys. Rev. A. - 1989. - V.40. - P.4322 -4329.3. N. Nasonov Borrmann effect in parametric X-ray radiation // Phys. Lett. - 1999. - V.A260. - P.391-394.4. Y.N. Adischev, S.V. Arishev, A.V. Vnukov et. al. Angular distribution o f X-ray radiation by 500 MeV electrons in a tungsten crystal/ / Nucl. Instr. Meth. B. - 2003. - V.201. - P. 114-122.5. Z. Pinsker Dynamical Scattering of X-rays in Crystals. -  Berlin: Springer, 1984. - 369 p.6. V. Bazylev, N. Zhevago Emission From Fast Particles Moving in a Medium and External Fields. -  Moscow: Nauka, 1987 -289 p. (in Russian).7. C.K. Gary, V.V. Kaplin, N.N. Nasonov, M.A. Piestrup, S.R. Uglov, V.N. Zabaev X-rays from relativistic crossing a multilayer nanostructure // Nucl. Instr. and Meth. B. - 2005. -V.227. - P.41-47.МЕТОД УВЕЛИЧЕНИЯ СПЕКТРАЛЬНО-УГЛОВОЙ ПЛОТНОСТИ ДИФРАГИРОВАННОГО ПЕРЕХОДНОГО ИЗЛУЧЕНИЯ РЕЛЯТИВИСТСКОГО ЭЛЕКТРОНА В МОНОКРИСТАЛЛЕС.В. Блажевич , А.В. Носков*  Белгородский государственный университет, Россия, Белгород, 308007, Студенческая 14.* *  Белгородский университет потребительской кооперации, Россия, Белгород, 308023, Садовая 116А.В рамках динамической теории дифракции рассматривается дифрагированное переходное излучение (ДПИ) релятивистского электрона, пересекающего монокристаллическую пластинку конечной толщины в геометрии Брэгга. Получены аналитические выражения для спектрально-угловых характеристик ДПИ с учетом различных ориентаций входной поверхности относительно дифрагирующих атомных плоскостей кристалла. Показано, что при фиксированном угле падения электрона на систему дифрагирующих плоскостей кристалла, спектрально-угловые характеристики ДПИ существенно зависят от ориентации входной поверхности.КЛЮЧЕВЫЕ СЛОВА: релятивистский электрон, переходное излучение, динамическая дифракция, угловоераспределение, монокристалл

A way to increase the spectral-angular density of diffracted transition radiation of the relativistic electron in single crystal

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A way to increase the spectral-angular density of diffracted transition radiation of the relativistic electron in single crystal

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