]>PLB34754S0370-2693(19)30458-710.1016/j.physletb.2019.06.063The Author(s)TheoryFig. 1a) Index of refraction n = n(ω) for Si (solid line) and C (diamond - dashed line) [44]. The regions of anomalous dispersion (∂n/∂ω < 0) are underlined; b) The refractive index derivative ∂n/∂ω for Si (solid line) and C (diamond - dashed line): the positions for ∂n/∂ω = 0 are indicated by vertical lines.Fig. 1Fig. 2Dependences of polar angles θ on ChCR-photon energies ħω calculated for Si (a) and diamond (b) crystals (electron energy 255 MeV). Dashed line presents the ChR-photon angles calculated according Eq. (5) with Ωii = 0 and n(ω) taken from Fig. 1.Fig. 2Fig. 3ChCR angular distributions for (220) channeled 255 MeV electrons a) for the ChCR-photons energy ħω ≃ 3.3 eV (wavelength λ ≃ 375.7 nm) in Si; b) for the ChCR-photons energy ħω ≃ 7.2 eV (wavelength λ ≃ 172.2 nm) in diamond (C).Fig. 3Fig. 4ChCR angular distributions for Si (220) channeled 255 MeV electrons for the ChCR-photons energy ħω ≃ 2.07 eV (wavelength λ ≃ 600 nm). The position of Cherenkov angle θCh ≃ 75.3∘ is indicated by arrow.Fig. 4Fig. 5ChCR-photons spectral distribution for Si (220) channeled 255 MeV electrons at polar angles θ ≃ 78.7∘ (a) and θ ≃ 80.3∘ (b). The points corresponding to the ChCR-photons energy ħω ≃ 2.07 eV are indicated by vertical lines.Fig. 5Fig. 6The function d/θC and the dechanneling length ld vs the electron energy E = γmc2.Fig. 6Fig. 7Transmittance T = T(ω) for Si (a) and diamond (b) crystals for five various thicknesses (compilation of experimental data [44]).Fig. 7Cherenkov-Channeling radiation from sub-GeV relativistic electronsK.B.Korotchenkoa⁎korotchenko@tpu.ruYu.L.PivovarovaY.TakabayashibS.B.DabagovcdeaNational Research Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, RussiaNational Research Tomsk Polytechnic University30 Lenin Ave.Tomsk634050RussiaNational Research Tomsk Polytechnic University 30 Lenin Ave., 634050 Tomsk, RussiabSAGA Light Source, 8-7 Yayoigaoka, Tosu, Saga 841-0005, JapanSAGA Light Source8-7 YayoigaokaSagaTosu841-0005JapanSAGA Light Source, 8-7 Yayoigaoka, Tosu, Saga 841-0005, JapancINFN - Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044, Frascati (RM), ItalyINFN - Laboratori Nazionali di FrascatiVia E. Fermi 40Frascati (RM)I-00044ItalyINFN - Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044, Frascati (RM), ItalydRAS - P.N. Lebedev Physical Institute, Leninsky Pr. 53, 119991 Moscow, RussiaRAS - P.N. Lebedev Physical InstituteLeninsky Pr. 53Moscow119991RussiaRAS - P.N. Lebedev Physical Institute, Leninsky Pr. 53, 119991 Moscow, RussiaeNR Nuclear University MEPhI, Kashirskoe Sh. 31, 115409 Moscow, RussiaNR Nuclear University MEPhIKashirskoe Sh. 31Moscow115409RussiaNR Nuclear University MEPhI, Kashirskoe Sh. 31, 115409 Moscow, Russia⁎Corresponding author.Editor: L. RolandiAbstractQuantization of transverse energy levels for planar channeled relativistic electrons together with crystal dispersion may drastically change Cherenkov radiation resulting in mixed Cherenkov-Channeling Radiation (ChCR). In this work we have developed a quantum theory of ChCR. We have shown that the main feature of ChCR is reflected by its angular distribution, which consists of narrow bands reflecting the quantization of the projectile transverse energy. Moreover, mixed ChCR is shifted compared to Cherenkov radiation to larger emission angles. To stimulate experimental investigation of predicted phenomenon, a numerical analysis is carried out for 255 MeV electrons planar channeled in C (diamond) and Si crystals.KeywordsCherenkov radiationChanneling radiationMixed Cherenkov-Channeling radiation1IntroductionOver the last years the series of successful experiments on interaction of relativistic electrons with aligned crystals have been performed at the SAGA-LS facility. The results of these experiments are reflected in publications on the observation of quantum effects for parametric X-ray radiation at channeling (PXRC) [1–3], on the study of Cherenkov radiation (ChR) in a diamond crystal [4], on the investigations of sub-GeV electrons scattering [5] and rainbow scattering [6] by thin and ultrathin crystals.Despite the number of