---
author:
- 'R G Polozkov, V K Ivanov, G B Sushko, A V Korol, and A V Solov’yov'
date: 'Received: date / Revised version: date'
title: Radiation Emission by Electrons Channeling in Bent Silicon Crystals
---

Introduction
============

The basic effect of the channeling process in a straight crystal is in an anomalously large distance which a charged projectile can penetrate moving along a crystallographic plane (the planar channeling) or an axis (the axial channeling) experiencing collective action of the electrostatic field of the lattice ions [@Lindhard]. The field is repulsive for positively charged projectiles so that they are steered into the inter-atomic region. Negatively charged particles, electrons in particular, move in the close vicinity of ion strings.

Channeling of a charged particle is accompanied by the channeling radiation [@ChRad:Kumakhov1976] which is due to channeling oscillations of the particle under the action of an interplanar or an axial field. The radiation intensity depends on the type of the projectile and on its energy as well as on the type of a crystal and a crystallographic plane (axis). By means of crystalline undulator it is feasible to produce monochromatic undulator-like radiation in the hundreds of keV up to the MeV photon energy range. The intensity and characteristic frequencies of the radiation can be varied by changing the type of channeling particles, the beam energy, the crystal type and the parameters of periodic bending (see recent review [@ChannelingBook2013] for more details).

In recent years, several experiments were carried out to measure the channeling parameters and the characteristics of emitted radiation of sub-GeV light projectiles. These include the attempts made [@BaranovEtAl_CU_2006] or planned to be made [@Backe_EtAl_2011a; @Backe_EtAl_2008] to detect the radiation from a positron-based crystalline undulator. More recently, a series of the experiments with straight, bent and periodically bent crystals have been carried out with 195–855 MeV electron beams at the Mainz Microtron (Germany) facility [@Backe_EtAl_2010; @Backe_EtAl_2011; @Backe_EtAl_2013]. The crystalline undulators, used in the experiment, were manufactured in Aarhus University (Denmark) using the molecular beam epitaxy technology to produce strained-layer Si$_{1-x}$Ge$_{x}$ superlattices with varying germanium content [@MikkelsenUggerhoj2000; @Darmstadt01].

Theoretical support of the ongoing and future experiments as well as accumulation of numerical data on channeling and radiative processes of ultra-relativistic projectiles in crystals of various content and structure must be based on an accurate procedure which allows one to simulate the trajectories corresponding to the channeling and non-channeling regimes. Recently, a universal code to simulate trajectories of various projectiles in an arbitrary scattering medium, either structured (straight, bent and periodically crystals, superlattices, nanotubes etc) or amorphous (solids, liquids) has been developed as a new module [@NewPaper_2013] of the MBN Explorer package [@MBN_ExplorerPaper; @MBN_ExplorerSite]. To simulate propagation of particles through media the algorithms used in modern molecular dynamics codes were utilized. Verification of the code against available experimental data as well as against predictions of other theoretical models were carried out for electron and positron channeling in straight Si(110) as well as in amorphous Si [@NewPaper_2013]. In the cited papers, critical analysis was carried out of the underlying physical model and the algorithm implemented in the recent code for electron channeling described in Refs. [@KKSG_simulation_straight]. It was shown, that the specific model for electron–atom scattering leads in a noticeable overestimation of the mean scattering angle In this paper we present new results on electron channeling and emission spectra in straight and uniformly bent silicon crystal. The electron channeling along Si(110) crystallographic planes are studied for the projectile energy ${{\varepsilon}}$ = 855 MeV for two lengths of a crystalline sample and for different bending curvatures.

Electron Channeling in Si (110) \[ElectronChanneling\]
======================================================

Simulation of the Channeling Process with MBN Explorer \[MBN\_algorithm\]
-------------------------------------------------------------------------

Numerical Results for ${{\varepsilon}}=855$ MeV Electron Channeling in Si(110) \[ChannelingResults\]
----------------------------------------------------------------------------------------------------

\[h\] ![ Channeling of 855 MeV electrons in a $L_1=25$ $\mu$m thick silicon crystal with bending radius $R = 1.3$ cm (the bending parameter $C \approx 0.1$). []{data-label="Figure.01"}](Figure_01.eps "fig:")

