Features of the transition radiation (TR) generated in the vacuum ultraviolet range by weakly-relativistic electrons escaping from a target at small (grazing) angles have been theoretically studied. A significant (more than tenfold) increase in the angular density of TR with decreasing escape angle is predicted. It is established that the spectral and angular distribution of the TR intensity significantly depends on the target permittivity and the electron energy and escape angleyesBelgorod State Universit

Kubankin, A. S.

DSpace at Belgorod State University

 ISSN 1063-7850, Technical Physics Letters, 2008, Vol. 34, No. 11, pp. 927–929. © Pleiades Publishing, Ltd., 2008.Original Russian Text © A.S. Kubankin, 2008, published in Pis’ma v Zhurnal Tekhnichesko œ  Fiziki, 2008, Vol. 34, No. 21, pp. 46–51. 927 Investigation of the angular distribution of the inten-sity of transition radiation (TR) generated by a nonrel-ativistic charged particle crossing the medium–vacuuminterface [1] showed that the ratio of the real ( ε '( ω )) andimaginary ( ε ''( ω )) parts of the complex dielectric per-mittivity strongly influences the magnitude and posi-tion of the maximum in the angular profile under con-sideration ( ω  is the TR quantum energy). It was estab-lished [1] that, for  ε ''( ω ) = 0, this maximum exactlycoincides corresponds to the angle of total internalreflection of light from the interface and the TR inten-sity at the maximum is greater than that in the case of ε ''( ω )  ≠  0. In the general case, the position of maximumin the angular distribution of TR for nonrelativisticcharged particle depends only on  ε '( ω ) and  ε ''( ω ). A much more interesting situation is observed forweakly relativistic electrons. In this case, the Coulombfield of the radiating particle is deformed and the angu-lar distribution of TR becomes sensitive to not only ε '( ω ) and  ε ''( ω ), but also to the particle energy. Themost interesting phenomena can be expected in the vac-uum ultraviolet (VUV) spectral range, where the ratioof  ε '( ω ) and  ε ''( ω ) can exhibit considerable variationsthat would lead to sharp changes in the spectral andangular characteristics of the TR. The results of investigation [2] of the TR of relativ-istic electrons in the soft X-ray range at small (grazing)angles of electron incidence on the medium–vacuuminterface showed the possibility of increasing theTR yield in cases where the relativistic particle movesat an angle on the order of the angle of total internalreflection of the TR from the interface. Below, the pos-sibility of sharply increasing the angular density ofTR for the grazing escape of weakly-relativistic elec-trons from the target is demonstrated and it is shownthat the mechanism of this growth is different from thatconsidered in [2]. The predicted effect can provide fora more than tenfold increase in the TR density in the10–100 eV range, which is of considerable interest for thedevelopment of effective VUV radiation sources. The spectral and angular distribution of the TR of anelectron crossing the medium–vacuum interface in theresponse plane determined by the vectors of the elec-tron velocity ( V ) and the normal ( n ) to the target surfacecan be described by the following equation:  (1)ωdNdωdΩ---------------e2π2----χ'2 χ''2+1 V ϕ( ) Θ( )sincos–( )2 V2 ϕ( ) Θ( )cos2sin–( )2---------------------------------------------------------------------------------------------------------------V2 ϕ( ) Θ( )cos2sin1 V ϕ( ) Θ( )sincos– V ϕ( )β'sin–( )2 V2 ϕ( )β''2sin+---------------------------------------------------------------------------------------------------------------------------=×V ϕ( ) Θ( )sin V ϕ( )cos–( )β'sin Θ( ) 1 V2 ϕ( )sin2– V ϕ( ) Θ( )sincos–( )sin–( )2 V2 ϕ( ) Θ( )sin V ϕ( )cos–( )2β''2sin2+1 χ'+( ) Θ( ) β'+cos( )2 χ'' Θ( )cos β''+( )2+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, Features