<figure><img src="image_56.png" /><figcaption>Fig. 52. The relative width of PXR photon spectra from various media as a function of the radiation frequency.</figcaption></figure>

At photon energies higher than 10 keV the nanotube ropes dominate over fullerite and carbon-tungsten MLS. For $\omega_B \simeq 3$ keV, the total number of photons (in the above-defined effective solid angle) per electron from a nanotube superlattice is about $10^{-5}$ which is comparable to that from the carbon-tungsten MLS, while ordinary crystals like diamond or germanium are unable to emit PXR at such low frequencies at all.

The relative widths of the photon spectra from various media calculated using (6.21) are illustrated in Fig. 52. One can see that PXR from ordinary crystals have much narrower photon energy distributions than from superlattices. However, it should be noted that for some applications, such as X-ray lithography, the whole number of photons is more important than high monochromaticity.

### 6.3.3. “Channeled” transition radiation

A transition radiation of quite different character may arise [91] at a grazing incidence angle smaller than the Fresnel angle. Let us consider a bundle of micro-capillaries which qualitatively describes a rope of thick and wide enough multi-wall nanotubes. Let a fast charged particle traverse a capillary at small angle $\psi_0$ to the axis and at impact parameter $b$ and consider more particularly the radiation arising when the charged particle impinges the capillary wall from inside of the capillary (vacuum). If the particle energy is high enough ($E \gg \psi^{-1}$), we may neglect the angular divergence of the equivalent photon beam compared to the incidence angle and present the frequency and angular distribution of radiated photons, due to the reflection from the wall, in the following form:

$$ \frac{d^2 N_s}{d\omega d\Omega} = \frac{e^2}{\pi^2 \omega} \left( \frac{P_s \theta}{\theta^2 + E^{-2}} \right)^2 R_s(\psi) . \qquad (6.22) $$

The reflection coefficient $R_s(\psi)$ depends on the actual angle of incidence $\psi = \psi_0(1 - b^2/r^2)^{1/2}$ to the wall, $r$ being the internal radius of the capillary. The small-angle reflection coefficient from an absolutely
smooth surface does not depend on the polarization and may be written in the simple form

$$R_s(\psi) = \left| \frac{\psi - (\chi + \psi^2)^{1/2}}{\psi + (\chi + \psi^2)^{1/2}} \right|^2, \qquad (6.23)$$

where $\chi$ is the complex dielectric susceptibility of the wall substance. Since the capillary diameter is considered as small compared to the diameter of the beam, expression (6.22) should be averaged over all impact parameters $b$. The frequency distribution of radiation is determined exclusively by the frequency dependence of the averaged coefficient $\langle R_s \rangle$ and may be written in the form

$$\frac{dN}{d\omega} = \frac{e^2}{\pi\omega} \langle R_s \rangle \left[ \ln(1 + (E\theta_m)^2) - \frac{(E\theta_m)^2}{1 + (E\theta_m)^2} \right]. \qquad (6.24)$$

The upper limit $\theta_m$ of the integration over $\theta$ is chosen to be comparable to the Fresnel angle $\psi_F$ (due to the logarithmic divergence of (6.24) at large $\theta_m$ there is no need to define $\theta_m$ more accurately).

In addition to backward transition radiation (BTR) when the electron leaves the hole, there is also the forward transition radiation (FTR) when it enters the hole. At $E \gg \psi^{-1}$ the two radiations are nearly equal. Inside the hole, FTR travels nearly parallel to the electron, is reflected on the opposite wall near the specular direction and thereafter interfere with BTR. The interference is destructive when the electron path in the hole is smaller than the FTR coherence length in vacuum, $l_{coh} = 2\pi/(\omega - \mathbf{k}\mathbf{v}) \approx 2\lambda/(\theta^2 + E^{-2})$, divided by 4. To avoid such an effect the inner diameter $d$ of the nanotube should be larger than $d_{min} = \psi E^2 \lambda/4$.

