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An investigation of the Cherenkov X-rays from relativistic electrons
Nuclear Instruments and Methods in Physics Research B 227 (2005) 95–103 www.elsevier.com/locate/nimb
An investigation of the Cherenkov X-rays from relativistic electrons C. Gary a, V. Kaplin b, A. Kubankin c, N. Nasonov
c,* ,
M. Piestrup a, S. Uglov
b
a Adelphi Technology, 2181 Park Blvd. Palo Alto, CA 94306, USA Nuclear Physics Institute, Tomsk Polytechnic University, P.O. Box 25, Tomsk 634050, Russia Laboratory of Radiation Physics, Belgorod State University, 14, Studencheskaya Street, Belgorod 308007, Russia b
c
Received 18 December 2003; received in revised form 8 June 2004
Abstract X-ray Cherenkov radiation is studied theoretically for grazing incidence of emitting electrons on thin foils of different materials. The growth of the angular density of emitted photons due to the modification of Cherenkov cone structure is shown. The characteristics of a possible Cherenkov X-ray source are discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 78.70.g; 78.70.Ck; 79.90.+t Keywords: Relativistic electron; Cherenkov X-ray emission; Absorption edge; Multiple scattering
1. Introduction Cherenkov emission mechanism allows to generate soft X-rays in the vicinity of atomic absorption edges, where the medium refractive index may exceed unity [1]. This theoretical prediction has been experimentally confirmed using 1200 [2], 75 [3] and 5 MeV electron beam [4] and silicon or carbon targets. The high efficiency of possible Cher-
*
Corresponding author. Tel.: +7 22 341477; fax: +7 22 341692. E-mail address: [email protected] (N. Nasonov).
enkov X-ray source has been best demonstrated in the experiment [4], where the yield 1 Æ 103 photon/electron has been obtained. On the other hand the experimental result [3] is of great physical interest since the Cherenkov emission on condition of grazing incidence of an emitting electron on the targets surface was studied in this work. Authors of the work [3] reasoned that the Cherenkov emission yield for grazing incidence can substantially exceed the yield at perpendicular incidence due to the suppression of photoabsorption in the discussed emission process as the main contribution to emission yield for grazing incidence was made from the part of
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.06.015
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emitting electron trajectory placed outside the target near to its surface. Cherenkov radiation for grazing incidence of electron beam on the target is analyzed in this work as well. We show that the Cherenkov radiation is generated from the part of the particles trajectory placed inside the target in contrast with conclusion [3]. Nevertheless, the particles trajectory outside the target plays an important role in the formation of Cherenkov emission yield due to the interference between emission amplitudes corresponding to different parts of the trajectory of emitting particle. Another effect of study is the strong change of the structure of Cherenkov cone taking place for small enough incidence angles. We show a possibility to increase substantially the emission angular density due to the effect in question. It should be particularly emphasized that only Si and C targets have been used for X-ray Cherenkov photon production. Because of this an analysis of the possibility to generate X-rays in the wide range of emitted photon energies on the basis of Cherenkov radiation from a variety of media is of immediate interest. Such analysis is performed in this work. A dependence of emission characteristics on the value of grazing incidence angle, electron energy, angular size of a photon collimator is studied in this work as well as an influence of multiple scattering of emitting particles.
2. General expressions Let us consider an emission from relativistic electrons penetrating a foil of amorphous medium. This task is well known (see for example [5]) so we can use the general results presented in [5]. We are interested in the emission process in soft X-ray range of the emitted photon energy x, where an influence of photoabsorption is very important. Assuming that the photoabsorption length lab 1/xv00 (x) (v00 is the imaginary part of the dielectric susceptibility) is less than the electron path in the target l/u (l is the thickness of the target, u is the grazing incidence angle, u 1) we use the simple model corresponding to the emission of
Fig. 1. The geometry of the emission process, n is the unit vector to the direction of emitted photon propagation, v is the emitting electron velocity, u is the incidence angle.
a fast electron flying from semi-infinite absorbing target to a vacuum where the emitted photons are recorded by X-ray detector (see Fig. 1). Since background in the small frequency range under study is determined in the main by transition radiation, we neglect the contribution of bremsstrahlung. In addition to this we consider the emission from electrons moving with uniform velocity v, assuming that the value of multiple scattering angle, achievablepon the ffi distance of the order of ffiffiffiffiffiffiffiffiffiffiffiffiffi lab ; Wms ðk =Þ lab =LR (k 21 MeV, is the energy of emitting electron, LR is the radiation length) is small relating to characteristic angle of pffiffiffiffiffiffiffiffiffiffiffi the Cherenkov cone v0 ðxÞ (v 0 is the real part of the dielectric susceptibility). Taking into account the above mentioned question we consider the structure of excited electromagnetic fields in a more detailed way in comparison with [5]. In accordance with [5] Fourier-transform of the excited in a vacuum (x > 0) 4 electromagnetic field Etrkx ¼ ð2pÞ Rtransverse 3 tr ixtikr dt d r E ðr; tÞe can be presented in the form Etrkx ¼
2 X
ekk Ekk ;
e1k ¼ ½ex kk =k k ;
k¼1
e2k ¼ ½k e1k =k; Ek;k ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pix e j þ a d k x2 k 2k ; kk kk x kx k k 2 x2 ð1Þ
where k = ki + kxex, ex Æ ki = 0, the unknown coefficients akki are determined by the ordinary boundary conditions for electromagnetic fields at the targets surface and have the form
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4pix a1kk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk x k x þ x2 k 2k 1 1 1 e1k jkx ; 2 k x2 k 2 x2 4pix2 ð2Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2kk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 x kk þ x kk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 Z 1 k x2 k 2k k x @ A þ dk x x2 k 1 1 1 2 e2k jkx ; k x2 k 2 x2 where (x) is the dielectric permeability of the targets material, jkx is the Fourier-transform of the current density of emitting electron. The result of integration over kx in (2) can be presented as the integral over time a1kk ¼
Z 1 ex 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q dteiðxkk rek Þ e1kk ve 4p2 x2 k 2 1 k 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 pffiffiffiffiffiffiffiffiffi2ffi i x2 k k xe 6 x kk x kk 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðtÞe x2 k 2k þ x2 k 2k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi2ffi 2 x2 k 2k 2 i x k k xe þ rðtÞe qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k rðtÞe
a2kk ¼
pffiffiffiffiffiffiffiffiffiffi2ffi
i
x2 k k xe
;
