The problem of transition radiation under impact of relativistic particles under small angle to the atomic string of a crystal is considered. The conditions under which the non-uniformity of the electron density along the string is not substantial are obtained. In this case the problem of transition radiation is reduced to the problem of the particle radiation on the thin fiber-like dielectric target. The formulae for the spectral-angular distribution of transition radiation under both regular and random collisions of a particle with the set of fiber-like targets are obtained. The radiation of the particle on a single atomic plane and on a set of planes in crystal is also considered

Shul’ga, N.F.

Syshchenko, V.V.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

POLARIZATION OF TRANSITION RADIATION ON SOME SORTS OF TARGETSN.F. Shul’ga1, V.V. Syshchenko21National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukrainee-mail: shulga@kipt.kharkov.ua 2Belgorod State University, Belgorod, Russian Federatione-mail: syshch@bsu.edu.ru The problem of transition radiation under impact of relativistic particles under small angle to the atomic string of a crystal is considered. The conditions under which the non-uniformity of the electron density along the string is not substantial are obtained. In this case the problem of transition radiation is reduced to the problem of the particle radiation on the thin fiber-like dielectric target.  The formulae for  the spectral-angular  distribution of  transition radiation under both regular and random collisions of a particle with the set of fiber-like targets are obtained. The radiation of the particle on a single atomic plane and on a set of planes in crystal is also considered.PACS: 41.60.-m1. INTRODUCTIONThe  transition  radiation  arises  under  crossing  by charged particle the boundary between two media with different dielectric properties (see [1-4] and references in  them).  For  relativistic  particle  this  radiation  is concentrated  in  the  region  of  small  angles  along  the direction of the particle motion. The process of radiation develops  in  large  spatial  region  along  the  particle velocity, that is called as the coherence length [2,5,6]. If the  particle  in  the  limits  of  this  region  crosses  some boundaries  of  different  media,  the  interference  of radiation emitted under crossing of every boundary is substantial. It was shown in [7] that for long waves not only longitudinal, but also transverse dimensions of the region of  radiation formation could have macroscopic sizes.  If  the  transverse  size  of  the  target  satisfies  the condition  γ λ≤⊥L ,  where  λ  is  the  length  of  the radiated  wave  and  2/12 )1( −−= vγ  is  the  particle's Lorentz-factor (we use the system of units, in which the velocity  of  light  is  equal  to  unit),  than the transverse sizes  of  the  target  and  its  geometrical  shape  make substantial influence on the transition radiation. In  the  presented  paper  the  problems  on  transition radiation by relativistic particle on dielectric targets of some particular geometries are considered. We consider the  region of  high  frequencies  of  radiated  waves,  for which  the  dielectric  function  of  the  medium  can  be presented in the form:221ωωε ωp−≈ ,    pωω > ,  (1)where mrne ep /)(42 piω =  is the plasma frequency, m is the electron mass, )(rne is the electron density in the target.2. TRANSITION RADIATION IN THE FRAMES OF PERTURBATION THEORYFor the case of high frequencies of radiated waves, pωω > > , the second term in the dielectric function of the  medium  (1)  can  be  considered  as  a  small perturbation. In the frameworks of perturbation theory which  was  built,  the  spectral-angular  density  of radiation  with  given  polarization  is  determined  by relation2224Ieddd piωοω=Ε, (2)where e  is the polarization vector, ke ⊥ , 1|| =e, k is the wave vector of the radiated wave, and the value I in our case can be written in the form)()(41 )0(23 rErerdI prki  ωωpi ω ∫≈ , (3)where  )()0( rEω  is  the Fourier  component  of  the non-disturbed  Coulomb  field  