X-Ray Cherenkov Radiation Under Conditions of Grazing Incidence of Relativistic Electrons Onto a Target Surface A
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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: 78.70 g; 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 10 3 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
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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 g 2 þ Y2 þ Y2 þ j2 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 2 0 2 2 0 2 study is mainly determined by transition radiation, we g w þ Yy þðj 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 ð k= 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 2 is the electron energy, k 21 MeV, and LR is the radiation can be realized if w 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 w04g 1, 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 2 2 2 2 2 2 2 d E0 e Yy þðj YxÞ ðg þ Yy þ Yx j Þ þ 4j Yy ¼ 2 2 2 2 2 2 2 2 do d Y p ðg þ Yy þðj YxÞ Þ Oð ÞOðþÞ w02 j2Y2 , ð3Þ x , 2 0 2 2 2 002 2 2 2 2 0 2 2 002 ðg w þ Yy þðj YxÞ Þ þ w ðg þ Yy þ Yx þ j 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 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 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 2 x x d E0 e w 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 Yx w , with that of the general distribution (1), 2 0 O ¼ g 2 þ Y2 þðj Y Þ2, ð Þ y x 3 ð Þ 2 0 2 d E e w g 4 B 1 ¼ @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and H ¼ exYx þ eyYy is the observation angle in the 2 2 002 2 0 2 dod Y max p w j j w g direction of n (see Fig. 1). An influence of multiple 1 2 1 C þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A . ð4Þ ðj w0 g 2Þ2 w0
It follows from Eq. (2) that maxima (4) is realized at the observation angles Yy ¼ 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 0 2 2 0 Yx ¼ Yx max ¼ ðj w g Þ 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 2 velocity of the emitting electron, and j is the incidence angle. j4 w þ w g . ARTICLE IN PRESS
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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 w0 g 2ojo w0 þ w0 g 2, just in accordance with Eq. (5). This maximum can exceed substantially the asymptotic value e2ðw0 g 2Þ=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. Yield of Cherenkov X-ray radiation from the Be target was calculated in the present work by using formula (1) and the dielectric susceptibility wðoÞ¼w0ðoÞþiw00ðoÞ determined experimentally by
Fig. 4. Spectrum of Cherenkov radiation from Be target at different incidence anglespffiffiffiffiffiffiffiffiffij. Curves have been calculated with 0 0 the parameters 1=g wmax ¼ 0:1p andffiffiffiffiffiffiffiffiffi wmax ¼ 0:05. Collimatorpffiffiffiffiffiffiffiffiffi 0 0 angular sizes were DYx ¼ 0:3 wmax andpffiffiffiffiffiffiffiffiffiDYy ¼ 0:3pffiffiffiffiffiffiffiffiffiwmax. 0 0 Curves 1; 2, andpffiffiffiffiffiffiffiffiffi 3 correspond to j ¼ 5 wmax, j ¼ 0:5 wmax, 0 and j ¼ 0:05 wmax, respectively.
Henke et al. (1993). Curves presented in Fig. 4 describe spectra of Cherenkov photons emitted from the Be target into a collimator of a finite angular size. In the performed calculations, the collimator center was placed Fig. 2. Amplification factor for the Cherenkov angular density at the angle corresponding to the maximum of the as a function of the incidence angle j. The curves 1; 2; 3 emission angular density. Angular size of the collimator correspond to the value of the parameter g2w0 ¼ 2; 5; 10, was chosen in such a way that the emission yield passing respectively. through the collimator was close to saturation for small ARTICLE IN PRESS
916 A. Kubankin et al. / Radiation Physics and Chemistry 75 (2006) 913–919 incidence angles j, when the emission angular distribu- tion over the azimuthal angle is strongly non-uniform. Presented curves demonstrate a substantial growth of the emission yield with a decrease of the incidence angle j. This effect has a simple geometric interpretation (Knulst, 2004). As clear from Fig. 1, photon emitted at ðþÞ the angle Yx max4j has a path Lph in the target which is shorter than the path Lel L=j of the emitting electron. ðþÞ According to (5), difference between Ypx maxffiffiffiffi and j is small for large orientation angles jb w0, when the photon yield is formed by part of the electron inner trajectory having a size of the order of the absorption 00 lengthpffiffiffiffiLab 1=ow . However, in the case of small 0 ðþÞ jo w , the ratio Lel=Lph Lel=Lab Yx max=j increases substantially in accordance with (4). One can say that the useful part of the electron trajectory, and conse- quently the photon yield, increases.