Z 1 ex 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q dteiðxkk rek Þ 4p2 x2 k 2 1 k 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k2 pffiffiffiffiffiffiffiffiffi2ffi x x2 k 2k k i x2 k k xe 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðtÞe 4e2k1 ve qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k pffiffiffiffiffiffiffiffiffi2ffi i x2 k k xe þ e2k2 ve rðtÞe pffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x2 k 2k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2k3 ve qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k rðtÞe
pffiffiffiffiffiffiffiffiffiffi2ffi
i
x2 k k xe
;
ð3Þ
97
where re(t) = rek + xeex is the emitting particle trajectory, ve ¼ dtd re is its velocity,qr(t) = 1 ffi if ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t > 0 and r(t) = 0 if t0; k1 ¼ kk x2 k 2k ex , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ¼ kk þ x2 k 2k ex ; k3 ¼ kk þ x2 k 2k ex , when integrating over kx we supposed that the particle penetrate the targets surface at the moment t = 0. To determine the emission amplitude Akn one should perform the inverse Fourier-transform for field components Ekk in (1). Using the stationary phase method one can obtain the following formula: Z eixr ; Ek ¼ d3 keiknr Ekk ! Akn r Z 1 e ixðtnre Þ Akn ¼ 2pix dte v e n a kxn e x kxnk ; 4p2 1 ð4Þ
where n = ni + nxex is the unit vector to the direction of emitted photon observation. Using formulae (3) and (4), we can perform the integration in (4) over t, assuming that ve(t) = v = const. In the case of grazing incidence under study the result of integration has the form e A1n ¼ ðWy Hy Þ p 1 s Hx 1 2Hx 1 ; XðÞ s þ Hx XðþÞ s þ Hx XðÞ e Wx Hx s Hx Wx þ Hx A2n ¼ p XðÞ s þ Hx XðþÞ 2Hx Wx s ; s þ Hx XðÞ 2
2
Xð Þ ¼ c2 þ ðWy Hy Þ þ ðWx Hx Þ ; 2
X ¼ c2 þ ðWy Hy Þ þ W2x þ H2x 2Wx s; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ H2x þ v0 þ iv00 ¼ s0 þ is00 ; ð5Þ where v is the dielectric susceptibility, angular variables Wx, Wy and Hx, Hy describe the electron beam angular spread and the angular distribution of emitted photon flux, respectively (see Fig. 1). It should be noted that the formula (5) coincides with well known result [6]. Let us consider the nature of different terms in the formulae (5) determining the total emission amplitudes Akn. The first terms
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proportional to X1 ðÞ correspond to fields excited by the fast electron moving in a vacuum outside the target. The second terms proportional to X1 ðþÞ correspond to ‘‘vacuum part’’ of emitting electron trajectory as well. Fields corresponding to these terms can be interpreted as the fields excited in a vacuum by the mirror-image of fast electron, moving inside the target in contrast to real electron, moving outside the target. The last terms describe the fields emitted from the electron trajectory, placed inside the target. It is this part of emitting electron trajectory which is responsible for Cherenkov X-ray generation, since the Cherenkov condition Re[Xk] = 0 can be valid for the denominator of last terms in (5) only. On the other hand ‘‘vacuum part’’ of the electrons trajectory contributes to Cherenkov radiation yield due to interference terms in the general formula for the emission spectral–angular distribution proportional to j Akn j2. The final expression for the emission spectral–angular distribution, obtained from (5), has the form x
d3 N 16e2 v02 þ v002 ¼ 2 2 p ðHx þ s0 Þ2 þ s002 dx d H
ðc2 þ ðWy Hy Þ2 þ H2x W2x Þ2 þ 4W2x ðWy Hy Þ2 X2ðÞ X2ðþÞ
W2x H2x
; ðc2 þ ðWy Hy Þ2 þ H2x þ W2x 2Wx s0 Þ2 þ 4W2x s002 rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 0 s ¼ pffiffiffi ðH2x þ v0 Þ þ v002 þ H2x þ v0 ; 2 rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 00 s ¼ pffiffiffi ðH2x þ v0 Þ þ v002 H2x v0 : 2 ð6Þ
The result (6) allows to examine the dependence of the total emission distribution on the value of incidence angle Wx. Within the range of small Wx under consideration the multiple scattering of emitting electrons in the target can influence substantially on emission properties. To take into account such influence one should average formula (6) over Wx and Wy. We use in this work the simple distribution function " # W2y þ ðu Wx Þ2 1 f ðWx ; Wy Þ ¼ exp ; ð7Þ W2l pW2l
where W2l ¼ 2k l=2 LR u is the square of multiple scattering angle of emitting electron on the electron path in the target, u is the average incidence angle, l is the thickness of the target, LR is the radiation length. Since the distribution function (7) includes negative values of Wx one should use this function in the range of incidence angles u, where u2 W2l , or u3 2k l=2 LR and the contribution of such Wx is small. The presented simple formulae (6) and (7) are the base of further analysis.
3. The effect of Cherenkov cone modification As will be apparent from (6) the presented spectral–angular distribution of total emission yield contains a maximum (Cherenkov maximum), determined by the condition c2 þ ðWy Hy Þ2 þ H2x þ W2x 2Wx s0 ¼ 0, which can be represented as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ð8Þ c2 v0 þ H2y þ u H2x þ v0 ¼ 0 in the frequency range of anomalous dispersion before absorption edge, where v00 is usually muchffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
less than q v 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . As this takes place s0 H2x þ v0 , ffi s00 12 v00 = H2x þ v0 and the Cherenkov radiation can be realized if v 0 c2 > 0 in accordance with (8) (this is well known Cherenkov threshold in X-ray range). An influence of multiple scattering is neglected in (8) as well. Let us consider the angular structure of the Cherenkov peak versus the orientation pffiffiffiffi angle u. From (6) for large enough u v0 > c1 the emission angular distribution comprises two symmetric cones x
d3 N 0 e2 v02 ¼ dx d2 H p2 ðc2 v0 þ H2y þ ðu Hx Þ2 Þ2 þ v002
H2y þ ðu Hx Þ
2 2 2
ðc2 þ H2y þ ðu Hx Þ Þ
;
ð9Þ
the first of them corresponding to the condition (8) describes Cherenkov radiation and another one describes the well known transition radiation. The structure of these cones is changed very essentially when decreasing of u. The distribution of
C. Gary et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 95–103
emission intensity over azimuth angle on the Cherenkov cone becomes strongly non-uniform in contrast with (9). To show this let us compare the magnitudes of the distribution (6) in the plane Hy = 0 at the points r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ð Þ ð10Þ u v0 c2 v0 ; Hx max ¼ from (8) and corresponding to maximum of the Cherenkov radiation intensity. Such magnitudes follow from (6) and (10): d3 N ð Þ 4e2 v0 c2 1 max x ¼ 2 2 p v002 u2 dx d H Hy ¼0 0 12 B @
u
1 1 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiA : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 v0 c2 ðu v0 c2 Þ v0 ð11Þ
First of all it is necessary to note that two maxima (11) can be realized theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi range of large enpffiffiffiffi in p ough values of u > v0 þ v0 c2 only, as it follows from (10). Obviously, the values of these pffiffiffiffi maxima are the same for large enough u v0 and are equal to those, which follow from (9) ð Þ ð Þ d3 N 0 max d3 N max e2 v0 c2 x dx d2 H jHy ¼0 ¼ x dx d2 H ¼ p2 v002 . In accordance with (10) only the maximum ðþÞ pffiffiffiffi0 the range v (the Cherenkov cone begins contact ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwith the target surface if pffiffiffiffito p u ¼ v0 þ v0 c2 ). The magnitude of this maximum can exceed substantially theasymptotic .