of  the  uniformly  moving relativistic particle,∫ −−−=rievviedrE κω κωδκωωκpiκ )()(223)0( . (4)The  spectral-angular  distribution  of  radiation summed over polarizations is determined by equation2241 Ikddd ×=Εpiοω. (5)Using  the  Fourier  transformation  of  the  electron density distribution )(rne,∫ −= rqieq ernrdn )(3 ,we can rewrite (5) in the form226kJkmeddd ×=Εωοω, (6)where ∫ −−−−−−= ))(()(2 2223vqkqkvqknqdJ qkωδωωpi. (7)3. TRANSITION RADIATION ON A DIELECTRIC FIBER AND ATOMIC STRINGLet us consider the transition radiation by relativistic particle incident on a thin dielectric fiber under small angle 1< <ψ  to its axis . The atomic string in crystal [6] or nanotube [8] can be treated as such fiber in the case when the length of radiation formation (the coherence length)  exceeds  in  much  the  atomic  string  thickness along the particle motion direction:PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, 135-137. 135ψθγωγ Rlcoh2112~ 222> >+, (8)where  R is the screening radius of the atomic potential (Thomas-Fermi  radius),  θ  is  the  radiation  angle  (the angle between the wave vector k of the radiated wave and the particle velocity). Let us take the electron density distribution in the fiber in Gaussian form,   −+−−= 22022 2)()(exp2)(RyyzxRnrn eeψpi, (9)where  z axis is directed along the particle velocity  v, the axis of the fiber is parallel to (x, z) plane, 0y  is the distance between the particle trajectory and the axis of the fiber,  en  is the electron density per unit length of the fiber. The Fourier transformation of this distribution has the form:×+= )(2 0 zxyiqeq qtgqenn y ψδpi   +−×2)(exp222 Rqq yx.  (10)The spectral-angular distribution of radiation (6) must be  averaged  over  all  possible  values  of  the  impact parameter 0y ,∫∞∞−Ε=Εοωοω ddyddyadddy)(1 00 , (11)where ya  is the distance between the atomic strings in the crystal along the  y axis. After substituting (7) with qn  determined by (10) into (6) and averaging according to  (11),  one  can  demonstrate  that  for  small  radiation angles  ( 1< <θ )  and  for  small  angles  of  incidence  (1< <ψ )  the  condition  (8)  leads  to  the  possibility  to neglect  the  second  exponential  factor  in  (10),  that corresponds to the case of zero thickness of the fiber. In other  words,  the  details  of  the  electron  density distribution in the fiber are not substantial under such conditions.For  the  case  of  infinitely  thin  dielectric  fiber  we obtain:),(2226ϕθω ψγοωFmanedddye=Ε,            (12)where2/322222221cos121cos21),( +−+ +−+=γ ψθγϕγ θγ ψθγϕγ θϕθF ,       (13)ϕ  is the azimuth angle (the angle between the  x axis and the projection of the wave vector k onto the plane (x,  y)). This function is plotted on the Fig. 1. One can see  that  the  angular  distribution  possesses  the  axial symmetry  relatively  to  the  axis  of  the  fiber  ( ψθ = , 0=ϕ ) (it  can be easy shown analytically from (13)). Near the axis of symmetry of the angular distribution the intensity has rather high level, but with incraesing of the  incidence  angle  ψ  the  minimum  in  the  center developes itself. For 110~ −γψ  the distribution has the shape of narrow double ring. Fig. 1. Surface  plot  (view  from  above)  of  the  function ),(100 ϕθF⋅  for 310−=ψ , 2000=γ  ( ϕθθ cos=x,  ϕθθ sin=y ). The lines of equal level of the surface are shownChoosing the polarization vectors in the formxxekeke ××=)1(, ω)1()2( eke ×= ,                         (14)we  find  that  radiation  is  partially  polarized  in  )1(e  direction with polarization22221cos211 +−+=γ ψθγϕγ θP.               (15)This function is plotted on Fig. 2.Fig. 2. Polarization of radiation determined by Eq. (15) at the same conditions as in Fig. 1When  our  particle  moves  through  the  cristall  it collides  with  the  periodical  set  of  parallel  atomic strings. The motion of the particle in this case can be both  regular  and  chaotic  [9].  If  the  particle  moves chaotically in the plane perpendicular to atomic strings (under the constant incidence angle ψ ), we can neglect the interference of radiation produced by interaction of the  particle  with  different  strings.  