The most interesting point is that radiation is emitted back into the capillary and propagates almost totally inside it, suffering multiple reflections from the wall (X-ray channeling). During that time the charged particle passes through the capillary wall and traverses many other capillaries. Since radiation does not pass through the capillary (MWNT) walls, it is not absorbed as much as PXR. The attenuation of radiation is determined only by a small difference of $R_s$ from unity. If $\psi \ll \psi_F$, then (6.23) gives $R_s \simeq 1 - \delta$, where $\delta = 2\psi\chi''/(\psi_F|\chi'|)$ and the effective number of reflections inside the capillary may be estimated as $N_e \simeq 1/\delta$. The number of capillaries passed by the charged particle during that time is comparable to $N_e$ in order of magnitude. Thus the resulting *channeled transition radiation* (CTR) from a bundle of capillaries can be $N_e$ times higher than transition radiation from a single capillary or MWNT. However, it is important that the grazing angle $\psi$ remains less than $\psi_F$ despite possible scattering of the particle by the atoms of the wall. Compared to the spectrum of PXR, the radiation spectrum of CTR is relatively broad, extending up to the cutoff frequency $\omega_{max} \simeq \omega_p/\psi$.

## 6.4. Nanotube undulator radiation (NUR)

The idea of using spatially periodic magnetic fields (undulators) for the production of monochromatic radiation belongs to Ginzburg [97]. It was developed by Korkhmazian [98] who analyzed the possibility of undulator radiation by ultra-relativistic particles in the X-ray band and gave the analytical expression for the spectral-angular distribution of radiation in an undulator with a transverse planar sinusoidal magnetic field $\mathbf{H}(z) = H_0\mathbf{n}_x \sin(2\pi z/\lambda_w)$ which depends on the longitudinal coordinate $z$. The spectrum of undulator radiation (UR) can be also calculated analytically for the helical magnetic field $\mathbf{H}(z) = H_0[\mathbf{n}_x \sin(2\pi z/\lambda_w) - \mathbf{n}_y \cos(2\pi z/\lambda_w)]$ [99] and for the more complicated case where the helical field is superimposed on a homogeneous longitudinal field [100].
In the theory of the X-ray undulator radiation a fundamental role is played by the *undulator parameter* $p$ which is the ratio of the maximum deviation angle $\psi$ of the electron to the effective angle of the radiation $\theta_{\text{eff}} = E^{-1}$. It is also the amplitude of the transverse momentum oscillation, in units of the electron mass. We recall (see Section 6.2.1) that in the limiting case $p \ll 1$ ($p \gg 1$) the radiation has a dipole (respectively synchrotron-like) character. In the case $p \lesssim 1$ one speaks of a *coherent wiggler*. For the planar sinusoidal magnetic field the undulator parameter can be written as $p = eH_0\lambda_w/2\pi$, where $H_0$ is the field amplitude and $\lambda_w$ the undulator period. The calculations [98–100] showed that the spectral-angular energy density of X-ray radiation from an undulator with fixed $\lambda_w$ and total number of periods $N$ radiation is maximum at $p \simeq 1$. Indeed, at small $p \ll 1$ the amplitude of electron oscillations is too small and at $p \gg 1$ the monochromaticity of the radiation, which is due to the constructive coherence of the different undulator periods, disappears. Besides, the radiation is spread in a larger angular domain and, for the helical undulator, it is no more peaked in the z-direction. This is why it is desirable to hold the undulator parameter near unity. If the magnetic field is measured in Tesla and the undulator period in centimeters, the parameter is given by $p = 0.1 H_0 \lambda_w$. One can see that it is a rather difficult task when $\lambda_w$ is below 1 cm. On the other hand, because the characteristic wavelength of the undulator radiation is proportional to the undulator period and to the inverse square $E^{-2}$ of the electron energy, shorter periods are desirable, if we want to achieve shorter wavelengths without elevating the electron energy.