d3 N max is realized in x dx d2 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 c2 u v0 þ v0 c2
3
99
cylindrical cone pwith ffiffiffiffi relatively large characteristic angle HCh v0 to the electron trajectory. To obtain a uniform angular distribution of the emitted photons one should extract a small part of Cherenkov cone by the photon collimator. As a consequence the used part of the total emission yield is reduced substantially. Distribution of Cherenkov photons over azimuth angle on the Cherenkov cone becomes strongly non-uniform. Therefore the possibility to increase the used part of emission yield is opened by placing of photon collimator at the point corresponding to the maximum of the angular density of Cherenkov radiation. Such possibility is demonstrated by Fig. 2, where the angular distribution of Cherenkov radiation calculated by the formula (6) for fixed photon energy is presented. As illustrated in Fig. 2, there is not only the effect of non-uniform distribution of emitted photons over azimuth angle, but the effect of decreasing of Cherenkov emission angle when decreasing of the incidence angle u as well. The last effect is demonstrated by the curves presented in Fig. 3. These curves, calculated by the Cherenkov condition (8) for fixed value of the parameter c2v 0 = 10 and different values of the incidence angle u, show the position of Cherenkov cone in the space of observation angles Hx, Hy. The effect in question has the simple geometrical interpretation [7]. As is clear from Fig. 1, a photon emitted at the angle HðþÞ x max > u has a
ð Þ
d N max 2 2 00 2 0 value e2(v c )/p v . The ratio x dx d2 H 3 x ddxNd0 2max , as the performed analysis of the forH
mula (11) implies, depends strongly on the parameter c2v 0 . For example, this ratio has a value of about 10 if c2v 0 = 5. The obtained result is of great importance for the problem of effective Cherenkov based X-ray source creation. Indeed, Cherenkov emission mechanism allows to generate very intensive soft X-ray beams (the yield of the order of 103 photon/electron has been obtained experimentally from a single foil [4]). On the other hand, the angular density of Cherenkov radiation is not high since Cherenkov photons are emitted in a hollow
Fig. 2. The dependence of Cherenkov cone structure on the incidence angle u. The presented spectral–angular distributions of Cherenkov radiation pffiffiffiffiffiffiffiffiffi have been calculated for Be target, x = 111.6 eV 1=c v0max ¼ 0:4, v0max ¼ 0:05. Distribution 1 corffi pffiffiffiffiffiffiffiffi responds to the value offfi u ¼ 0:17 v0max . Distribution 2 pffiffiffiffiffiffiffiffi corresponds to u ¼ 3 v0max .
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Fig. 3. The position of Cherenkov cone in the space of pffiffiffiffi pffiffiffiffi observation angles. c2v 0 =10; (1) u= v0 ¼ 0:2, (2) u= v0 ¼ 1:9, pffiffiffiffi0 (3) u= v ¼ 3.
shorter path Lph in the target than that of Lel emitting electron. This effect is small for large orientapffiffiffiffi tion angles u v0 . The photon yield in this case is formed at the part of electrons inside trajectory 00 of the order of absorption length Lab1/v pffiffiffiffi0 . On the other hand in the range of small u v the ratio Lel =Lph Lel =Lab HðþÞ x max =u is increased substantially in accordance with (11), which is to say that the useful part of electron trajectory and consequently the photon yield are increased. Geometrical interpretation allows us to explain both azimuthal non-uniformity of the angular distribution of emitted photons (photon path in the target is increased with increasing azimuth angle) and the effect of ‘‘Cherenkov angle’’ decreasing. It should be noted that the degree of non-uniformity in the emission angular distribution over azimuth angle depends strongly on the value of incidence angle u. The great importance of correct choice of u is demonstrated by the curves presented in. Fig. 4. These curves describing the ratio 3
ðþÞ
d N max x dx d2 H
3
x ddxNd0 2max H
as a function of u have
been calculated by the formula (11) for different values of the parameter c2v 0 . As it is evident from Fig. 4 the local emission angular density increases very substantially when increasing emitting particle energy, but this growth is followed by decreasing of optimum value of incidence angle u. Along that the influence of multiple scattering of emitting electrons in the target increases as well. Obviously such influence must constrain the discussed growth of the emission angular density.
Fig. 4. The amplification factor for Cherenkov angular density as a function of the incidence angle u. The curves 1, 2 and 3 correspond to the value of the parameter c2v 0 = 2, 5 and 10, respectively.
4. Characteristics of possible Cherenkov X-ray sources Emission characteristics of most practical importance are the intensity and the spectrum of photons, emitted into the collimator with given angular size. The special property of Cherenkov radiation is its narrow spectral width of the order of units eV. On the other hand this kind of radiation is not tunable in contrast with, for example, parametric X-ray radiation, therefore it is very interesting to study the possibilities to generate Cherenkov X-rays in a wide range of emitted photon energies by the use of radiators consisting of different materials. Cherenkov X-ray radiation yield from Be, C, Ti, Fe and other targets have been calculated in this work by the use of formula (6) and dielectric susceptibilities v 0 (x) and v00 (x) determined experimentally [8]. The curves presented in Figs. 5–8 describe the spectra of Cherenkov photons, emitted from the above mentioned targets into the collimator with finite angular size. The collimators center was placed in performed calculations at the point corresponding to maximum of the emission angular density. Its angular size was chosen so that the emission yield in such a collimator was close to saturation for small incidence angles u when the emission angular
C. Gary et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 95–103
Fig. 5. The spectrum of Cherenkov radiation from Be target as a function of the of the incidencep angle have been ffiffiffiffiffiffiffiffiffi u. The curves 0 0:1; v calculated for Be target, 1=c v0maxp¼ ¼ 0:05, theffi max pffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi collimator angular sizes DHx ¼ 0:3 v0max ; DHy ¼ 0:3 v0max . pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi Curves 1–3 correspond to u ¼ 5 v0max ; 0:5 v0max and pffiffiffiffiffiffiffiffi ffi 0 0:05 vmax , respectively.