of the Transition Radiation Generatedby Weakly-Relativistic Electrons in the Regime of Grazing Escape from a Flat Target  A. S. Kubankin  Belgorod State University, Belgorod, Russiae-mail: kubankin@bsu.edu.ru Received April 2, 2008  Abstract —Features of the transition radiation (TR) generated in the vacuum ultraviolet range by weakly-rela-tivistic electrons escaping from a target at small (grazing) angles have been theoretically studied. A significant(more than tenfold) increase in the angular density of TR with decreasing escape angle is predicted. It is estab-lished that the spectral and angular distribution of the TR intensity significantly depends on the target permit-tivity and the electron energy and escape angle.PACS numbers: 78.70.-g  DOI:  10.1134/S1063785008110084 928 TECHNICAL PHYSICS LETTERS        Vol. 34        No. 11       2008 KUBANKIN where  Θ  is the observation angle (measured relative tothe outer normal to the surface of the medium) and  ϕ  isthe angle of electron escape from then medium (mea-sured from the surface plane of the medium, and  χ  = χ ' +  i χ '' is the complex susceptibility. The numerators in the first and second factors in theright-hand side of Eq. (1) determine the maxima of theangular distribution of the TR, which appear due to atransformation of the Coulomb field of the radiatingparticle in the vacuum and medium, respectively. Thedenominator of the second term for  χ '( ω ) > 0 has a res-onant character, which determines the position of themaximum in the spectral and angular distribution of theVavilov–Cherenkov radiation intensity. The denomina-tor of the third term in Eq. (1) determines the maximumof the angular distribution of radiation that is related tothe total internal reflection of the radiation quanta fromthe medium–vacuum interface, which is described indetail in [1]. Since only the third term of Eq. (1) enters into theexpression for the spectral and angular distribution ofthe TR of nonrelativistic electrons ( V     1), the shape ofβ' 12-- Θ( )cos2 χ'+( )2 χ''2+ Θ( )cos2 χ'+ + ,=β'' 12-- Θ( )cos2 χ'+( )2 χ''2+ Θ( )cos2– χ'– ,= the angular distribution is determined entirely by the χ '(ω) and χ''(ω) values: (2)The maximum in the angular distribution occurs nearthe total internal reflection angle (determined by therelation cos(Θ) = ) and this position is indepen-dent of the orientation of momentum of the radiatingparticle. As the electron energy increases, the Coulomb fieldof the radiating particle exhibits transformation and theshape of the spectral and angular distribution of the TRbecomes dependent not only on the permittivity of themedium, but on the electrum momentum as well. Themaxima in the angular distribution of the TR of aweakly-relativistic electron are determined by twomechanisms. According to the first mechanism, whichis related to the transformation of the Coulomb field ofthe electron, the maxima are observed for |χ'|  1 andϕ  (V +  near the angles Θ ≈  – ϕ ± (V + .The second mechanism is related to the total internalreflection and the corresponding maxima are near theangles Θ ≈ ± . As a rule, the maxima determined by the two mech-anisms are independent. However, at a certain angle ofelectron incidence on the interface, these positions maycoincide and then the angular density of TR will signif-icantly increase. Figure 1 shows the angular distributions of TR witha fixed quantum energy ω for various angles ϕ of elec-tron incidence on the interface, which were calculatedfor experimentally determined susceptibility compo-nents χ'(ω) and χ''(ω) for carbon [4] and an electronenergy of ε = 0.5 MeV. The position of maximum in theoptimum case (curve 3) corresponds to the total internalreflection angle. Figure 2 shows the energy dependence of the ratioof the maxima of the angular distributions for the nor-mal and grazing incidence: G = ω /ω  ωdNdωdΩ---------------e2π2---- χ'2 χ''2+( )=×V2 ϕ( ) Θ( ) Θ( )sin2cos2sin21 χ'+( ) Θ( )cos β'+( )2 χ'' Θ( )cos β''+( )2+-------------------------------------------------------------------------------------------------------.