In was shown in Section 3.4 that chiral nanotubes bent with the Tsyganov radius are equivalent to a very strong (588 T) magnetic field. By screwing the nanotube rope one may obtain a bunch of spiral nanotubes which obviously act as a spiral magnetic field [46]. Alternatively, periodically curved channels can be produced by an intense ultrasonic plane wave propagating through a crystal [101,102]. As is known, in ordinary crystals charged particles cannot follow the curved channels over long enough path because incoherent multiple scattering is important. However, the results of Section 5.2 give the hope that this is possible at least for positrons in nanotubes. In this subsection we analyze how the *nanocrystal undulator* (NU) can indeed help us to solve the short-period problem. To be more specific, we will consider the case of a helically bent nanotube.

Most of the theoretical results [98–100] for the magnetic undulators can be applied to NU. However, there are several major differences between a magnetic undulator (MU) and NU:

* The helix radius and period of NU cannot be chosen independently since helix curvature radius must be larger than Tsyganov critical radius, otherwise particles cannot follow the helix for long enough time.

* There are short-wave oscillations of the channeled particles in the nanotube continuum potential which may affect the coherence conditions of NUR.

* Since NUR occurs in a medium one should take into account the possible influence of the dielectric polarization of the medium [103].

* In NU, the deviation angle $\psi$ is imposed through the rope helix radius and period, therefore the undulator parameter $p = E\psi$ strongly depends on the particle energy, whereas in MU $\psi \propto E^{-1}$ and $p$ is energy independent.

* The effective NU length is limited by particle dechanneling.

### 6.4.1. Limitations due to the critical bending and channeling adiabaticity

Let us denote by $R_w$ the radius of the nanotube spiral. In the case of the planar sinusoidal bending of the channel caused by an ultrasonic wave, $R_w$ will mean the amplitude of the sinusoid. From (3.29)
(there $R_w$ was denoted by A) we have

$$ \psi = \omega_w R_w, \quad R_b \approx 1/(\omega_w^2 R_w), $$

where $\omega_w = 2\pi/\lambda_w$. For a given helix radius $R_w$ and curvature radius $R_b$ of the channel the NU period $\lambda_w$ is determined by the equality

$$ \lambda_w = 2\pi(R_b R_w)^{1/2} . $$

The NU period is limited from below by the condition $R_b > R_c$ where $R_c = d/\theta_L^2$ is the Tsyganov critical radius. Thus the period $\lambda_w$ cannot be shorter than

$$ \lambda_w^{(\min)} = \pi \left( \frac{2d R_w E}{U_0} \right)^{1/2} . \qquad (6.25) $$

Taking this value for $\lambda_w$, the deviation angle $\psi$ is maximum, therefore the largest possible undulator parameter is $p^{(\max)} = (2EU_0R_w/d)^{1/2}$.

The NU period should be also significantly larger than the periods of channeling oscillations, which is itself larger than $\lambda_{ch} = 2d/\theta_L$. The condition $\lambda_w \gg \lambda_{ch}$ guarantees that the channeling states evolve adiabatically and also that there is no interference between channeling radiation and undulator radiation (the influence of channeling oscillations on the averaged longitudinal velocity and the Doppler shift will be discussed below). This condition can be rewritten in the form

$$ \pi \left( \frac{R_b R_w}{d R_c} \right)^{1/2} \gg 1 . $$

If the helix curvature radius $R_b$ is equal to $R_c$ (i.e., the period is minimum, $\lambda_w = \lambda_w^{(\min)}$) then the helix radius must be large enough compared to the channel diameter, $R_w \gg \pi^{-2}d$. However, the inequality $R_w \gg d/2$ is also needed to consider channeling oscillations as a small perturbation of a smooth particle trajectory along nanotube deformation. Thus the above condition is satisfied automatically.

### 6.4.2. Kinematics of NUR

The influence of other factors mentioned above on NUR can be understood from simple considerations of NUR kinematics.