101
Fig. 7. The same but for Ti target (v0max ¼ 0:007).
Fig. 8. The same but for Fe target (v0max ¼ 0:0043).
Fig. 6. The same but for C target (v0max ¼ 0:011).
distribution over azimuth angle was strongly non-uniform. The presented curves demonstrate the possibility of both intense X-ray generation in the wide region of photon energies from 100 to 700 eV on the basis of Cherenkov emission mechanism and substantial growth of the emission yield when decreasing the incidence angle u.
In the performed calculations the angle u was chosen equal to optimum value demonstrated by Fig. 4. Since this optimum value of u changes substantially with changing of emitted electron energy the angle u must be controlled carefully in an experiment. In order to show this we present in Fig. 9 the spectra of Cherenkov radiation from Si target, calculated for two different emitting electron energies with and without corresponding tuning of the angle u of upffiffiffiffi is changing within the range pvalue pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi(the v0 v0 c2 u v0 þ v0 c2 , where the effect of Cherenkov cone transformation takes place).
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Fig. 9. The dependence of Cherenkov radiation yield on the energy of emitting particle (Siffi target, v0max ¼ 0:046, collimator pffiffiffiffiffiffiffiffi angular sizes DHxp¼ffiffiffiffiffiffiffiffi 0:3ffi v0max ¼ DHy ).p The ffiffiffiffiffiffiffiffifficurve 1 has been u ¼ 0:5 v0max . The curve 2 has calculated for 1=c v0max ¼ ffiffiffiffiffiffiffiffi ffi p0:5; been calculated for 1=c v0max ¼ 0:1 and the same value of incidence pffiffiffiffiffiffiffiffiffi angle u. The curve 3 has been calculated for 1=c v0maxp¼ffiffiffiffiffiffiffiffi 0:1ffi and the optimum value of the angle u ¼ 0:05 v0max .
It should be noted that the influence of emitting electron multiple scattering on the emission properties has not been taken into account in performed calculations. Such influence may be important for small incidence angles u. We have calculated the
Fig. 11. An influence of the multiple scattering on the Cherenkov radiation spectral distribution. The curves 1 and 2 have been calculated with and without account of the multiple scattering, respectively. The curves have been calculated for Be pffiffiffiffiffiffiffiffiffi 0:11; v0max ¼ 0:05, target, 1=c v0max the p¼ pffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffi ffi collimator angular 0 0 ; DH and incidence angle sizes DH ¼ 0:2 v ¼ 0:3 v x y max max pffiffiffiffiffiffiffiffiffi u ¼ 0:1 v0max .
spectral–angular distribution of Cherenkov radiation from the Be target on the basis of the formula (6) averaged over beam spread at the exit of the target using the distribution function (7). The result of calculations, presented in Fig. 10, shows a strong suppression of the angular density of Cherenkov emission due to multiple scattering (this is because of very small angular width of Cherenkov cone proportional to v00 as follows from (6)). On the other hand the yield fixed by a photon collimator with finite angular size is not changed very substantially as it is evident from Fig. 11. 5. Conclusion
Fig. 10. An influence of the multiple scattering on the Cherenkov radiation spectral–angular distribution. The curves 1 and 2 have been calculated with and without account of the multiple pffiffiffiffiffiffiffiffiffiscattering, respectively. pffiffiffiffiffiffiffiffiffi The curves calculated for 1=c v0max ¼ 0:04; u ¼ 0:08 v0max .
As the performed analysis implies, the angular density of Cherenkov X-ray radiation from relativistic electrons can be substantially increased on condition of grazing incidence of emitting particle at the targets surface. A specialized study has shown that this growth of the emission angular density becomes possible due to the modification of Cherenkov cone structure in the condition under consideration (Cherenkov emission angle is decreased and the distribution of emitted photons over azimuthal angle on the Cherenkov cone becomes strongly non-
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uniform). The substantial change in the structure of Cherenkov cone takes place when the orientation angle u becomes smaller p than ffiffiffiffi the characteristic Cherenkov radiation angle v0 . The possible enhancement of the Cherenkov radiation angular density increases with increasing of an emitting electron energy, but a maximum of this enhancement depends very strongly on the orientation angle u, so that the value of u must be chosen very carefully in the experiment. The performed calculations have shown the possibility to increase the emission angular density 10-fold and more. In accordance with obtained numerical results there is the possibility to create an effective source of quasimonochromatic (the spectral width is about units eV) and intense (the yield is about 103 photon per electron) X-rays in a wide range of emitted photon energies from 100 eV to 1 KeV on the base of Cherenkov radiation from relativistic electrons (the energy of emitting particles is about units or tens MeV), penetrating thin foils of different materials. Acknowledgements The authors are grateful to the referee for very kind and useful criticism. This work was sup-
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ported by the United States National Institute for Health under Small Business Innovation Research (SBIR) program (Grant no. 2R33CA086545-02) and the United States National Science Foundation under the SBIR program (Grant no. DMI-0214819) and by the Russian Foundation of Basic Research (Grant no. 03-0296431). One of the authors (A.K.) is grateful to Ministry of Education RF and Belgorod region Administration for financial support (Grant No. GM-18/3).