theoretical and experimental works on channeling radiation (CR), see e.g. [7–14], some features of CR are not studied up to date. Indeed, to outline both spectral and angular distributions of large-angle (close to Cherenkov one) optical and ultraviolet CR by relativistic channeled electrons in optically transparent crystals is still opened task. In the meantime, the optical radiation from channeled particles was first discussed in [15] revealing the general formulas, while in [16] the modified CR theory, taking into account the experimentally obtained crystal dispersion, was presented. The latter becomes important for getting theoretical basics to planned experiments.As known, relativistic electron moving in a medium by rectilinear trajectory emits ChR in optical and ultraviolet regions, various behaviors of which have been discussed in many publications (see, for instance, in[17–27]. In aligned crystals the electron trajectory may become undulating with high frequency and small amplitude that is realized at channeling conditions. Thus, the projectile motion in a channeling regime changes the rectilinear trajectory to quasi-periodic one. Due to the channeling undulation, the projectile (electron, in our case) emits the CR-photons. Typically, CR is associated with the emission in forward direction (respect to the projectile longitudinal momentum along the crystal channeling direction). However, CR can be also observed at large angles close to Cherenkov angle. Since these two types of radiations cannot be separated in that case, in principle, at large angles the radiation by channeled particle is composed by proper CR in combination with ChR at channeling (the possibility for ChR at channeling has been first mentioned in [28]), i.e., we deal with a new type of radiation, mixed Cherenkov-Channeling Radiation (ChCR).For example, let remind that similar mixed radiation, i.e. Synchrotron-Cherenkov Effect, has been studied both theoretically and experimentally in detail (see, in [29–32])). Two types of mixed radiation by relativistic heavy ions (RHI), i.e. transition-bremsstrahlung and Cherenkov-bremsstrahlung have been considered in [33] and in [34–40], respectively. The first theoretical model of ChCR by RHI based on classical electrodynamics is discussed in [41] paying attention to necessary corrections to be done for ordinary ChR.And finally, in [16] the quantum theory of ChCR by sub-GeV electrons has been developed. In that work within the frame of quantum electrodynamics new formulae describing the large-angle photon emission by channeled electrons was first obtained taking into account the refractive index dispersion. However, all calculations for both the matrix element and the form-factor were performed within first order approximation, so-called dipole approximation. Thus, it was not explicitly shown that newly described radiation is a mixed radiation, i.e. a term ChCR was not introduced.In this work we demonstrate the applicability of general expressions derived for describing both CR and ChR in two limiting cases. The features of angular and spectral distributions of ChCR are underlined and successfully illustrated via numerical calculations of ChCR in ultrathin diamond and Si crystals, making up in such a way future experimental studies.2Basics of the ChCR quantum theoryIn the first order perturbation theory the matrix element for the probability of ChCR can be presented by the following general expression(1)dwif=e22πħPi(θ0)∑τ|eταif|2δ(κif−κ)d3κκ, where e and ħ are the electron charge and Planck constant, correspondingly, τ=(σ,π) is an index denoting the components of polarization vector eτ for CR photon, κ=|κ|=ω/c⁎, c⁎ħκif=(Ei−Ef) with the phase velocity of CR photon in a crystal c⁎=c/n(ω), and Ei(f)≃εi(f)+c(pi(f)z2+m2c2)1/2 is the total relativistic energy of channeled electron in initial i and final f quantum transverse states with corresponding transverse energies εi(f). The vector components αif for the case of planar channeling are defined as follows [16]:(2)(αif)x=−i〈x〉ifΩifc,〈x〉if=〈f|xe−iκxx|i〉,(αif)y=0,(αif)z=−i〈F〉ifβ,〈F〉if=i〈f|e−iκxx|i〉 with ħΩif=εi−εf, 〈F〉if is the transition form-factor, which at i=f