The polarization effect in the X-ray region was first considered in [79] for the case of channeling radiation (see Section 6.2. Note that classical channeling can be treated as a kind of electric undulator) and then in [104] for the gas-filled magnetic undulators. Let us first show how the polarization and channeling effects influence the X-ray energy, using the conservation laws during the radiation. Ignoring the relatively rapid (if $2R_w \gg d$) oscillations of the channeling motion we may consider the channel curvature as an equivalent magnetic field. Since a magnetic field does not change the total particle energy by itself, the latter changes only due to the radiation of the photon with energy $\omega$ and we may write the conservation of energy as

$$ E_1 - E_2 = \omega, \qquad (6.26) $$

where $E_1$ and $E_2$ are the particle energy before and after the radiation of the photon, respectively. Since the equivalent magnetic field is periodical along $z$, the particle has a quasi-momentum $p_z$ which conserves
modulo the reciprocal vector $\omega_w = 2\pi/\lambda_w$ of the magnetic lattice, therefore we may write the conservation of longitudinal quasi-momentum as

$$ \Delta p_z = p_{z1} - p_{z2} = k_z + \frac{2\pi}{\lambda_w} n, \qquad (6.27) $$

where $k_z$ is z-component of the photon momentum and $n$ denotes the integer harmonic number of NUR. Since $\omega \ll E$ (we do not consider the possibility of quantum recoil effects), the energy difference can be presented as

$$ E_1 - E_2 \approx \frac{\partial E}{\partial p_z} \Delta p_z, \qquad (6.28) $$

where $\partial E/\partial p_z = \langle v_z \rangle$ is the mean (over the period) particle velocity along the NU axis. The substitution of $E_1 - E_2$ in (6.28) by $\omega$, and $\Delta p_z$ by the right-hand side of (6.27) gives for the frequencies of NUR the following result

$$ \omega \approx \frac{2\pi n \langle v_z \rangle}{\lambda_w} \frac{1}{1 - \langle v_z \rangle \sqrt{\epsilon'(\omega)} \cos \theta}, \qquad (6.29) $$

where $\theta$ is the angle of NUR with respect to the axis of the undulations. The influence of the polarization is taken into account through the relation $k = \omega\sqrt{\epsilon'(\omega)}$ where $\epsilon'(\omega)$ is the real part of the dielectric permeability of the medium. For the first harmonic $n = 1$ the denominator in (6.29) express the Doppler shift of the radiation frequency $\omega$ from the oscillation frequency $2\pi\langle v_z \rangle/\lambda_w \approx \omega_w$ of the particle in NU.

According to the standard theory of helical magnetic undulators [98–100], in the ultrarelativistic limit the mean longitudinal velocity can be written as

$$ \langle v_z \rangle \approx 1 - (E^{-2}/2)(1 + p^2). \qquad (6.30) $$

For an ordinary (magnetic) helical undulator, $p = E\psi = E\omega_w R_w$. As expected, the larger is the transverse velocity $v_T = \psi$, the slower is the longitudinal one. In the nanotube undulator case, as noted in Section 3.4, we have also the short-wave oscillations of the channeling motion, which are superimposed on the long-wave oscillations of the periodically bent nanotube axis. These channeling oscillations increase $\langle v_T^2 \rangle$, therefore make a further decrease of $\langle v_z \rangle$. The effective value of $p^2$ is then

$$ p^2 = \langle \mathbf{p}_T^2 \rangle \approx (E\omega_w R_w)^2 + 2E\epsilon_{\text{kin}}, \qquad (6.31) $$

where $\mathbf{p}_T(t) = E\mathbf{v}_T(t)$ is the time-dependent transverse momentum with respect to the undulator axis and $\epsilon_{\text{kin}}$ is the averaged (over the short-wave oscillations) kinetic part of the transverse energy $\epsilon$ in channeling motion. One may equally consider the channeled electron or positron as a particle of effective mass squared $m_{\text{eff}}^2 = m^2 + 2\epsilon_{\text{kin}}E$. Due to the flatness of the continuum potential for positrons in nanotubes (except the vicinity of the wall), most of channeled positrons have $\epsilon_{\text{kin}} \approx \epsilon$.