References [1] V.A. Bazylev, V.I. Glebov, E.I. Denisov, N.K. Zhevago, A.S. Khlebnikov, JEPT Lett. 24 (1976) 371. [2] V.A. Bazylev, V.I. Glebov, E.I. Denisov, N.K. Zhevago, M.A. Kumakhov, A.S. Khlebnikov, V.G. Tsinoev, Sov. Phys. JEPT 54 (1981) 884. [3] M.J. Moran, B. Chang, M.B. Schneider, X.K. Maruyama, Nucl. Instr. and Meth. B 48 (1990) 287. [4] W. Knulst, O.J. Luiten, M.J. van der Wiel, J. Verhoeven, Appl. Phys. Lett. 79 (2001) 2099. [5] G.M. Garibian, C. Yang, X-ray Transition Radiation, Armenian Academy of Science, Erevan, 1983. [6] N.E. Pafomov, Proc. Lebedev Phys. Inst. 44 (1969) 28. [7] W. Knulst, Ph.D. Thesis, 2004. [8] B.L. Henke, E.M. Gullikson, J.C. Davis, At. Data Nucl. Data Tables 54 (1993) 18. www-cxro.lbl.gov.
An investigation of the Cherenkov X-rays from relativistic electrons C. Gary a, V. Kaplin b, A. Kubankin c, N. Nasonov
c,* ,
M. Piestrup a, S. Uglov
b
a Adelphi Technology, 2181 Park Blvd. Palo Alto, CA 94306, USA Nuclear Physics Institute, Tomsk Polytechnic University, P.O. Box 25, Tomsk 634050, Russia Laboratory of Radiation Physics, Belgorod State University, 14, Studencheskaya Street, Belgorod 308007, Russia b
c
Received 18 December 2003; received in revised form 8 June 2004
Abstract X-ray Cherenkov radiation is studied theoretically for grazing incidence of emitting electrons on thin foils of different materials. The growth of the angular density of emitted photons due to the modification of Cherenkov cone structure is shown. The characteristics of a possible Cherenkov X-ray source are discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 78.70.g; 78.70.Ck; 79.90.+t Keywords: Relativistic electron; Cherenkov X-ray emission; Absorption edge; Multiple scattering
1. Introduction Cherenkov emission mechanism allows to generate soft X-rays in the vicinity of atomic absorption edges, where the medium refractive index may exceed unity [1]. This theoretical prediction has been experimentally confirmed using 1200 [2], 75 [3] and 5 MeV electron beam [4] and silicon or carbon targets. The high efficiency of possible Cher-
*
Corresponding author. Tel.: +7 22 341477; fax: +7 22 341692. E-mail address: [email protected] (N. Nasonov).
enkov X-ray source has been best demonstrated in the experiment [4], where the yield 1 Æ 103 photon/electron has been obtained. On the other hand the experimental result [3] is of great physical interest since the Cherenkov emission on condition of grazing incidence of an emitting electron on the targets surface was studied in this work. Authors of the work [3] reasoned that the Cherenkov emission yield for grazing incidence can substantially exceed the yield at perpendicular incidence due to the suppression of photoabsorption in the discussed emission process as the main contribution to emission yield for grazing incidence was made from the part of
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.06.015
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emitting electron trajectory placed outside the target near to its surface. Cherenkov radiation for grazing incidence of electron beam on the target is analyzed in this work as well. We show that the Cherenkov radiation is generated from the part of the particles trajectory placed inside the target in contrast with conclusion [3]. Nevertheless, the particles trajectory outside the target plays an important role in the formation of Cherenkov emission yield due to the interference between emission amplitudes corresponding to different parts of the trajectory of emitting particle. Another effect of study is the strong change of the structure of Cherenkov cone taking place for small enough incidence angles. We show a possibility to increase substantially the emission angular density due to the effect in question. It should be particularly emphasized that only Si and C targets have been used for X-ray Cherenkov photon production. Because of this an analysis of the possibility to generate X-rays in the wide range of emitted photon energies on the basis of Cherenkov radiation from a variety of media is of immediate interest. Such analysis is performed in this work. A dependence of emission characteristics on the value of grazing incidence angle, electron energy, angular size of a photon collimator is studied in this work as well as an influence of multiple scattering of emitting particles.
2. General expressions Let us consider an emission from relativistic electrons penetrating a foil of amorphous medium. This task is well known (see for example [5]) so we can use the general results presented in [5]. We are interested in the emission process in soft X-ray range of the emitted photon energy x, where an influence of photoabsorption is very important. Assuming that the photoabsorption length lab 1/xv00 (x) (v00 is the imaginary part of the dielectric susceptibility) is less than the electron path in the target l/u (l is the thickness of the target, u is the grazing incidence angle, u 1) we use the simple model corresponding to the emission of
Fig. 1. The geometry of the emission process, n is the unit vector to the direction of emitted photon propagation, v is the emitting electron velocity, u is the incidence angle.
a fast electron flying from semi-infinite absorbing target to a vacuum where the emitted photons are recorded by X-ray detector (see Fig. 1). Since background in the small frequency range under study is determined in the main by transition radiation, we neglect the contribution of bremsstrahlung. In addition to this we consider the emission from electrons moving with uniform velocity v, assuming that the value of multiple scattering angle, achievablepon the ffi distance of the order of ffiffiffiffiffiffiffiffiffiffiffiffiffi lab ; Wms ðk =Þ lab =LR (k 21 MeV, is the energy of emitting electron, LR is the radiation length) is small relating to characteristic angle of pffiffiffiffiffiffiffiffiffiffiffi the Cherenkov cone v0 ðxÞ (v 0 is the real part of the dielectric susceptibility). Taking into account the above mentioned question we consider the structure of excited electromagnetic fields in a more detailed way in comparison with [5]. In accordance with [5] Fourier-transform of the excited in a vacuum (x > 0) 4 electromagnetic field Etrkx ¼ ð2pÞ Rtransverse 3 tr ixtikr dt d r E ðr; tÞe can be presented in the form Etrkx ¼