[1.00]{}[@ccccccc]{} $R$ & $C$ &${{\cal A}}$ & $L_{p1}$ & $L_{p2}$ & ${{L_{\rm ch}}}$ & $N_0$\
(cm) & & & ($\mu$m) & ($\mu$m) & ($\mu$m) &\
$\infty$& 0.0 & 0.66 &$11.72\pm 0.32$&$10.88\pm 0.15$&$26.24\pm 0.49$ & 12600\
70 & 0.002 & 0.65 &$11.71\pm 0.33$&$10.86\pm 0.16$&$25.78\pm 0.50$ & 11950\
25 & 0.006 & 0.64 &$11.62\pm 0.33$&$10.86\pm 0.16$&$24.59\pm 0.50$ & 11950\
13 & 0.01 & 0.65 &$11.43\pm 0.32$&$10.87\pm 0.17$&$21.79\pm 0.50$ & 12000\
7 & 0.02 & 0.64 &$11.55\pm 0.34$&$11.06\pm 0.21$&$16.35\pm 0.55$ & 12000\
2.5 & 0.06 & 0.55 &$10.23\pm 0.30$&$10.16\pm 0.27$&$07.00\pm0.27$ & 12000\
1.3 & 0.1 & 0.44 &$08.67\pm0.28$&$08.71\pm0.27$&$04.14\pm0.21$ & 12050\
0.75 & 0.2 & 0.34 &$07.03\pm0.41$&$07.03\pm0.40$&$02.41\pm0.27$ & 4300\
0.45 & 0.3 & 0.22 &$05.32\pm0.33$&$05.33\pm0.33$&$01.18\pm0.17$ & 4500\

Radiation Emitted by Electrons in Si (110) \[Radiation\]
========================================================

The simulated trajectories in straight and periodically bent Si(110) channels were used to compute spectral distribution of the emitted radiation. To this end, for each set of simulated trajectories of the total number $N_{0}$ the spectral distribution emitted within the cone $\theta\leq\theta_0$ with respect to the incident beam was calculated as follows: $$\begin{aligned}
{{{\rm d}}E(\theta\leq\theta_0) \over \hbar\, {{\rm d}}{{\omega}}}
=
{1 \over N_0}
\sum_{j=1}^{N_0} 
\int\limits_{0}^{2\pi}
\int\limits_{0}^{\theta_0}
{{{\rm d}}^3 E_j \over \hbar\,{{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}}
\label{Radiation:eq.01} $$ Here, ${{\rm d}}^3 E_j/\hbar{{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}$ stands for the spectral-angular distribution emitted by a particle which moves along the $j$th trajectory. The sum is carries out over [*all*]{} simulated trajectories, i.e. its takes into account the contribution of the channeling segments of the trajectories as well as of those corresponding to the non-channeling regime.

To calculate ${{\rm d}}^3 E_j/\hbar{{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}$ one can use a general quasi-classical method developed by Baier and Katkov [@Baier67]. The details of the formalism, as well as its application to a variety of radiative processes, can be found in Ref. [@Baier] (see also [@Uggerhoj2011]). A remarkable feature of this method is that it allows one to combine the classical description of the motion in an external field and the quantum effect of radiative recoil, i.e. the change of the projectile energy due to the photon emission. Its role is governed by the ratio $\hbar {{\omega}}/{{\varepsilon}}$. In the limit $\hbar \omega /\varepsilon \ll 1$ a purely classical description of the radiative process can be used (see, e.g., [@Landau2; @Jackson]). For $\hbar \omega /\varepsilon \le 1$ quantum corrections must be accounted for. The quasi-classical approach explicitly takes into account the quantum corrections due to the radiative recoil. The method is applicable in the whole range of the emitted photon energies, except for the extreme high-energy tail of the spectrum $\left(1-\hbar{{\omega}}/{{\varepsilon}}\right)\ll 1$.

Within the framework of quasi-classical approach the spectral distribution of energy radiated in given direction ${{\bf n}}$ by an ultra-relativistic particle is given by the the following expression (see Ref. [@Baier] for the details):

$$\label{Radiation:eq.02} {{{\rm d}}^3 E \over \hbar{{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}}
=
\alpha \,
{ q^2\omega^2  \over 8\pi^2 }
\int\limits_{-\infty}^{\infty} {{\rm d}}t_1\!
\int\limits_{-\infty}^{\infty} {{\rm d}}t_2\,
{{\rm e}}^{{{\rm i}}\,\omega^{\prime} \left(\psi(t_1) -\psi(t_2)\right)}
\left[
\left( 1+(1+u)^2 \right)
\left(
{{{\bf v}}_1\cdot{{\bf v}}_2 \over c^2}  -1
\right)
+{u^2 \over \gamma^2}
\right],
$$