At frequencies above the binding energy of K-shell electrons (which is about 284 eV for carbon) the real part of the dielectric permeability $\epsilon'(\omega)$ is determined by the “plasma formula” $\epsilon'(\omega) = 1 - \omega_p^2/\omega^2$, therefore Eq. (6.29), connecting the frequency and the angle of NUR, can be represented as

$$ \theta^2 = \frac{2n\omega_w}{\omega} - \omega_p^2/\omega^2 - E^{-2}(1 + p^2) \qquad (6.32) $$
in the small-angle approximation (the big Doppler shift requires $\theta \ll 1$). In $p^2$, let us forget temporarily the term $2E\epsilon_{\text{kin}}$, which is small for energies $E \lesssim E_{\text{cr}}^{(1)} = 1/U_0$ determining the applicability of the dipole approximation to CR (as shown in 6.2.1 for chiral nanotubes $E_{\text{cr}}^{(1)} = 4.3$ GeV). Note that for a given spirally bent nanotube $R_w$ and $\psi$ are fixed, therefore the NU parameter $p = E\psi$ is energy dependent. This is in contrast to the case of a magnetic undulator, where $\psi \propto E^{-1}$ and $p$ occurs to be independent of energy. Eq. (6.32) has solutions for real $\theta$ and $\omega$ only if $E$ is higher than the threshold (called critical energy in [77])

$$E_{\text{c}}^{(n)} = \frac{\omega_p}{\omega_w [n^2 - (\omega_p R_w)^2]^{1/2}}. \qquad (6.33)$$

Taking into account the “channeling term” $2E\epsilon_{\text{kin}}$ in $p^2$ makes this threshold still higher. In the dipole approximation ($p \ll 1$) we omit the $R_w$ term in Eq. (6.33) and obtain a threshold energy similar to (6.4) for the case of quantum channeling radiation.<fnref n="1" />

If the particle energy is below $E_c^{(n)}$, then all radiation up to $n$th harmonic is totally suppressed by the polarization effect. The critical energy $E_c^{(n)}$ tends to infinity when the helix radius $R_w$ tends to $R_w^{(\text{max})}(n) = n\lambda_p$, where $\lambda_p = 1/\omega_p$ is the plasma wavelength which, as noted in Section 4.2, is about 85 Å for a rope of (11, 9) SWNTs ($\lambda_p$ increases with the diameter of the nanotube like $(d + g)d^{-1/2}$ where the inter-tube gap $g$ is about 3.15 Å. Let us estimate now the lowest possible value of the critical energy, $\inf E_c^{(n)}$. For this purpose we should take for $\omega_w$ in (6.33) the largest possible value $\omega_w^{(\text{max})} = 2\pi/\lambda_w^{(\text{min})}$, which is fixed by (6.25). We find

$$\inf E_{\text{c}}^{(n)} = \frac{1}{2U_0} \frac{dR_w}{\lambda_p^2 [n^2 - (\omega_p R_w)^2]} . \qquad (6.34)$$

For chiral (11, 9) nanotube according to (2.1) $d = 13.4$ Å, $U_0 = 60$ eV (Fig. 3) and supposing $R_w = 50$ Å we find $\inf E_c^{(1)} = 307$ MeV.

For electrons instead of the nanotube diameter $d$ we should use the width of the potential well which according to Fig. 4 is about 2 Å. The critical energy (6.33) for electrons happens to be one order of magnitude lower than for positrons. Unfortunately, as shown in Section 5.2.1, electrons have much shorter dechanneling length than positrons and their ability to pass lots of NU periods is rather questionable.

The band of X-ray emission at $n$th harmonic is determined by the condition that the right side of (6.32) be positive:

$$\begin{aligned} & \omega_n^{(\text{min})} \le \omega \le \omega_n^{(\text{max})}, \\ & \omega_n^{(\text{max})/(\text{min})} = \gamma_z^2 \{n\omega_w \pm [(n\omega_w)^2 - \omega_p^2\gamma_z^{-2}]^{1/2}\}. \end{aligned} \qquad (6.35)$$

Here $\gamma_z = E(1 + p^2)^{-1/2}$ with $p^2$ given by (6.31). The extreme frequencies $\omega_n^{(\text{min})}$ and $\omega_n^{(\text{max})}$ are always radiated at zero angle $\theta$. The maximum possible angle of X-ray emission at $n$th harmonic is

$$\theta_n^{(\text{max})} = [(n\omega_w/\omega_p)^2 - E^{-2}(1+p^2)]^{1/2}. \qquad (6.36)$$