3. Cherenkov X-rays from relativistic electrons crossing a Fig. 5. Cherenkov X-ray radiation from a multilayer X-mirror. multilayer nanostructure e1 is the axis of the emitting electron beam, e2 is the axis of the emitted photon flux, y0 is the orientation angle describing possible turning of the crystalline target by the goniometer, g is Let us consider X-ray emission from relativistic the reciprocal lattice vector, Yk and Ck are components of the electrons moving in a medium withP a periodic dielectric angular variables H and W parallel to the plane determined by 0 iðg;rÞ susceptibility wðo; rÞ¼w0ðoÞþ g wgðoÞe . Under the vectors e1 and e2. conditions of Bragg scattering of a fast electron by the Coulomb field of periodic heterogeneities of the distribution of the emitted energy: medium, the angle between electron velocity v and 3 X2 direction n of emitted photons can be large. This feature d E ¼ jA j2, allows to arrange an irradiated sample in the immediate 2 l dod Y l¼1 vicinity of the source and consequently increase the 0 00 photon density at the sample surface. In case of a one- e ðw þ iw Þal A ¼ Y g g dimensional structure consisting of alternative layers l p l D w0 þ d0 iðw00 þ d00Þ "#0 l 0 l with thicknesses a and b and with susceptibilities waðoÞ ð Þ ð Þ ð Þ 1 1 and wb o , respectively, the quantities w0 o and wg o , ð7Þ 2 2 0 00 2 2 are determined by the expressions g þ Y D dl þ idl g þ Y a b where w ðoÞ¼ w þ w , 0 T a T b j o j 1 eiðg;aÞ D ¼ 2sin2 1 þðy0 þ Y Þ cot , w ðoÞ¼ ðw w Þ, ð6Þ 2 o k 2 g igT a b B where T ¼ a þ b is the period of multilayer structure, a1 ¼ 1; a2 ¼ cos j, g ¼ exg, g gn ¼ð2p=TÞn, n ¼ 0; 1; ..., and ex is the rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi normal to the target surface (see Fig. 5). 0 signðDlÞ 2 2 d ¼ pffiffiffi C þ D þ Cl, Embarking on a study of emission properties, one l 2 l l should note that only the Bragg scattering geometry can rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be realized in the case under consideration. Since soft qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 1ffiffiffi 2 2 X-rays are strongly absorbed in a dense medium, a dl ¼ p Cl þ Dl Cl, 2 simple model of semi-infinite multilayer nanostructure can be used for calculations. Willing to discuss only 2 C ¼ðD w0 Þ2 ðw0 w002Þa2 w002, fundamental aspects of the problem, we shall restrict our l 0 g g l 0 consideration to a specific case of the emitting electron D ¼ ½ 00ð 0 Þþ 0 00 2 moving with a constant velocity. Using general results of l 2 w0 D w0 wgwgal , Caticha (1989) and Nasonov et al. (2003), one can 0 2 2 2 obtain the following expression for the spectral-angular Y1 ¼ Y?; Y2 ¼ 2y þ Yk; Y ¼ Y1 þ Y2. ARTICLE IN PRESS
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0 00 Also w0 and w0 are real and imaginary parts of the where average dielectric susceptibility w0, oB ¼ g=½2 sinðj=2Þ is the Bragg frequency, and 2 0 2 0 00 00 D0 þðwgalÞ 00 2D0wgal d ¼ w w al 0 1 0 0 l 0 2 ð 0 Þ2 g 2 ð 0 Þ2 wg ¼ p sinðpa=TÞðwa wbÞ, D0 wgal D0 wgal w00 ¼ p 1 sinðpa=TÞðw00 w00Þ. g a b and Let us consider a parameter domain, in which the vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! distribution (7) is maximum. To the moment, we neglect u 2=3 u 00 00 00 00 0 t w0 wgal 0 an influence of photoabsorption and take w0 ¼ wg ¼ 0. D0o wgal 1 2 0 wgal. Then the equation that determines radiation wgal maximum, Obviously, density (12) and that of ordinary Cherenkov g 2 þ Y2 D d0 ¼ 0, l radiation are of the same magnitude in the frequency 0 or range with jD0j4jwgalj. Thus, the discussed emission qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mechanism based on self-diffracted Cherenkov radiation 0 0 02 2 2 D0 D signðD Þ D wgal ¼ 0, generated in a multilayer nanostructure is indeed of interest for creation of X-ray sources. 