where $\alpha= e^2/ \hbar\, c$ is the fine structure constant, $q$ is the charge of a projectile in units of the elementary charge, ${{\bf v}}_{1,2} ={{\bf v}}(t_{1,2})$ denote the velocities, and the phase function reads as $\psi(t) = t - {{\bf n}}\cdot{{\bf r}}(t)/ c$. The quantities ${{\omega}}^{\prime}$ and $u$ account for the radiative recoil: $$\begin{aligned}
{{\omega}}^{\prime}
=
(1+u)\, {{\omega}},
\qquad
u =
{\hbar {{\omega}}\over {{\varepsilon}}- \hbar {{\omega}}}.
\label{Radiation:eq.03} $$ In the classical limit $u\approx \hbar{{\omega}}/{{\varepsilon}}\to 0$ and ${{\omega}}^{\prime} \to {{\omega}}$, so that (\[Radiation:eq.02\]) reduces to the classical formula [@Landau2; @Jackson].

Eq. (\[Radiation:eq.02\]) allows us to compute the emission spectrum for each simulated trajectory.

The simulated trajectories were used to compute spectral distribution of the emitted radiation following the algorithm described in Refs. [@NewPaper_2013]. The calculations were carried out for 25 and 75 $\mu$m thick straight and bent crystals. The integration over the emission angle in (\[Radiation:eq.01\]) was performed for the aperture $\theta_{0}$ = 2.4 mrad which greatly exceeds the natural emission angle $\gamma^{-1} \approx 0.6$ mrad. The averaging was carried out over $N_0\approx 2000$ simulated trajectories.

Emission Spectra for $L=25$ $\mu$m
----------------------------------

In Fig. \[Figure.02\] the calculated spectral distribution (\[Radiation:eq.01\]) of the channeling radiation in straight Si(110) crystal (solid curve) is compared with the spectral intensity in an amorphous silicon (dashed curve) obtained within the Bethe-Heitler (BH) approximation (see, for example, Ref. [@Tsai1974]). It is seen that the intensity of radiation in the oriented crystal greatly exceeds (in the maximum located at $\hbar{{\omega}}\approx 4$ MeV there is an order of magnitude excess) that in the amorphous medium. The enhancement is due to the emission by the particles moving along quasi-periodic channeling trajectories, which bear close resemblance with the undulating motion. As a result, constructive interference of the waves emitted from different but similar parts of the trajectory increases the intensity.

\[h\] ![ Solid curve represents the calculated spectral distribution of radiation by 855 MeV [electrons]{} in oriented straight Si(110). Dashed line shows the Bethe-Heitler for amorphous silicon. The data refer to the aperture $\theta_{0}$ = 2.4 mrad and to the crystal thickness $L=25$ $\mu$m. []{data-label="Figure.02"}](Figure_02.eps "fig:")

Spectral enhancement factor, i.e. the spectral distribution ${{\rm d}}E/{{\rm d}}(\hbar{{\omega}}) $ of radiation emitted in the crystalline medium normalized to that in amorphous silicon, calculated for 855 MeV electrons is presented in Fig. \[Figure.03\]. Solid curve represents the dependence obtained for straight Si(110), two other curves stand for uniformly bent channel: the broken curve corresponds to $R=2.5$ cm, the chained one – to $R=1.3$ cm.

Let us note several specific features of the emission spectra formed in bent channels.

First, the bending gives rise to the synchrotron radiation, since the channeled particle experiences the circular motion in addition to the channeling oscillations. This leads to the increase of the intensity in the photon energy range $\lesssim 10^2$ keV. For these energies the radiation yield from the bent channel exceeds that from the straight channel.

\[h\] ![ Enhancement factor for 855 MeV [electron]{} channeled in $L=25$ $\mu$m straight (solid curve) and bent Si crystals along (110) planes. Broken curve corresponds to the bending radius $R=2.5$ cm (the bending parameter $C=0.06$), chained curve – to $R=1.3$ cm ($C=0.1$). The data refer to the aperture $\theta_{0}$ = 2.4 mrad. []{data-label="Figure.03"}](Figure_03.eps "fig:")