2 X
ekk Ekk ;
e1k ¼ ½ex kk =k k ;
k¼1
e2k ¼ ½k e1k =k; Ek;k ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pix e j þ a d k x2 k 2k ; kk kk x kx k k 2 x2 ð1Þ
where k = ki + kxex, ex Æ ki = 0, the unknown coefficients akki are determined by the ordinary boundary conditions for electromagnetic fields at the targets surface and have the form
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4pix a1kk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk x k x þ x2 k 2k 1 1 1 e1k jkx ; 2 k x2 k 2 x2 4pix2 ð2Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2kk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 x kk þ x kk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 Z 1 k x2 k 2k k x @ A þ dk x x2 k 1 1 1 2 e2k jkx ; k x2 k 2 x2 where (x) is the dielectric permeability of the targets material, jkx is the Fourier-transform of the current density of emitting electron. The result of integration over kx in (2) can be presented as the integral over time a1kk ¼
Z 1 ex 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q dteiðxkk rek Þ e1kk ve 4p2 x2 k 2 1 k 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 pffiffiffiffiffiffiffiffiffi2ffi i x2 k k xe 6 x kk x kk 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðtÞe x2 k 2k þ x2 k 2k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi2ffi 2 x2 k 2k 2 i x k k xe þ rðtÞe qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k rðtÞe
a2kk ¼
pffiffiffiffiffiffiffiffiffiffi2ffi
i
x2 k k xe
;
Z 1 ex 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q dteiðxkk rek Þ 4p2 x2 k 2 1 k 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k2 pffiffiffiffiffiffiffiffiffi2ffi x x2 k 2k k i x2 k k xe 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðtÞe 4e2k1 ve qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k pffiffiffiffiffiffiffiffiffi2ffi i x2 k k xe þ e2k2 ve rðtÞe pffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x2 k 2k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2k3 ve qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k 2k þ x2 k 2k rðtÞe
pffiffiffiffiffiffiffiffiffiffi2ffi
i
x2 k k xe
;
ð3Þ
97
where re(t) = rek + xeex is the emitting particle trajectory, ve ¼ dtd re is its velocity,qr(t) = 1 ffi if ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t > 0 and r(t) = 0 if t0; k1 ¼ kk x2 k 2k ex , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ¼ kk þ x2 k 2k ex ; k3 ¼ kk þ x2 k 2k ex , when integrating over kx we supposed that the particle penetrate the targets surface at the moment t = 0. To determine the emission amplitude Akn one should perform the inverse Fourier-transform for field components Ekk in (1). Using the stationary phase method one can obtain the following formula: Z eixr ; Ek ¼ d3 keiknr Ekk ! Akn r Z 1 e ixðtnre Þ Akn ¼ 2pix dte v e n a kxn e x kxnk ; 4p2 1 ð4Þ
where n = ni + nxex is the unit vector to the direction of emitted photon observation. Using formulae (3) and (4), we can perform the integration in (4) over t, assuming that ve(t) = v = const. In the case of grazing incidence under study the result of integration has the form e A1n ¼ ðWy Hy Þ p 1 s Hx 1 2Hx 1 ; XðÞ s þ Hx XðþÞ s þ Hx XðÞ e Wx Hx s Hx Wx þ Hx A2n ¼ p XðÞ s þ Hx XðþÞ 2Hx Wx s ; s þ Hx XðÞ 2
2
Xð Þ ¼ c2 þ ðWy Hy Þ þ ðWx Hx Þ ; 2
X ¼ c2 þ ðWy Hy Þ þ W2x þ H2x 2Wx s; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ H2x þ v0 þ iv00 ¼ s0 þ is00 ; ð5Þ where v is the dielectric susceptibility, angular variables Wx, Wy and Hx, Hy describe the electron beam angular spread and the angular distribution of emitted photon flux, respectively (see Fig. 1). It should be noted that the formula (5) coincides with well known result [6]. Let us consider the nature of different terms in the formulae (5) determining the total emission amplitudes Akn. The first terms
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proportional to X1 ðÞ correspond to fields excited by the fast electron moving in a vacuum outside the target. The second terms proportional to X1 ðþÞ correspond to ‘‘vacuum part’’ of emitting electron trajectory as well. Fields corresponding to these terms can be interpreted as the fields excited in a vacuum by the mirror-image of fast electron, moving inside the target in contrast to real electron, moving outside the target. The last terms describe the fields emitted from the electron trajectory, placed inside the target. It is this part of emitting electron trajectory which is responsible for Cherenkov X-ray generation, since the Cherenkov condition Re[Xk] = 0 can be valid for the denominator of last terms in (5) only. On the other hand ‘‘vacuum part’’ of the electrons trajectory contributes to Cherenkov radiation yield due to interference terms in the general formula for the emission spectral–angular distribution proportional to j Akn j2. The final expression for the emission spectral–angular distribution, obtained from (5), has the form x
d3 N 16e2 v02 þ v002 ¼ 2 2 p ðHx þ s0 Þ2 þ s002 dx d H
ðc2 þ ðWy Hy Þ2 þ H2x W2x Þ2 þ 4W2x ðWy Hy Þ2 X2ðÞ X2ðþÞ
W2x H2x
; ðc2 þ ðWy Hy Þ2 þ H2x þ W2x 2Wx s0 Þ2 þ 4W2x s002 rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 0 s ¼ pffiffiffi ðH2x þ v0 Þ þ v002 þ H2x þ v0 ; 2 rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 00 s ¼ pffiffiffi ðH2x þ v0 Þ þ v002 H2x v0 : 2 ð6Þ
The result (6) allows to examine the dependence of the total emission distribution on the value of incidence angle Wx. Within the range of small Wx under consideration the multiple scattering of emitting electrons in the target can influence substantially on emission properties. To take into account such influence one should average formula (6) over Wx and Wy. We use in this work the simple distribution function " # W2y þ ðu Wx Þ2 1 f ðWx ; Wy Þ ¼ exp ; ð7Þ W2l pW2l
where W2l ¼ 2k l=2 LR u is the square of multiple scattering angle of emitting electron on the electron path in the target, u is the average incidence angle, l is the thickness of the target, LR is the radiation length. Since the distribution function (7) includes negative values of Wx one should use this function in the range of incidence angles u, where u2 W2l , or u3 2k l=2 LR and the contribution of such Wx is small. The presented simple formulae (6) and (7) are the base of further analysis.