2 2 0 D0 ¼ g w0 þ Y ; D ¼ D w0, (8) The angular structure of the emitted photon flux has the solution calculated with the use of general formula (7) for a Be-C multilayer nanostructure is illustrated in Fig. 6. The D2 þ w2a2 period of nanostructure, T, and the orientation angle D0 ¼ 0 g l , (9) 2D0 j=2 have been chosen so that the Bragg frequency oB and the frequency at the maximum in real part of the in two non-overlapping ranges of the parameter D . The 0 dielectric susceptibility of Be are close to each other. first of them is determined by the inequality Distribution presented in Fig. 6 has been calculated for D04jwgalj. (10) fixed energies of emitting electrons and emitted photons. It is crucial for the purposes of X-ray source creation It corresponds to the branch of ordinary parametric that the presented distribution is strongly non-uniform. X-ray radiation (PXR), including the contribution of This feature is analogous to that discussed in the diffracted transition radiation (DTR). Properties of previous section and it shows how to increase the yield PXR and DTR from relativistic electrons crossing a of strongly collimated radiation. multilayer nanostructure with the negative average 2 2 dielectric susceptibility w0ðoÞ¼ op=o have been studied in detail in our previous work (Nasonov et al., 2003). The second branch is given by the inequality
D0o jwgalj. (11) It corresponds to the Cherenkov branch of PXR. This radiation can only appear if the Cherenkov condition D0o0 is fulfilled. 0 Since jD j4jwgalj in accordance with (8), both the branches are realized outside the region of anomalous dispersion. Radiation corresponding to these branches can nevertheless appear inside the anomalous dispersion 0 region jD jojwgalj due to effects of photoabsorption. But the performed analysis showed that the yield of such radiation is small. Let us estimate the greatest possible spectral-angular density for the discussed mechanism of emission. 005 0 005 0 Assuming that w0 w0, wg wg, one can obtain from (7) the following simple formula that describes the Cherenkov branch of PXR Fig. 6. Spectral-angular distribution of Cherenkov radiation 3 2 w0 a 2 2 from a multilayer X-mirror.pffiffiffiffiffiffiffiffiffiffiffiffi Calculationspffiffiffiffiffiffiffiffiffiffiffiffi are performed using d El e g l Yl 0 0 0 the parameters 1=g w ¼ 0:2, y = w ¼ 0:7, oB ¼ o ¼ 2 2 00 , (12) a max a max dod Y p D0 d 0 max l 111:6 eV, j ¼ 44 , a=T ¼ 0:9, wa max ¼ 0:05 (beryllium). ARTICLE IN PRESS
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4. On focusing Cherenkov X-rays with grazing-angle optics
Let us consider the next possibility to increase the angular density of Cherenkov radiation using the grazing-angle optics. Returning to the scheme described in Section 2, one should estimate a possible growth of the Cherenkov angular density which can be achieved using a simple cylindrical mirror placed on the in- vacuum side of the target so that the axis of the mirror coincides with the axis of the emitting electron beam. It is clear that, in the considered case, primary Cherenkov photons emitted from the target are focused on the point of the emitting-electron trajectory that lies at the ðþÞ distance Lk 2R=ðYx max jÞ from the point where the electron leaves the target. Here R is the radius of the ðþÞ cylindrical X-mirror. The angles j and Yx max are defined as before. Distribution of photons with a fixed energy in the image plane located perpendicular to the electron-beam axis at the above focus point has been calculated in the present work using geometric optics and usual expres- Fig. 8. Spectral-angular distributions of Cherenkov radiation sions for the reflection coefficient that describes interac- from a Be target focused with a cylindrical carbon X-mirror. tion of X-rays with the X-mirror. In order to facilitate a Parameters are the same as in Fig. 7. comparison of focused and non-focused photon beams, the calculated distribution was expressed in terms of under consideration. Therefore, X-mirrors consisting of observation angles identical to those describing the light elements offer a few advantages over those made of initial non-focused angular distribution of Cherenkov heavy elements. photons. Experimental values of real and imaginary In our calculations, we considered a Cherenkov X-ray parts of the X-mirror dielectric susceptibility have been source with a beryllium radiator and a carbon X-mirror. used in the performed calculations. According to Results of computations are shown in Figs. 7 and 8, obtained results, the influence of photoabsorption in where spectral-angular distributions for non-focused the X-mirror is very essential in the range of soft X-rays Cherenkov radiation and for the focused photon flux, respectively, are given. Obviously, the effect of X-ray focusing is very essential.