is simply the form-factor of i-th quantum state for planar channeled electron, 〈x〉if is the transition matrix element, κx=|κ|sinθcosφ.Although both the transition form-factor 〈F〉if and the matrix element 〈x〉if are not explicitly influenced by the frequency Ωif, the presence of δ-function in Eq. (1) makes them frequency dependent. Thus, integrating Eq. (1) over all azimuthal φ and polar θ∈(θ,θ+Δθ) angles we can derive the expression for the number of photons emitted by channeled electron per unit path at its transition i→f between the quantum states of transverse motion(3)dNifω=1cħdwifdωΔθ==αcħΩif22c2n2β3ωPi(θ0)G(ω)Θ(WΔ)Θ(W0)××[(2c2βFif2−xif2Ω2if)×[(1−2ωΩifm+(ωΩifm)2))−2n2β2ωΩif2(1−ω2Ωifm)]−−2n2β2ω2xif2)],G(ω)=1+ωn(ω)∂n∂ω,Ωifm=Ωif1−n2(ω)β2.Here, Pi(θ0) stands for the initial population of i-th transverse quantum state as a function of the angle of incidence θ0=arctan[px/pz] with respect to the channeling planes, α is the fine-structure constant. The arguments of the Heaviside's functions Θ(...) are defined by the dependences(4)WΔ=Ωif1−n(ω)βcos(θ+Δθ)−ω,W0=ω−Ωif1−n(ω)βcosθSince we deal with the radiation emission, i.e. W0≥0, the emission angle should fulfill the condition(5)cosθ≤1n(ω)β(1−Ωifω), which differs from the standard Cherenkov angle cosθCh=1/n(ω)β (the resonance case) in the deviation parameter Ωif/ω.Eq. (3) is presented in the form that permits performing simple transition from ChCR to CR. Indeed, if we take both functions 〈F〉if and 〈x〉if in dipole approximation, after some algebra it reduces to the expression for CR-photon number first reported in [8]. However, the presence in Eq. (3) of the form-factor 〈F〉if allows rewriting this equation in a different, but analytically equivalent, form(6)dNifω=αcħβPi(θ0)G(ω)××[(1−1n2β2)Fif2+Fif2n2β2(2Ωifω−Ωif2ω2)++xif2Ωif22β2c2(n2β2+1−2Ωifω+Ωif2ω2)1n2β2], where for the convenience of perception the Heaviside's functions Θ(...) are not shown. To transform further this general expression to that, which corresponds to the standard ChR formula, we have to consider the intraband transitions i→i with Ωii→0 (i.e. κx→0), and Fii2≃1 as reduced from (2). As a result, Eq. (6) can be simplified to(7)dNiiω≃αocħβPi(θ0)(1−1n2β2)G(ω), which at the condition (5) takes the form(8)dNiiω≃αcħβPi(θ0)sin2θChG(ω), where θCh is the Cherenkov angle.After summing up over populated channeling quantum states ∑iPi(θ0)=1 one obtains the formula, which exactly (at G(ω)=1) coincides with the one for ChR angular distribution within standard theory (see, for instance, in [17–19,42,43]).Hence, the results obtained prompt the conclusion on a new type of radiation, i.e. mixed Cherenkov-Channeling Radiation (ChCR).3Index of refraction for Si and C (diamond)The experimentally measured dependences of refractive index n=n(ω) of Si and diamond (C) crystals for optical and near ultraviolet frequencies (Si - red line, diamond (C) - blue dashed line) in the photon energy in range ħω∈(1,12) eV are shown in Fig. 1a [44].In Fig. 1a one may see that the variation range n(ω) for Si, n(ω)=1÷6.7 is much greater than that for C, 0.46÷3.5. Additionally, we have to point out, the value for index of refraction remains constant only for C in the range ħω∈(0.2,3) eV, while becomes rapidly increasing for ħω∈(1.3,3.3) eV in Si and ħω∈(3.3,7.2) eV in C, as well as - rapidly decreasing for ħω∈(3.3,6) eV in Si and ħω∈(11.2,13) eV in C.We have also to underline the presence for both crystals the regions of anomalous dispersion ∂n/∂ω<0 [15,20,45–50], which are separately emphasized in Fig. 1. To analyze ChCR spectral and angular properties (see in Eqs. (3), (6)) aforementioned features of the refraction indexes for both crystals should be correctly taken into account.4Kinematics for ChCRThe conditions for ChCR have been defined in Sec. 2, main features for which come from simple formulas (4). However, in order to get the band structure due to the projectile channeling, it becomes necessary carrying out rather routine calculations to solve the quantum equation of motions [16,51,52]. The latter provides us with the wave functions and energies of the quantum levels for channeled projectiles and then with the transition energies Ωif.We have