<footnote n="1">¹ Similar threshold effects also exist for radiation in a medium with dielectric permeability varying periodically in space (resonance transition radiation) [80].</footnote>
A single (for a fixed harmonic) frequency $\omega_n(\theta_n^{(\text{max})}) = \omega_p^2/(n\omega_w)$ is emitted at this angle. Corresponding to each fixed emission angle $\theta < \theta_n^{(\text{max})}$, there are two different frequencies, determined by (6.32), in the allowed X-ray frequency band.

### 6.4.3. Intensity of spontaneous NUR

Assuming $\lambda_w \gg \lambda_{\text{ch}}$, the undulator radiation emitted in a periodically bent nanotube is not sensitive to the channeling oscillations, except that the longitudinal motion is slower due to the effective mass squared $m_{\text{eff}}^2 = m^2 + 2\epsilon_{\text{kin}}E$. We can therefore use the general formulae for the radiation from the helical undulator in a medium [99], which at small polar angles $\theta \ll 1$ and X-ray frequencies give the spectral-angular distribution of NUR energy

$$ \frac{d^2 W}{d\omega d\Omega} = \frac{e^2 \omega^2}{\omega_w^2} \sum_{n=1}^{\infty} \left\{ \left[ \psi \frac{dJ_n(\eta)}{d\eta} \right]^2 + \left| \theta - \frac{n\omega_w}{\omega\theta} \right|^2 J_n^2(\eta) \right\} \frac{\sin^2(N\xi_n)}{\xi_n^2}, \quad (6.37) $$

where $N \gg 1$ is the total number of NU periods, $J_n$ is the Bessel function of $\eta = \omega\theta R_w$ and

$$ \xi_n = \frac{\pi\omega}{2\omega_w} \left( \theta^2 + \frac{\omega_p^2}{\omega^2} + E^{-2} + \omega_w^2 R_w^2 + \frac{2\epsilon_{\text{kin}}}{E} \right) - n\pi. \quad (6.38) $$

The function

$$ S_n(\omega) = \frac{\sin^2(N\xi_n)}{\xi_n^2} \quad (6.39) $$

describes the shape of the spectral line radiated at fixed $\theta$. It has a sharp maximum equal to $N^2$ at $\xi_n = 0$ which corresponds to a totally constructive interference between the radiations from the NU oscillations, and a width $\Delta\xi_n = \pi/N$. The natural spectral width of the line is

$$ \Gamma_n = \left( \frac{\partial \xi_n}{\partial \omega} \right)^{-1} \Delta \xi_n = \frac{\omega}{nN} \left( 1 - \frac{\omega_p^2}{n\omega\omega_w} \right)^{-1}. \quad (6.40) $$

For an infinite undulator we have

$$ \lim_{N \to \infty} S_n(\omega) \to \pi N \delta(\xi_n), $$

where $\delta(\xi_n)$ is the Dirac delta-function, i.e. the frequency and polar angle $\theta$ of the radiation are indeed connected by the relation (6.32) which gives the zeros of $\xi_n$. The effective number of NU periods $N_{\text{eff}}$ is limited by the ratio of the dechanneling length $l_{\text{dc}}$, for the estimation of which we can use (5.5), to the NU period $\lambda_w^{(\text{min})}$ (6.25).

The line-shape function (6.39) refers to a particle with definite total $E$ and transverse channeling energy $\epsilon$. For a particle beam we should take into account the spread of $E$ and $\epsilon_{\text{kin}}$. The spread of $E$ consists of the primarily spread (which may be very low) and the spread attained due to the energy loss in the medium. The spread of $\epsilon_{\text{kin}}$ cannot be eliminated in principle and is always of the order of the depth $U_0$ of the continuum potential (see Figs. 3 and 4). It gives a so-called *inhomogeneous broadening* of the radiation line $\Delta\omega_{\text{inh}} = 2\omega E U_0/(1 + p^2)$, which adds to inhomogeneous broadening coming from the spread of other parameters determining $\xi_n$, like $R_w$, $\lambda_w$ and NU axis direction.
For example, in the case of a twisted nanotube rope, the helix radius is not the same for all nanotubes, but has relatively large spread. Inhomogeneous broadening can be much higher than the natural width (6.40), thus changing the line shape (6.39) and reducing the peak value of spectral-angular density (6.35) proportionally to Γ/Δω<sub>inh</sub>.