Second, there is a noticeable decrease in the intensity of the channeling radiation with the decrease of the bending radius. The ratio $\xi$ of the maximum value of the spectrum in the bent channel to that in the straight one is $0.66$ for $R=2.5$ cm and $0.49$ for $R=1.3$ cm. The statistical uncertainties of these values due to the finite number of the simulated trajectories is on the level of 10 per cent. The decrease of $\xi$ can be explain as follows. In the vicinity of the maximum, the main contribution to the channeling radiation intensity comes from those segments of a trajectory where the particle stays in the same channel experiencing channeling oscillations. The intensity of radiation emitted from either of this segments and integrated over the emission angles is proportional to the segment length $l$ and to the square of the undulator parameter $K$ associated with the channeling oscillations. The undulator parameter $K$ is related to the mean square of the transverse velocity: $K^2 = 2\gamma^2 \,\langle v^2_{\perp}\rangle /c^2$ (see, e.g., [@Baier]). Then, to estimate the right-hand side of Eq. (\[Radiation:eq.01\]) one can consider the contribution to the sum coming from the primary channeling segments. Hence $$\begin{aligned}
{{{\rm d}}E(\theta\leq\theta_0) \over \hbar\, {{\rm d}}{{\omega}}}
\propto 
{{\cal A}}\, K^2\, L_{\mathrm p1}\,.
\label{Radiation:eq.04} \end{aligned}$$ For bending radii much larger than the critical radius $R_{\rm c}={{\varepsilon}}/F_{\max} = 0.15$ the factor $K^2$ depends weakly on $R$. Hence, the factor $\xi$, defined above, can be calculated at the ratio of the products ${{\cal A}}\, L_{\mathrm p1}$ calculated for the bent and the straight channels. Using the data from Table \[Table\_lengths\] one estimates: $\xi = 0.72 \pm 0.03$ and $0.48 \pm 0.02$ for $R=2.5$ and 1.3 cm, respectively. These values correlate with the ones quoted above. Thus, the decrease in the acceptance ${{\cal A}}$ and in the penetration length $L_{\mathrm p1}$ with $R$ is the main reason for lowering the intensity of channeling radiation.

Finally, comparing the curves in Fig. \[Figure.03\] one notices that the position ${{\omega}}_{\max}$ of the maximum shifts to higher photon energies with the growth of the crystal curvature $1/R$. This feature is specific for the electron channeling (more generally, for channeling of negatively charged projectiles) and is due to strong anharmonicity of the channeling oscillations. As a result, the frequency ${{\Omega}}_{\rm ch}$ of the oscillations varies with the amplitude $a_{\rm ch}$. Typically, smaller vales of $a_{\rm ch}$ correspond to larger frequencies (see, for example, the results of simulations presented in [@Sub_GeV_2013]). As the curvature increases, the allowed values of $a_{\rm ch}$ decrease due to the action of the centrifugal force [@BiryukovChesnokovKotovBook]. Hence, on average, the frequency of the channeling increases and so does the frequency of the emitted photons ${{\omega}}_{\max} \propto {{\Omega}}_{\rm ch}$.

The spectra presented in Figs. \[Figure.02\] and \[Figure.03\] are obtained from Eq. (\[Radiation:eq.01\]) by averaging over all simulated trajectories. The main contribution to the radiation enhancement comes from the electrons which stay mostly in the channeling regime. In this context it is of interest to analyze the spectral distribution of radiation produced only by electrons propagating through the whole bent crystal staying in a single channel or in a few different channels (i.e. changing the channels due to the rechanneling effects). Such trajectories, although being quite rare, allow one to visualize the influence of the channeling oscillations as well as of the rechanneling effect on the synchrotron-like part of the spectrum.

\[h\] ![ Spectral distribution of radiation by 855 MeV [electrons]{} in a $L=25$ $\mu$m thick Si(110) crystal bend with the radius $R=1.3$ cm. The data refer to the aperture $\theta_{0}$ = 2.4 mrad. Curves 1 and 2 correspond to the enhancement factor averaged over two specially selected sets of trajectories. The broken represents the spectrum of synchrotron radiation normalized to the Bethe-Heitler background. See also explanation in the text. []{data-label="Figure.04"}](Figure_04a.eps "fig:")

The increase of the enhancement factor at small photon energies is associated with the synchrotron radiation. To visualize this we calculated the spectral distribution of radiation due to motion along a $25$ $\mu m$ arc of a circle of the radius 1.3 cm, i.e. along the bent channel centerline ignoring the channeling oscillations. The result, normalized to the Bethe-Heitler spectrum, is presented in Fig. \[Figure.04\] by the broken line. Within the statistical errors (not indicated) the curves 1 and 2 coincide with the synchrotron spectrum at $\hbar{{\omega}}\lesssim 10^{-1}$ MeV.