analyzed the case of 255 MeV channeled electrons (the linac of SAGA-LS accelerator facility). In our calculations we have presented every band as a set of 11 subbands. Hence, since the number of energy bands equals 15 for (110) Si and 11 for (110) C, the total number of quantum states is defined by 15×11 for (110) Si and 11×11 for (110) C. The results of calculations performed for a photon beam collimator of Δθ=0.3 mrad are shown in Fig. 2 as contour maps.The lowermost lines of finite width in Fig. 2 correspond exceptionally to i→i transitions (intraband transitions).The vertical black line (left) drawn at ħω=2.07 eV (wavelength λ≃600 nm) intersects allowed emission angles regions and well pronounced gaps: from θ≃75∘ in the case of Si and θ≃65∘ in the case of C. Then one may suggest, the ChCR angular distribution should consist of well pronounced several broad peaks (the first of them is located just near the corresponding to λ≃600 nm Cherenkov angle for Si) and almost continuous (gapless) spectrum at greater angles. This simple prediction is confirmed by exact calculation for Si (Fig. 3).5ChCR by sub-GeV electrons in a thin Si crystalThe use of Eq. (3) or Eq. (6) requires very time-consuming computer calculations. First we have to determine the wave functions and band structure for the transverse energy levels of channeled electrons. Then it will be followed by calculation of the transition form-factor Fif and matrix elements xif as defined by Eq. (2). Due to the small energy of ChCR-photon with respect to the energy of relativistic channeled electron, the matrix elements xif and Fif are calculated in dipole approximation. Thus, the number of ChCR-photons emitted at the polar angles θ≠θCh is obtained in dipole approximation (i≠f), while the number of ChCR-photons when the initial state coincides with the final one (i=f) - using Eq. (7).Considering all these factors we can get a final formula to calculate the intensity of ChCR spectral distribution within a solid angle Δo that is similar to Eq. (14) in [16]dIifdωΔo=e2xif2Ωif24c3πβ3n2(ω)Pi(θ0)G(ω)ωΘ(WΔ)Θ(W0)×(9)×[Δ+(1−2ωΩifm+(ωΩifm)2)+n2(ω)β2Δ−] Summing up over all transitions between quantum states of a channeled electron, we successfully can define the photon number emitted by electron per unit path within a cone Δo(10)dN=∑i,f1cħ1ħωdIifdωΔo As seen, the number of ChCR-photons is a function of two parameters, the polar emission angle θ and the energy of ChCR photons dN=f(θ,ħω). The Figs. 3-5 make evident these specific dependences.Moreover, the positions for large peaks in Figs. 3 and 4, are within the area of allowed emission angles that is in accordance with the graphs in Fig. 2. Similar distribution for photon energy ħω≃2.07 eV (wavelength λ≃600 nm) is shown in Fig. 4, from which we can resolve two main peaks angularly separated in accordance with Fig. 2a.The spectral distributions of ChCR-photons emitted by 255 MeV (220) electrons planar channeled in Si crystal at fixed polar angles θ≃78.7∘ and θ≃80.3∘ (corresponding to the case of Fig. 4) are formed by single peaks of finite width, respectively, revealing wherein specific behaviors of ChCR (Fig. 5).Using Eq. (7) one can calculate the number of ChCR-photons emitted at intraband transitions (i→i). The number of these photons (i.e. ChR-photons) with the energy ħω≃2.07 eV emitted at the angle θ≃75.3∘ is dN≃0.16 ph/(eV cm) that more than 40 times less of ChCR-photons at the same conditions.6Discussion on characteristic lengthsIn sections 4-5 we have evaluated some ChCR features for fixed electron beam energy and chosen crystal thickness. These results confirm the need to study the new type of radiation analyzing in detail its dependence on main parameters, such as the beam energy, the crystal thickness and the radiation emission wavelength. It becomes also important to understand the relationship between these parameters in definition of ChCR.Let apply the concept of specific characteristic lengths for our analysis. Wherein we can establish two types of the length, one is related to the behaviors of electron scattering, other - to those of ChCR formation and attenuation.6.1Thin and ultrathin crystals at channelingLet