5. Conclusions
The performed studies of the process of Cherenkov X-ray radiation from relativistic electrons moving through a dense medium showed some possibilities to increase the angular density of emitted photons. The emission angular density can be increased substantially under conditions of grazing incidence of emitting electrons on a target surface, when the structure of radiation in the Cherenkov cone is strongly modified. According to performed calculations, such an approach allows to achieve an enhancement by a factor of 10250. High photon density at the surface of an irradiated sample can be obtained with the use of a periodic nanostructure as a Cherenkov radiator. An advantage of Fig. 7. Spectral-angular distribution of non-focused Cherenkov this scheme consists in a possibility to generate radiation from a Be target. Calculations have beenpffiffiffiffiffiffiffiffiffi done with Cherenkov X-rays at large angles relative to the emitting 0 the followingpffiffiffiffiffiffiffiffiffi parameters: o ¼ 111:6 eV, g wmax ¼ 0:1, electron velocity and, consequently, to arrange the 0 0 j ¼ 0:1 wmax, wmax ¼ 0:05. irradiated sample in the immediate vicinity of the ARTICLE IN PRESS
A. Kubankin et al. / Radiation Physics and Chemistry 75 (2006) 913–919 919 radiator. The performed analysis shows the possibility Bazylev, V., Glebov, V., Denisov, E., Zhevago, N., Kumakhov, to obtain, in this scheme, the angular density of emitted M., Khlebnikov, A., Tsinoev, V., 1981. Ro¨ntgen Cherenkov photons close to that achievable in the ordinary scheme radiation: theory and experiment. Zh. Eksp. Teor. Fiz. 81 using a homogeneous radiator. (11), 1664–1680 (Sov. Phys. JETP 54, 884) (in Russian). Substantial growth of the angular density of Cherenkov Caticha, A., 1989. Transition-diffracted radiation and the Cherenkov emission of X-rays. Phys. Rev. A 40, photons can be achieved by focusing emitted flux with 4322–4329. the use of grazing-angle optics. In accordance with Garibian, G., Yang, C., 1983. X-ray Transition Radiation. obtained numerical results, the emission density can be Armenian Academy of Science, Erevan. increased by the factor of 100 in a simple scheme using Gary, C., Kaplin, V., Kubankin, A., Nasonov, N., Piestrup, a cylindrical X-mirror. M., Uglov, S., 2005. An investigation of the Cherenkov X-rays from relativistic electrons. Nucl. Instr. Meth. Phys. Res. B 227, 95–103. Acknowledgements Henke, B., Gullikson, E., Davis, J., 1993. X-ray interactions: photoabsorption, scattering, transmission and reflection at This work was supported by the United E ¼ 50230 000 eV, Z ¼ 1292. At. Data Nucl. Data Tables 54, 181 URL: http://www-crxo.lbl.gov. States National Institute for Health under Small Knulst, W., Luiten, O., van der Wiel, M., Verhoeven, J., 2001. Business Innovation Research (SBIR) program (grant Observation of narrow band Si L-edge Cherenkov radiation 2-R33CA086545-02), by the United States National generated by 5 MeV electrons. Appl. Phys. Lett. 79, Science Foundation under the SBIR program (grant 2999–3001. DMI-0214819), by the Russian Foundation of Basic Knulst, W., van der Wiel, M., Luiten, O., Verhoeven, J., 2003. Research (grant 04-02-16583), by Programs DOPFIN, High-brightness, narrow band and compact soft X-ray University of Russia (grant 02.01.002), and by Ministry Cherenkov sources in the water window. Appl. Phys. Lett. of Education and Science of Russia. One of the authors 83, 4050–4052. (AK) is grateful to Ministry of Education RF and Knulst, W., 2004. Cherenkov radiation in the soft X-ray region: Belgorod region Administration for financial support toward a compact narrow-band source. Ph.D. Thesis, Eindhoven. (grant GM-18/3). Knulst, W., van der Wiel, M., Luiten, O., Verhoeven, J., 2004. High-brightness compact X-ray source based on Cherenkov radiation. Proc. SPIE Int. Soc. Opt. Eng. 5196, 393. References Moran, M., Chang, B., Schneider, M., Maruyama, X., 1990. Grazing-incidence Cherenkov X-ray generation. Nucl. Instr. Bazylev, V., Glebov, V., Denisov, E., Zhevago, N., Khlebnikov, Meth. Phys. Res. B 48, 287–290. A., 1976. Cherenkov radiation as intense Ro¨ntgen source. Nasonov, N., Kaplin, V., Uglov, S., Piestrup, M., Gray, C., Pisma Zh. Eksp. Teor. Fiz. 24 (7), 406–409 (JETP Lett. 24, 2003. X-rays from relativistic electrons in a multilayer 371) (in Russian). structure. Phys. Rev. E 68, 036504, 7 pp.