3. The effect of Cherenkov cone modification As will be apparent from (6) the presented spectral–angular distribution of total emission yield contains a maximum (Cherenkov maximum), determined by the condition c2 þ ðWy Hy Þ2 þ H2x þ W2x 2Wx s0 ¼ 0, which can be represented as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ð8Þ c2 v0 þ H2y þ u H2x þ v0 ¼ 0 in the frequency range of anomalous dispersion before absorption edge, where v00 is usually muchffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
less than q v 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . As this takes place s0 H2x þ v0 , ffi s00 12 v00 = H2x þ v0 and the Cherenkov radiation can be realized if v 0 c2 > 0 in accordance with (8) (this is well known Cherenkov threshold in X-ray range). An influence of multiple scattering is neglected in (8) as well. Let us consider the angular structure of the Cherenkov peak versus the orientation pffiffiffiffi angle u. From (6) for large enough u v0 > c1 the emission angular distribution comprises two symmetric cones x
d3 N 0 e2 v02 ¼ dx d2 H p2 ðc2 v0 þ H2y þ ðu Hx Þ2 Þ2 þ v002
H2y þ ðu Hx Þ
2 2 2
ðc2 þ H2y þ ðu Hx Þ Þ
;
ð9Þ
the first of them corresponding to the condition (8) describes Cherenkov radiation and another one describes the well known transition radiation. The structure of these cones is changed very essentially when decreasing of u. The distribution of
C. Gary et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 95–103
emission intensity over azimuth angle on the Cherenkov cone becomes strongly non-uniform in contrast with (9). To show this let us compare the magnitudes of the distribution (6) in the plane Hy = 0 at the points r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ð Þ ð10Þ u v0 c2 v0 ; Hx max ¼ from (8) and corresponding to maximum of the Cherenkov radiation intensity. Such magnitudes follow from (6) and (10): d3 N ð Þ 4e2 v0 c2 1 max x ¼ 2 2 p v002 u2 dx d H Hy ¼0 0 12 B @
u
1 1 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiA : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 v0 c2 ðu v0 c2 Þ v0 ð11Þ
First of all it is necessary to note that two maxima (11) can be realized theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi range of large enpffiffiffiffi in p ough values of u > v0 þ v0 c2 only, as it follows from (10). Obviously, the values of these pffiffiffiffi maxima are the same for large enough u v0 and are equal to those, which follow from (9) ð Þ ð Þ d3 N 0 max d3 N max e2 v0 c2 x dx d2 H jHy ¼0 ¼ x dx d2 H ¼ p2 v002 . In accordance with (10) only the maximum ðþÞ pffiffiffiffi0 the range v (the Cherenkov cone begins contact ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwith the target surface if pffiffiffiffito p u ¼ v0 þ v0 c2 ). The magnitude of this maximum can exceed substantially theasymptotic .
d3 N max is realized in x dx d2 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 c2 u v0 þ v0 c2
3
99
cylindrical cone pwith ffiffiffiffi relatively large characteristic angle HCh v0 to the electron trajectory. To obtain a uniform angular distribution of the emitted photons one should extract a small part of Cherenkov cone by the photon collimator. As a consequence the used part of the total emission yield is reduced substantially. Distribution of Cherenkov photons over azimuth angle on the Cherenkov cone becomes strongly non-uniform. Therefore the possibility to increase the used part of emission yield is opened by placing of photon collimator at the point corresponding to the maximum of the angular density of Cherenkov radiation. Such possibility is demonstrated by Fig. 2, where the angular distribution of Cherenkov radiation calculated by the formula (6) for fixed photon energy is presented. As illustrated in Fig. 2, there is not only the effect of non-uniform distribution of emitted photons over azimuth angle, but the effect of decreasing of Cherenkov emission angle when decreasing of the incidence angle u as well. The last effect is demonstrated by the curves presented in Fig. 3. These curves, calculated by the Cherenkov condition (8) for fixed value of the parameter c2v 0 = 10 and different values of the incidence angle u, show the position of Cherenkov cone in the space of observation angles Hx, Hy. The effect in question has the simple geometrical interpretation [7]. As is clear from Fig. 1, a photon emitted at the angle HðþÞ x max > u has a
ð Þ
d N max 2 2 00 2 0 value e2(v c )/p v . The ratio x dx d2 H 3 x ddxNd0 2max , as the performed analysis of the forH
mula (11) implies, depends strongly on the parameter c2v 0 . For example, this ratio has a value of about 10 if c2v 0 = 5. The obtained result is of great importance for the problem of effective Cherenkov based X-ray source creation. Indeed, Cherenkov emission mechanism allows to generate very intensive soft X-ray beams (the yield of the order of 103 photon/electron has been obtained experimentally from a single foil [4]). On the other hand, the angular density of Cherenkov radiation is not high since Cherenkov photons are emitted in a hollow
Fig. 2. The dependence of Cherenkov cone structure on the incidence angle u. The presented spectral–angular distributions of Cherenkov radiation pffiffiffiffiffiffiffiffiffi have been calculated for Be target, x = 111.6 eV 1=c v0max ¼ 0:4, v0max ¼ 0:05. Distribution 1 corffi pffiffiffiffiffiffiffiffi responds to the value offfi u ¼ 0:17 v0max . Distribution 2 pffiffiffiffiffiffiffiffi corresponds to u ¼ 3 v0max .
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C. Gary et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 95–103
Fig. 3. The position of Cherenkov cone in the space of pffiffiffiffi pffiffiffiffi observation angles. c2v 0 =10; (1) u= v0 ¼ 0:2, (2) u= v0 ¼ 1:9, pffiffiffiffi0 (3) u= v ¼ 3.
shorter path Lph in the target than that of Lel emitting electron. This effect is small for large orientapffiffiffiffi tion angles u v0 . The photon yield in this case is formed at the part of electrons inside trajectory 00 of the order of absorption length Lab1/v pffiffiffiffi0 . On the other hand in the range of small u v the ratio Lel =Lph Lel =Lab HðþÞ x max =u is increased substantially in accordance with (11), which is to say that the useful part of electron trajectory and consequently the photon yield are increased. Geometrical interpretation allows us to explain both azimuthal non-uniformity of the angular distribution of emitted photons (photon path in the target is increased with increasing azimuth angle) and the effect of ‘‘Cherenkov angle’’ decreasing. It should be noted that the degree of non-uniformity in the emission angular distribution over azimuth angle depends strongly on the value of incidence angle u. The great importance of correct choice of u is demonstrated by the curves presented in. Fig. 4. These curves describing the ratio 3
ðþÞ
d N max x dx d2 H
3
x ddxNd0 2max H
as a function of u have
been calculated by the formula (11) for different values of the parameter c2v 0 . As it is evident from Fig. 4 the local emission angular density increases very substantially when increasing emitting particle energy, but this growth is followed by decreasing of optimum value of incidence angle u. Along that the influence of multiple scattering of emitting electrons in the target increases as well. Obviously such influence must constrain the discussed growth of the emission angular density.
Fig. 4. The amplification factor for Cherenkov angular density as a function of the incidence angle u. The curves 1, 2 and 3 correspond to the value of the parameter c2v 0 = 2, 5 and 10, respectively.