start with the length characterizing the electrons channeling. In the literature one can find several definitions such as a thin crystal when the projectile dechanneling is neglected, a thick crystal when the dechanneling should be taken into account, a ultrathin crystal, indeed a very thin crystal, when a particle makes a part of its complete oscillation in a planar channel during its motion down to the crystal exit. For instance, a Half-Wave Crystal (HWC), at which the projectile trajectory in a crystal is a half of the channeling length, is a very recent trend in high-energy channeling as it allows the beams deflection/splitting (see, e.g. in Refs. [53–56]) practically without essential scattering. We do not deal with HWC in our discussion below.Assuming the channeling regime is established, we consider the electron with many oscillations done in a planar channel during its motion in a crystal. At classical description the electron oscillation period can be estimated as T≃d/βcθC, Lz=βcT=πd/θC, therefore N≃L/Lz=(L/d)θC≫1. Here we have used the definitions for the characteristic channeling length Lz, which corresponds to the channeling period T, and the number of oscillations N. Thus, at the well established channeling regime the crystal thickness should be L≫d/θC, where θC is the planar channeling critical angle(11)θC=2U0E=2U0γmc2 with the electron energy E, relativistic factor γ, electron rest mass m and the depth of planar channeling potential U0, which equals to U0≃21.2 eV for Si (110) and U0≃23.6 eV for C (110).On the other hand, for effective channeling the thickness L of the crystal should not exceed the dechanneling length ld defined as a length, at which scattering at channeling θC and multiple scattering [42](12)θMS(l)=13.6MeVβcpZlX0[1+0.038loglX0] result in equal scattering angle(13)θC=θC(ld)→Z2ldX0=(βcp13.6MeV)22U0γmc2=2U0β4γmc2(13.6MeV)2,ld=X0Z22U0β4γmc2(13.6MeV)2 The radiation lengths X0 for the crystals under consideration are ≃ 9.4 cm (Si) and ≃ 12.13 cm (C) that for the case γmc2=255MeV, β≃1, Z=1 gives us the following dechanneling lengths, ld≃5.5μm (Si) and ld≃7.9μm (C).11It means the thin Si crystal (L=0.76μm) already used in a Rainbow Scattering experiment [56] is quite suitable for observation of the ChCR features predicted.Therefore, we can conclude that to observe ChCR the crystal length should be limited by two specific lengths, the 1st one to provide stable particle channeling (L≫d/θC), while the 2nd one avoids its dechanneling (L<ld). For our case:–5.5μm >L≫0.3μm (C);–7.9μm >L>>0.5μm (Si). For clarity, Fig. 6 shows the dependence d/θC as well as the dechanneling length ld on the electron energy E=γmc2.6.2The length of radiation emissionThe concept of formation length for radiation was first introduced by Frank (see in Ref. [57,58]) to interpret the origin of Cherenkov radiation. Originally, it was called a Fresnel zone and defined as the path of a charged particle moving with a constant velocity, for which the phase difference for the waves emitted at the ends of trajectory is less than(14)l=πβcω[1−βn(ω)cosθ] From general considerations we cannot define a moment (point) of emission, hence, as pointed by Frank, l→∞, which is valid at the ChR condition 1−βn(ω)cosθ=0.Later on in [59,60] this idea was extended for small-angle radiation by high-energy particles moving along arbitrary trajectory in the following(15)lph=∫−∞∞vexp[i(ωt−kr(t))]dt,k=ωn(ω)/c In many cases this integral reduces to the contribution of a small number of the Fresnel zones, and the expressions (14) and (15) become of the same order of magnitude. The concept of formation length makes quick qualitative and quantitative conclusions on the features of radiation for given trajectory r(t) and velocity v(t) easily done.The radiation intensity defined by(16)dW(ω,k)dωdΩ=Z2e24π2c2|k×lph|2 are higher at larger lph. It is important to underline that for high-energy particles and small-angle scattering lph can reach macroscopic values [59,60].In our case of large-angle emission and |v(t)|≃vz one can obtain the well-known formula for angular distribution of ChR(17)dW(ω,k)dωdΩ=Z2e24π2c2(lnωcsinθ)2,l=∫0Texp[i(ωt−n(ω)cosθ)]dt For the quantum