\[h\] ![ Low-energy part of radiation spectrum formed by 855 MeV electrons in a $L=25$ $\mu$m thick silicon crystal with bending radius $R = 1.3$ cm. The curves correspond to different apertures, which are integer multiples of the natural emission angle $\gamma^{-1}\approx 0.6$ mrad. Full curves stand for the spectra averaged over the electrons channeling through whole crystal in the same channel, Broken curves represents the spectra of synchrotron radiation. []{data-label="Figure.05"}](Figure_05.eps "fig:")

In Ref. [@Sub_GeV_2013] the influence of the detector aperture $\theta_0$ on the form of the spectral distribution of the channeling radiation in straight and bent Si(110) channels was explored. In Fig. \[Figure.05\] we present results of the similar analysis of the low-energy part of the spectrum. Full curves correspond to the spectra (\[Radiation:eq.01\]) averaged over the simulated trajectories corresponding to the motion through the whole crystal staying in the same channel (“group 1”). Broken curves represent the distribution of the synchrotron radiation due to the motion along the circle arc. The calculations were performed for $\theta_0$ = 0.6, 1.2, 2.4 and 3.6 mrad which are integer multiples of the natural emission angle $\theta_{\gamma} = \gamma^{-1}\approx 0.6$ mrad. The latter characterizes the cone (along the instantaneous velocity) which accumulates most of the radiation emitted by an ultra-relativistic projectile.

Two features of the presented dependences can be mentioned.

First, it is seen that the intensity is quite sensitive to the detector aperture. This effect is more pronounced for the lower values of $\theta_0$ and within the $10\dots 100$ keV photon energy range where most of radiation is emitted via the synchrotron mechanism. For example, a two-fold change in the aperture from $\theta_{\gamma}$ to $2\theta_{\gamma}$ results in a nearly four-fold increase of the intensity at in the lowest-energy part of the spectrum. Such a behaviour can be understood if one compares the quoted values of $\theta_0$ with the angle of rotation of the bent crystal centerline, $\theta_{L} = L/R \approx 1.9$ mrad. For apertures smaller than $\theta_{L}$ the radiation within the cone $\theta_0$ along the incident velocity will be effectively emitted only from the initial part of the crystal, the length $l_0$ of which can be estimated as $l_0 \approx R\theta_0 = (\theta_0/\theta_{L})L < L$. Thus, the raise of the aperture from $0.6$ to $1.2$ mrad results in the two-fold increase of $l_0$ which, in turn, leads to additional augmentation of the emitted intensity. This effects becomes less pronounced for larger apertures, $\theta_0 > \theta_{L} \gg \theta_{\gamma}$, which collects virtually all radiation emitted within $L$.

Another feature to be mentioned is that the influence of the channeling motion on the emitted spectrum becomes more pronounced over wider range of photon energies with the increase of the aperture. Indeed, the deviation of the total emission spectrum (full curves) from the spectrum of synchrotron radiation (broken curves) starts at $\hbar{{\omega}}\approx 250$ keV for $\theta_0=0.6$ mrad but at $\hbar{{\omega}}\approx 50$ keV for $\theta_0=3.6$ mrad. To explain this one recalls that channeling motion bears close resemblance with the undulating motion. As a result, constructive interference of the waves emitted from different but similar parts of the trajectory increases the intensity. For each value of the emission angle $\theta$ the coherence effect is most pronounced for the radiation into harmonics. The frequency of the lowest (first) harmonic can be estimated as ${{\omega}}_{1}(\theta) = 2\gamma^2\, {{\Omega}}_{\rm ch}/\left(1 + \gamma^2 \theta^2 + K^2/2\right)$ (see, e.g., [@Baier]). Anharmonicity of the electron channeling oscillations results in the variation of ${{\Omega}}_{\rm ch}$ along the trajectory. Therefore, the radiation will be emitted within some frequency band which will form the main peak in the spectral distribution of the channeling radiation (see Fig. \[Figure.02\]). For small apertures, when $(\gamma\theta_0)^2 \ll 1$, the emission of low-energy photons with ${{\omega}}\ll {{\omega}}_{1}(0)$ is strongly suppressed. For larger apertures a big part of energy is radiated into the cone $\gamma^{-1} < \theta < \theta_{0}$. For $(\gamma\theta_{0})^2\gg 1$ the harmonic energy is strongly red-shifted ${{\omega}}_{1}(\theta_0) \ll {{\omega}}_{1}(0)$. As a result, the contribution of the channeling radiation to the low-energy part of the spectrum increases with the aperture.