As follows from (6.37), in forward direction (θ = 0) only the first circularly polarized [99] harmonic can be emitted, with the spectral-angular density

$$ \frac{d^2 W_0}{d\omega d\Omega} = \frac{e^2 \omega^2 R_w^2 \sin^2(N\xi_1)}{2 \xi_1^2} . \qquad (6.41) $$

In this case the peak (or resonance) radiation frequencies are equal to ω<sub>1</sub><sup>(max)</sup> and ω<sub>1</sub><sup>(min)</sup> in (6.35).

An interesting situation occurs at E = E<sub>c</sub><sup>(1)</sup>. Here the two frequencies come to a common value ω<sub>c</sub> = nω<sub>w</sub>γ<sub>z</sub><sup>2</sup> = γ<sub>z</sub>ω<sub>p</sub> and (6.36) gives θ<sub>1</sub><sup>(max)</sup> = 0. It means that the X-ray photons are emitted in a very narrow forward cone and are nearly monochromatic. The group velocity of the photon (which has an effective mass ω<sub>p</sub>) is equal to ⟨v<sub>z</sub>⟩, i.e. we are at the kinematical (photon + final electron) threshold. The half-aperture Δθ of the cone can be defined as the angle θ where ξ<sub>1</sub> attains its half-width value π/(2N) (for ω = ω<sub>c</sub>). This gives

$$ \Delta\theta = (1/E_c^{(1)}) \left[ \frac{1+p^2}{N} \right]^{1/2} . $$

The half-width Δω/2 in frequency at θ = 0 can be defined similarly. Expanding ξ<sub>1</sub> up to second order in ω − ω<sub>c</sub>, one gets

$$ \Delta\omega = 2\omega_c N^{-1/2} . $$

Note that expression (6.40) for the width goes to infinity and cannot be applied in this case (more generally, when θ ≃ θ<sub>n</sub><sup>(max)</sup>). Integrating over ω one obtains an angular intensity dW<sub>0</sub>/dΩ at θ = 0 larger than usual by a factor ~ N<sup>1/2</sup>.

The spectral-angular density (6.41) at a fixed ξ<sub>1</sub> is proportional to the squared helix radius which, as shown above, is limited by λ<sub>p</sub>. Thus the maximum spectral-angular density of NUR is e<sup>2</sup>ω<sup>2</sup>N<sup>2</sup>/(2ω<sub>p</sub><sup>2</sup>) multiplied by the factor Γ/Δω<sub>inh</sub> describing the influence of the inhomogeneous broadening of the line. The effective number of periods N<sub>eff</sub> is limited by the ratio of the dechanneling length (5.5) to λ<sub>w</sub><sup>(min)</sup> (6.25). One can see that N<sub>eff</sub> is proportional to E<sup>1/2</sup>. Besides dechanneling, the X-ray absorption length may be the main factor limiting N<sub>eff</sub>.

At non-zero polar angles θ ≠ 0 the higher harmonics n ≫ 2 can be also emitted. They have significant spectral-angular density if p ≳ 1.