theory of ChCR the valuable for calculations matrix elements (see Eq. (2) are reduced in the quasi-classical limit to the equations containing similar to (15) lengths with obvious replacement, reflecting the influence of channeling (see Eq. (4))(18)ω→ω−Ωif Then, the formation length lph (photon energies are of order of several eV) becomes large for both emissions at Cherenkov angle and at the angles greater than Cherenkov one. Finally, for these emission angles lph practically does not depend on relativistic factor γ, while d/θC and ld reveal rapid increase with γ. Comparing these lengths allows us to conclude that the condition d/θC≪L<ld for sub-GeV electrons is stronger than L<lph.6.3Thin and ultrathin crystals for optical propertiesOnce ChCR is emitted in a crystal, it would be detectable only if not absorbed by the crystal. At theoretical analysis this fact can be included into consideration via crystal transmittance function T(ω,l), which depends on the photon energy and the length of the photon path in a crystal. In a simplest way in order to perform all calculations taking into account the radiation absorption, Eqs. (3) and (6) could be multiplied by the empirically drawn transmittance function T(ω,l). ChCR waves are continuously emitted by electron along all its trajectory inside the crystal-radiator. Obviously, corresponding photons cover different distances Lph=L/cosθ inside the radiator. Hence, for the crystal thickness L=0.74μm ChCR photons travel inside the crystal within the distance range Lph∈(0.8,1.2)μm.The transmittance functions T(ω) based on experimental data [44] are plotted in Fig. 7 at various crystal thicknesses for Si and diamond crystals. The dependences are shown for the energy range ħω∈(2,7) eV used for our analysis in previous sections.As seen, a Si crystal of L<5μm thickness is transparent for photons ħω<2.5 eV (λ>200 nm) (Fig. 6a), while a diamond (C) crystal independently on its thickness becomes transparent for higher energies ħω<5.6 eV (Fig. 6b). Moreover, the 50% transmittance and more, T(ω)>0.5, can be detected for photons ħω≃5.6÷6.8 eV in crystals of L=5÷0.5μm.This analysis proves the fact that both C and Si crystals can be used for ChCR detection. However, it should be underlined that ChCR-photons per unit path is greater in Si (Fig. 3) because of greater values of the refractive index n(ω) for Si crystal with respect to C one (Fig. 1).For example, following recent experimental studies, let choose ChCR-photons with energy ħω=2.07 eV (λ=600 nm) emitted in ultrathin crystal L=0.74μm [4,38]. At the radiation emission angle θ=75.4∘ the number of photons per unit length will be dN≃13 ph/eV and dNT≃8.34 ph/eV with the crystal transmittance T(ω) taken into account.The discussion on various characteristic lengths is important to stress their influence on possible experimental arrangements to study ChCR features.7ResumeThe large-angle radiation emission by relativistic channeled electrons in optically transparent thin crystals may result in a new type of radiation, mixed Cherenkov-Channeling Radiation (ChCR). New phenomenon can be explained within both classical and quantum approximations. However, specific features of ChCR are due to the quantization of transverse energy levels of planar channeled relativistic electrons as well as the crystal dispersion. We have shown that ChCR angular distribution consists of the series of peaks shifted to larger emission angles with respect to known ChR. General expressions derived from developed theory for ChCR have revealed in their limiting approximations the transition between “pure” CR and ChR. The numerical analyses done for the case of 255 MeV electron beam planar channeled in Si and C (diamond) crystals proves the feasibility of experimental studies for the features of ChCR.AcknowledgementsThe research is carried out at Tomsk Polytechnic University within the framework of Tomsk Polytechnic University Competitiveness Enhancement Program grant. One of the authors (SBD) would like to acknowledge the support by the Competitiveness Program of National Research Nuclear University MEPhI (Moscow). 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