ARTICLE IN PRESS Radiation Physics and Chemistry 75 (2006) 913–919 www.elsevier.com/locate/radphyschem X-ray Cherenkov radiation under conditions of grazing incidence of relativistic electrons onto a target surface A. Kubankina, N. Nasonova,Ã, V. Kaplinb, S. Uglovb, M. Piestrupc, C. Garyc aLaboratory of Radiation Physics, Belgorod State University, 14 Studencheskaya str., Belgorod 308007, Russia bNuclear Physics Institute, Tomsk Polytechnic University, P.O.Box 25, Tomsk 634050, Russia cAdelphi Technology Inc., 2181 Park Boulevard. Palo Alto, CA 4306, USA Abstract X-ray Cherenkov radiation in the vicinity of the photoabsorption edge of a target is considered in this work. A possibility of substantial increase in the yield of emitted photons under conditions of grazing incidence of emitted electrons onto the target surface is shown. We discuss peculiarities in the process of X-ray Cherenkov radiation from a multilayer nanostructure as well as possibilities of focusing emitted X-rays with the use of grazing-angle optics. r 2006 Published by Elsevier Ltd. PACS: 78.70Àg; 78.70.Ck Keywords: Cherenkov X-rays; Absorption edge; Relativistic electrons 1. Introduction the hollow cone. The above-mentioned circumstances lead to the following question: how to increase the The Cherenkov effect allows to produce soft X-rays in angular density for the discussed X-ray source? the vicinity of atomic absorption edges, where the Some possible ways to solve this problem are medium refractive index may exceed unity (Bazylev considered in our work theoretically. In Section 2, we et al., 1976). This theoretical prediction has been discuss a possibility to increase the emission angular confirmed experimentally (Bazylev et al., 1981; Moran density by using grazing incidence of emitting electrons et al., 1990; Knulst et al., 2001, 2003, 2004). The onto the target. The next section is devoted to a study of obtained experimental results (Knulst et al., 2001, 2003, X-ray Cherenkov radiation from a multilayer nanos- 2004) demonstrate a possibility to create an effective tructure. A possibility to focus the Cherenkov photon quasi-monochromatic X-ray source with intensity of the flux is considered in Section 4 on the basis of grazing- order of 10À3 photon=electron. On the other hand, the angle optics. Our conclusions are presented in the last average angular density of emission from a possible section. X-ray Cherenkov source is not high because of both a large value of the Cherenkov emission angle and an angular spread of the Cherenkov photon flux close to 2. Grazing-incidence Cherenkov radiation ÃCorresponding author. Tel.: +7 22 341 477; Let us consider emission from relativistic electrons fax: +7 22 341 629. crossing a foil of an amorphous medium. This theore- E-mail address: [email protected] (N. Nasonov). tical task is well known (see, for example, Garibian and 0969-806X/$ - see front matter r 2006 Published by Elsevier Ltd. doi:10.1016/j.radphyschem.2005.12.016 ARTICLE IN PRESS 914 A. Kubankin et al. / Radiation Physics and Chemistry 75 (2006) 913–919 Yang (1983)), so we can use general results presented in scattering is neglected in Eq. (1) for simplicity. This the cited work. We assume that the photoabsorption influence has been studied by Gary et al. (2005). 00 length Lab 1=ow ðoÞ is less than the electron path L=j Formula (1) describes the spectral-angular distribu- in the target. Here w00 is imaginary part of the dielectric tion of emission and allows to study its dependence on susceptibility w ¼ w0 þ iw00 of the target material, L is the the incidence angle j. The distribution (1) apparently thickness of the target, and j51 is the grazing incidence contains a maximum (Cherenkov maximum) determined angle. Then we arrive at a simple model: emission from a by the condition fast electron that moves out a semi-infinite absorbing gÀ2 þ Y2 þ Y2 þ j2 À 2jt0 ¼ 0, target to vacuum, where the emitted photon flux is y x measured by an X-ray detector (see Fig. 1). which can be recast as Since background in the small frequency range under qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À2 0 2 2 0 2 study is mainly determined by transition radiation, we g À w þ Yy þðj À Yx þ w Þ ¼ 0, (2) neglect a contribution of bremsstrahlung. In addition, we consider emission from electrons moving with a in the most interesting frequency range of anomalous dispersion before the absorption edge. In this range, w00 uniform velocity v and thusp assumeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that