In  this  case  the symmetry of angular distribution of radiation intensity, described above, leads to the same form of the angular distribution (13), as in the case of the single string. The total  intensity  of  radiation will  be proportional  to  the number of strings under collision.If the particle motion in the crystal  is  regular,  the account of interference effects is necessary.1364. TRANSITION RADIATION ON ATOMIC PLANE IN A CRYSTALNow let  us consider the transition radiation of the particle on the atomic plane in crystal in the case when this  atomic  plane  can  be  treated  as  a  uniform  thin dielectric plate. The problem of radiation on a plate is not  new in  the  theory  of  transition  radiation  (see  for example [1-3]), but our perturbation theory permits to obtain results in a rather simple way.The Fourier  transformation of  the electron density distribution  under  condition (8)  can  be  written in  the form)()()2( 2 zxyeq qtgqqnn += ψδδpi .               (16)Here en  denotes the electron density per unit area of the plane.  Substitution (16) into general  formulae (6),  (7) gives  us  the  following  result  for  the  spectral-angular distribution of transition radiation:),(4 222226ϕθψωγοωpe Fmneddd=Ε,                   (17)where22222222221cos)sin(121cos)sin(),( +−++ +−+=γ ψθγϕθγϕθγγ ψθγϕθγϕθγϕθpF.(18)The angular distribution (18) is plotted on the Fig. 3 for the case 14 −= γψ . One can see that the distribution is symmetrical relatively to the reflection by the atomic plane. The shape of distribution looks like two empty cones, one of which is directed almost along the particle velocity,  and  the  second is  "reflected"  by  the  atomic plane.Fig. 3. 3-dimentional plot of the function ),( ϕθpF  (18) ( ϕθθ cos=x , ϕθθ sin=y ) for 3102 −⋅=ψ , 2000=γThe  polarization  of  radiation  is  determined  by Stocks parameters:2222222221211cos21cossin2 +++− +−=ψ γθγψθγϕθθγψ γθγϕγ θϕγ θξ , 22222222222223211cos21cossin +++− +−−=ψ γθγψθγϕθθγψ γθγϕγ θϕθγξ .   (19)The radiation is linearly polarized with polarization 2321 ξξ +=P .  It  is  easy can be seen that  for  Stocks parameters (19) we have 1=P  for all values of θ  and ϕ .  The  angle  between  vector  )1(e  and  direction  of maximum polarization  α  is determined in our case of 100%  linear  polarization  by  relation  αξ 2sin1 = , αξ 2cos3 =  (see Fig. 4). Fig. 4. Vector plot of polarization of radiation at the same  conditions  as  on  Fig. 3.  Directions  of  arrows indicate  the  direction  of  maximal  polarization  (respectively to )1(e  direction)This work is  supported in part by INTAS (project 97-30392).REFERENCES1. V.L. Ginzburg,  V.N. Tsytovich.  Transition Radiation and Transition Scattering. Bristol, New York: A. Hilger, 1990, 360 p.2. M.L. Ter-Mikaelian.  High-Energy  Electro-dynamic  Processes  in  Condensed  Matter.  New York: “Wiley Interscience”, 1972, 457 p.3. G.M. Garibian,  C. Yang.  X-ray  Transition Radiation.  Yerevan:  Publ.  Acad.  of  Science  of Armenia, 1983, 320 p (in Russian).4. P. Rullhusen,  X. Artru,  P. Dhez.  Sourses Using  Relativistic  Electrons.  Singapure:  “World Scientific”, 1998, 202 p.5. B.M. Bolotovskij,  The  length  of  formation and its meaning for radiation by moving charges // Proc. P. N. Lebedev Phys. Inst. 1982, v. 140, p. 95-140 (in Russian).6. A.I. Akhiezer,  N.F. Shul’ga.  High-Energy Electrodynamics in  Matter.  Amsterdam: “Gordon and Breach”, 1996, 388 p.7. S.N. Dobrovol’skii,  N.F. Shul’ga.  Theory of relativistic electron transition in a thin metal target. // JETP, 2000, v. 117, p. 668-672.1378. J.W. Mintmire,  B.I. Dunlap,  C.I. White.  Are fullerem tubulus metallic? // Phys. Rev. Lett. 1992, v. 68, p. 631-634.9. A.I. Akhiezer,  V.I. Truten',  N.F. Shul'ga. Dynamical chaos in the motion of charged particles through a  crystal  //  Phys.  Reports. 1991,  v. 203, p. 289-343.138

Polarization of transition radiation on some sorts of targets

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Polarization of transition radiation on some sorts of targets

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)