4. Characteristics of possible Cherenkov X-ray sources Emission characteristics of most practical importance are the intensity and the spectrum of photons, emitted into the collimator with given angular size. The special property of Cherenkov radiation is its narrow spectral width of the order of units eV. On the other hand this kind of radiation is not tunable in contrast with, for example, parametric X-ray radiation, therefore it is very interesting to study the possibilities to generate Cherenkov X-rays in a wide range of emitted photon energies by the use of radiators consisting of different materials. Cherenkov X-ray radiation yield from Be, C, Ti, Fe and other targets have been calculated in this work by the use of formula (6) and dielectric susceptibilities v 0 (x) and v00 (x) determined experimentally [8]. The curves presented in Figs. 5–8 describe the spectra of Cherenkov photons, emitted from the above mentioned targets into the collimator with finite angular size. The collimators center was placed in performed calculations at the point corresponding to maximum of the emission angular density. Its angular size was chosen so that the emission yield in such a collimator was close to saturation for small incidence angles u when the emission angular
C. Gary et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 95–103
Fig. 5. The spectrum of Cherenkov radiation from Be target as a function of the of the incidencep angle have been ffiffiffiffiffiffiffiffiffi u. The curves 0 0:1; v calculated for Be target, 1=c v0maxp¼ ¼ 0:05, theffi max pffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi collimator angular sizes DHx ¼ 0:3 v0max ; DHy ¼ 0:3 v0max . pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi Curves 1–3 correspond to u ¼ 5 v0max ; 0:5 v0max and pffiffiffiffiffiffiffiffi ffi 0 0:05 vmax , respectively.
101
Fig. 7. The same but for Ti target (v0max ¼ 0:007).
Fig. 8. The same but for Fe target (v0max ¼ 0:0043).
Fig. 6. The same but for C target (v0max ¼ 0:011).
distribution over azimuth angle was strongly non-uniform. The presented curves demonstrate the possibility of both intense X-ray generation in the wide region of photon energies from 100 to 700 eV on the basis of Cherenkov emission mechanism and substantial growth of the emission yield when decreasing the incidence angle u.
In the performed calculations the angle u was chosen equal to optimum value demonstrated by Fig. 4. Since this optimum value of u changes substantially with changing of emitted electron energy the angle u must be controlled carefully in an experiment. In order to show this we present in Fig. 9 the spectra of Cherenkov radiation from Si target, calculated for two different emitting electron energies with and without corresponding tuning of the angle u of upffiffiffiffi is changing within the range pvalue pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi(the v0 v0 c2 u v0 þ v0 c2 , where the effect of Cherenkov cone transformation takes place).
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Fig. 9. The dependence of Cherenkov radiation yield on the energy of emitting particle (Siffi target, v0max ¼ 0:046, collimator pffiffiffiffiffiffiffiffi angular sizes DHxp¼ffiffiffiffiffiffiffiffi 0:3ffi v0max ¼ DHy ).p The ffiffiffiffiffiffiffiffifficurve 1 has been u ¼ 0:5 v0max . The curve 2 has calculated for 1=c v0max ¼ ffiffiffiffiffiffiffiffi ffi p0:5; been calculated for 1=c v0max ¼ 0:1 and the same value of incidence pffiffiffiffiffiffiffiffiffi angle u. The curve 3 has been calculated for 1=c v0maxp¼ffiffiffiffiffiffiffiffi 0:1ffi and the optimum value of the angle u ¼ 0:05 v0max .
It should be noted that the influence of emitting electron multiple scattering on the emission properties has not been taken into account in performed calculations. Such influence may be important for small incidence angles u. We have calculated the
Fig. 11. An influence of the multiple scattering on the Cherenkov radiation spectral distribution. The curves 1 and 2 have been calculated with and without account of the multiple scattering, respectively. The curves have been calculated for Be pffiffiffiffiffiffiffiffiffi 0:11; v0max ¼ 0:05, target, 1=c v0max the p¼ pffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffi ffi collimator angular 0 0 ; DH and incidence angle sizes DH ¼ 0:2 v ¼ 0:3 v x y max max pffiffiffiffiffiffiffiffiffi u ¼ 0:1 v0max .
spectral–angular distribution of Cherenkov radiation from the Be target on the basis of the formula (6) averaged over beam spread at the exit of the target using the distribution function (7). The result of calculations, presented in Fig. 10, shows a strong suppression of the angular density of Cherenkov emission due to multiple scattering (this is because of very small angular width of Cherenkov cone proportional to v00 as follows from (6)). On the other hand the yield fixed by a photon collimator with finite angular size is not changed very substantially as it is evident from Fig. 11. 5. Conclusion
Fig. 10. An influence of the multiple scattering on the Cherenkov radiation spectral–angular distribution. The curves 1 and 2 have been calculated with and without account of the multiple pffiffiffiffiffiffiffiffiffiscattering, respectively. pffiffiffiffiffiffiffiffiffi The curves calculated for 1=c v0max ¼ 0:04; u ¼ 0:08 v0max .
As the performed analysis implies, the angular density of Cherenkov X-ray radiation from relativistic electrons can be substantially increased on condition of grazing incidence of emitting particle at the targets surface. A specialized study has shown that this growth of the emission angular density becomes possible due to the modification of Cherenkov cone structure in the condition under consideration (Cherenkov emission angle is decreased and the distribution of emitted photons over azimuthal angle on the Cherenkov cone becomes strongly non-
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uniform). The substantial change in the structure of Cherenkov cone takes place when the orientation angle u becomes smaller p than ffiffiffiffi the characteristic Cherenkov radiation angle v0 . The possible enhancement of the Cherenkov radiation angular density increases with increasing of an emitting electron energy, but a maximum of this enhancement depends very strongly on the orientation angle u, so that the value of u must be chosen very carefully in the experiment. The performed calculations have shown the possibility to increase the emission angular density 10-fold and more. In accordance with obtained numerical results there is the possibility to create an effective source of quasimonochromatic (the spectral width is about units eV) and intense (the yield is about 103 photon per electron) X-rays in a wide range of emitted photon energies from 100 eV to 1 KeV on the base of Cherenkov radiation from relativistic electrons (the energy of emitting particles is about units or tens MeV), penetrating thin foils of different materials. Acknowledgements The authors are grateful to the referee for very kind and useful criticism. This work was sup-
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ported by the United States National Institute for Health under Small Business Innovation Research (SBIR) program (Grant no. 2R33CA086545-02) and the United States National Science Foundation under the SBIR program (Grant no. DMI-0214819) and by the Russian Foundation of Basic Research (Grant no. 03-0296431). One of the authors (A.K.) is grateful to Ministry of Education RF and Belgorod region Administration for financial support (Grant No. GM-18/3).
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