the multiple qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 2 scattering angle Csc ðk=Þ Lab=LR corresponding to is usually much less than w , so that t Y þ w0 and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a distance of the order of Lab is small relative to the pffiffiffiffiffiffiffiffiffiffiffi 00 00 2 0 1 0 characteristic angle w ðoÞ of the Cherenkov cone. Here t 2 w = Yx þ w . In accordance with (2), Cherenkov 0 À2 is the electron energy, k 21 MeV, and LR is the radiation can be realized if w À g 40. This is well- radiation length. known Cherenkov threshold in the X-ray range. A solution to the considered task is well known Let us consider the angular structure of the Cherenkov (Garibian and Yang, 1983), so we put here only a final peak versusp theffiffiffiffi orientation angle j. For large enough result for the emission distribution at small angles: angles jb w04gÀ1, the distribution of emission in- tensity over the azimuthal angle of the Cherenkov cone is d3E 16e2 w02 þ w002 ¼ uniform in accordance with the asymptotic formula 2 2 0 2 002 dod Y p ðYx þ t Þ þ t 3 2 2 2 À2 2 2 2 2 2 2 d E0 e Yy þðj À YxÞ ðg þ Yy þ Yx À j Þ þ 4j Yy ¼ Â 2 2 À2 2 2 2 2 2 do d Y p ðg þ Yy þðj À YxÞ Þ OðÞOðþÞ w02 j2Y2 Â , ð3Þ Â x , À2 0 2 2 2 002 À2 2 2 2 0 2 2 002 ðg À w þ Yy þðj À YxÞ Þ þ w ðg þ Yy þ Yx þ j À 2jt Þ þ 4j t ð1Þ that follows from (1). On the other hand, the distribution (1) becomespffiffiffiffi strongly non-uniform in the range of small where g is the electron Lorentz factor, angles j w0. In order to show this in some detail, let us rqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi compare the maximum of the asymptotic distribution (3) 0 1 2 2 2 2 t ¼ pffiffiffi ðY þ w0Þ þ w00 þ Y þ w0, 3 2 0 À2 x x d E0 e w À g 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dod2Y p2 w002 1 max 00 pffiffiffi 2 0 2 002 2 0 t ¼ ðYx þ w Þ þ w À Yx À w , with that of the general distribution (1), 2 0 O ¼ gÀ2 þ Y2 þðj Æ Y Þ2, ðÞ y x 3 ðÞ 2 0 À2 d E e w À g 4 B 1 ¼ @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and H ¼ exYx þ eyYy is the observation angle in the 2 2 002 2 0 À2 dod Y max p w j j Æ w À g direction of n (see Fig. 1). An influence of multiple 1 À2 1 C þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A . ð4Þ ðj Æ w0 À gÀ2Þ2 À w0 It follows from Eq. (2) that maxima (4) is realized at the observation angles Yy ¼ 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðÞ 0 À2 2 0 Yx ¼ Yx max ¼ ðj Æ w À g Þ À w . (5) Note that the two maxima (4) exist together only at large Fig. 1. Geometry of the process of Cherenkov radiation. n is enough incidence angles the unit vector in the direction of the emitted photon, v is the pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 À2 velocity of the emitting electron, and j is the incidence angle. j4 w þ w À g . ARTICLE IN PRESS A. Kubankin et al. / Radiation Physics and Chemistry 75 (2006) 913–919 915 At smaller j, the Cherenkov cone begins to contact the target surface, so that only one maximum, ðþÞ dE 2 , do d Y max is realized in the angular range pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w0 À w0 À gÀ2ojo w0 þ w0 À gÀ2, just in accordance with Eq. (5). This maximum can exceed substantially the asymptotic value e2ðw0 À gÀ2Þ=p2w002. The ratio 3 ðþÞ 3 d E d E0 2 2 , do d Y max dod Y max Fig. 3. Dependence of the Cherenkov cone structure on the calculated with formula (4) for different values of the incidence angle j. Presented spectral-angular distributions of 2 0 parameter g w is presented in Fig. 2 as a function of j. Cherenkov radiation have beenp calculatedffiffiffiffiffiffiffiffiffi for Be target with 0 0 Two main conclusions follow from Fig. 2. The parameters o ¼ 111:6 eV, 1=g wmax ¼ 0:4, and wpmaxffiffiffiffiffiffiffiffiffi¼ 0:05. 0 discussed j-dependence is close to the resonant one Distributionspffiffiffiffiffiffiffiffiffi 1 and 2 correspond to j ¼ 0:17 wmax and 0 0 for emitting electrons of high enough energies (g2w b1). j ¼ 3 wmax, respectively. Therefore, it is necessary to choose the optimum value of the incidence angle j very carefully. Enhancement in the emission angular density that is achievable under conditions of grazing incidence of emitting electrons on the target surface can be very substantial. The effect of strong modification of the structure of the Cherenkov cone under the conditions considered is illustrated by Fig. 3 which shows the angular distribu- tion of Cherenkov radiation for a fixed value of the emitted-photon energy.