Emission Spectra for $L=75$ $\mu$m
----------------------------------

\[h\] ![ Enhancement factors for 855 MeV [electrons]{} channeled in $L=75$ $\mu$m straight and bent Si(110) crystals. The numbers indicate the values of the bending radius in cm. The data refer to the aperture $\theta_{0}$ =2.4 mrad, []{data-label="Figure.06"}](Figure_06.eps "fig:")

Figure \[Figure.06\] illustrates the modification of the spectral dependence of the enhancement factor in the vicinity of the maximum of channeling radiation. Comparing the presented dependences with those calculated for a shorter crystal, see Fig. \[Figure.03\], one can state that in both cases the profiles of the enhancement factor are similar, and the maximum values of the enhancement factor are close in both cases. In both figures there are two curves calculated for the same values of the bending radius: $R=\infty$ (straight crystal) and $R=1.3$ cm. Comparing these one notices that the maximum values are larger (by factors $\approx 1.4$ and $1.6$, respectively) for a shorter crystal. To explain this one can use the following arguments. In a straight crystal, the intensity of channeling radiation is proportional to the total length of the channeling segments, $L_{\rm ch}$, whereas the intensity of the incoherent bremsstrahlung scales with the crystal length $L$. Hence, the enhancement factor is proportional to $L_{\rm ch}/L$. For a $75$ $\mu$m thick crystal $L_{\rm ch}\approx 26$ $\mu$m (see Table \[Table\_lengths\]). For $L=25$ $\mu$m we found $L_{\rm ch}\approx 12.6$ $\mu$m. Hence the ratio of the $L_{\rm ch}/L$ values calculated for $L=25$ and $75$ $\mu$m straight Si(110) crystals is $3/2$ which correlates with the factor quoted above. Similar arguments can be applied to the case of a bent crystal. The only difference is that the intensity of the background bremsstrahlung radiation integrated over the aperture $\theta_0$ is proportional to the effective length $l_0 \sim \min\left\{L, \theta_0 R\right\}$.

The synchrotron radiation influences the total spectrum in the photon energy range well below the maximum of channeling radiation. Fig. \[Figure.07\] illustrates the behaviour the enhancement factor in the low-energy part of the spectrum.

It is seen from the figure that the enhancement due to the synchrotron radiation is a non-monotonous function of the bending radius. At small curvatures the enhancement increases with $1/R$. The maximum values are achieved at $R\approx 0.75$ cm and then the enhancement decreases with the curvature. This feature is due to the finite aperture which for sufficiently small values of $R$ introduces the length $l_0 < L$ of the crystal where the radiation detected within the cone $\theta \leq \theta_0$ is effectively formed, and thus reduces the enhancement.

\[h\] ![ Enhancement factors for the low-energy part of radiation spectrum formed by 855 MeV electrons in a $L=75$ $\mu$m straight and bent Si(110). The numbers indicate the values of the bending radius $R$ in cm. The data refer to the aperture $\theta_{0}$ =2.4 mrad, []{data-label="Figure.07"}](Figure_07.eps "fig:")

Using a newly developed code [@NewPaper_2013], which was implemented as a module in the MBN Explorer package [@MBN_ExplorerPaper], we have performed numerical simulations of trajectories of ultra-relativistic electrons in oriented straight and bent single Si(110) crystals.

The simulated trajectories were used as the input data for numerical analysis of the intensity of the emitted radiation. In the case of straight crystals the channeling radiation appears atop the incoherent bremsstrahlung background. In a bent channel the spectrum is enriched by the synchrotron radiation due to the circular motion of the projectile along the bent centerline.

The calculation of the spectra as well as the numerical analysis of channeling conditions and properties (acceptance, dechanneling length) have been carried out for the beam of 855 MeV electrons for two lengths of Si crystal. The obtained and presented results are of interest in connection with the ongoing experiments with electron beams at Mainz Microtron [@Backe_EtAl_2011].

Acknowledgments
===============

We are grateful to Hartmut Backe and Werner Lauth for fruitful and stimulating discussions. The work was supported by the European Commission CUTE-IRSES project (grant GA-2010-269131). The possibility of performing complex computer simulations at the Frankfurt Center for Scientific Computing is gratefully acknowledged.

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