χ'–14--arccos V2 8+ )π2--14--arccos V2 8+ )χ'–( )arccosdNgrazingdωdΩ-------------------- maxdNnormaldωdΩ------------------- max10801.5Θ, rad1.00.50–0.5–1.0–1.5642dNdωdΩ---------------106 phot./ el. eV ster.( ),1234Fig. 1. Increase in the angular density of TR at ω = 50 eVfor electrons with an energy of ε = 0.5 MeV incident on themedium–vacuum interface at various angles ϕ = π/2 (1),0.7 (2), 0.45 (3), and (4) 0.2. TECHNICAL PHYSICS LETTERS      Vol. 34      No. 11      2008FEATURES OF THE TRANSITION RADIATION GENERATED 929(for the grazing geometry, ϕ was selected equal to theaforementioned optimum value), where curves 1 and 3correspond to the nonrelativistic and relativistic cases,respectively. An insignificant increase in the angulardensity of TR in the nonrelativistic case is explained bya weak dependence of the shape of the angular distribu-tion on the orientation angle as described by Eq. (2). Inthe weakly relativistic case (described in [2]), the angu-lar distribution of TR is determined primarily by thedegree of transformation of the Coulomb field of theradiating particle. In the region of manifestation of thephenomenon under consideration (i.e., for χ'' = 0 in thelimit as γ  ∞, where γ is the Lorentz factor of theradiating electron), the ratio G of the TR intensities atthe maxims of angular distributions for the grazing andnormal incidence is about 4. Thus, the results of this investigation predict thepossibility of significantly increasing the spectral andangular density of the TR of weakly relativistic elec-trons crossing the medium–vacuum interface at a graz-ing angle. A more than tenfold increase in the TR inten-sity is manifested near the angle of total internal reflec-tion of the radiation quanta from the interface (Θ ≈) at an electron energy on the order of100 keV. The increase in the angular density is mostpronounced at 10–100 eV range in the case where theposition of maximum in the angular distribution of TR(formed due to the transformation of the Coulomb fieldof the radiating electron) is close to the angle of thetotal internal reflection of TR from the medium–vac-uum interface. It should be emphasized that this effectis not manifested for nonrelativistic and ultrarelativisticelectrons. Acknowledgments. This study was supported inpart by the Presidential Program of Support for YoungScientists (project no. 2555.2007.2) and the RussianFoundation for Basic Research (project no. 06-02-16942). REFERENCES 1. A. P. Kobzef, S. Mikhalyak, E. Rutkovski, and I. M. Frank,Yad. Fiz. 18, 865 (1973). 2. C. Coullaud, Nucl. Instrum. Meth. Phys. Res. A 495, 171(2002). 3. G. M. Garibyan and Yan Shi, Transition Radiation(Acad. Sci. Armenian SSR, Yerevan, 1983) [in Russian]. 4. B. L. Henke, E. M. Gullikson, and J. C. Davis, At. DataNucl. Data Tables 54, 2 (1993). Translated by P. Pozdeevχ'–( )arccos10070ω, eV605040512315GFig. 2. Suppression of the effect of increase in the angulardensity of TR as manifested by the ratio G of the TR inten-sities at the maxima of angular distributions for the grazing(optimum ϕ) and normal incidence angles at an electronenergy ε of (1) 0.05, (2) 0.5, and (3) 5 MeV. 

Features of the transition radiation generated by weakly-relativistic electrons in the regime of grazing escape from a flat target

http://dspace.bsu.edu.ru/bitstream/123456789/4523/1/Kubankin_Features%20of%20the%20Transition_2008.pdf

Features of the transition radiation generated by weakly-relativistic electrons in the regime of grazing escape from a flat target

Abstract

Similar works

Full text

Available Versions

DSpace at Belgorod State University