### 6.4.4. Induced NUR as X-ray amplifier

It is well known that the stimulated undulator radiation is used in the so-called free-electron lasers (FEL) for the generation of intense coherent microwave or even ultraviolet radiation. The principles of FEL action can be understood from the following semi-classical consideration. Let w<sup>(sp)</sup>(E, ω, n) = d<sup>3</sup>N<sub>phot</sub>/(dt dω dΩ) be the differential spectrum of spontaneous NUR photons per unit of time, i.e. expression (6.37) divided by the photon energy ω and by the NU length L = Nλ<sub>w</sub>. The argument E is the initial charged particle energy and n is the unit vector in the photon direction. When an external beam
of photons propagates in this direction it can either stimulate the radiation or be absorbed by the beam. The cross section of the induced radiation $\sigma^{(\text{ind})}(E, \omega, \mathbf{n})$ and the cross section of photon absorption $\sigma^{(\text{abs})}(E, \omega, \mathbf{n})$ are connected to $w^{(\text{sp})}(E, \omega, \mathbf{n})$ through the relations

$$ \sigma^{(\text{ind})}(E, \omega, \mathbf{n}) = \sigma^{(\text{abs})}(E - \omega, \omega, \mathbf{n}) = 2\pi\lambda^2 w^{(\text{sp})}(E, \omega, \mathbf{n}). \qquad (6.42) $$

Note that these equalities relate transitions between the same upper and lower energy levels of the particle, $E$ and $E-\omega$. Besides, due to the natural width of NUR, $\omega$ may not exactly equal to the resonance frequency $\omega_{\text{res}}$ which satisfies (6.32). The photon beam is amplified if the induced cross section is larger than the absorption one for the same initial particle energy $E$. In the small-signal ($G \lesssim 1$), cold-beam (no energy or angular spread of the beam is taken into account) approximation, the gain coefficient $G$ is given by

$$ \begin{aligned} G &= \rho_e[\sigma^{(\text{ind})}(E, \omega, \mathbf{n}) - \sigma^{(\text{abs})}(E, \omega, \mathbf{n})] \\ &= 2\pi\lambda^2 \rho_e[w^{(\text{sp})}(E, \omega, \mathbf{n}) - w^{(\text{sp})}(E + \omega, \omega, \mathbf{n})], \end{aligned} \qquad (6.43) $$

where $\rho_e=IL/(eS)$ is the number of beam particles per unit transverse area which are inside the undulator at a given time, $I$ and $S$ are the current and cross section of the charged particle beam, respectively.

Due to the smallness of $\omega/E$, the gain is usually small, except near the resonance where $w^{(\text{sp})}(E, \omega, \mathbf{n})$ is rapidly varying with $E$. In (6.37) we may neglect the $E$ dependence of all factors except the last one. This gives

$$ \begin{aligned} w^{(\text{sp})}(E, \omega, \mathbf{n}) - w^{(\text{sp})}(E + \omega, \omega, \mathbf{n}) &\approx -L^{-1}N^{-2} \frac{d^2 W(\omega = \omega_{\text{res}})}{d\omega d\Omega} \frac{d}{d\xi_n} \left[ \frac{\sin^2(N\xi_n)}{\xi_n^2} \right] \frac{d\xi_n}{dE}. \end{aligned} \qquad (6.44) $$

For a photon beam propagating along the axis ($\theta=0$), if one ignores the variation of $\varepsilon_{\text{kin}}/E$ in expression (6.38) for $\xi_1$, we have

$$ \frac{d\xi_1}{dE} = -\frac{\omega\lambda_w}{2m\gamma^3} \qquad (6.45) $$

and gathering (6.41), (6.44) and (6.45), we obtain the gain

$$ \begin{aligned} G &= \frac{\pi}{2} \frac{I}{I_0 S} \omega p^2 N^3 \lambda_w^3 \gamma^{-5} f(\zeta), \\ f(\zeta) &= \frac{d}{d\zeta} \frac{\sin^2 \zeta}{\zeta^2}, \quad \zeta = N\xi_1, \end{aligned} \qquad (6.46) $$

where $I_0 = mc^3/e$ and $\gamma = E/(mc^2)$. If in addition we ignore the polarization effect ($E \gg E_c^{(1)}$), we may replace $\gamma$ by $[(1 + p^2)\lambda_w/(2\lambda)]^{1/2}$ and recover the well-known result for the helical undulator [105,106]

$$ G = \frac{I\pi^2}{I_0 S} N^3 \lambda_w^{1/2} \lambda^{3/2} \frac{p^2}{(1 + p^2)^{5/2}} f(\zeta), \qquad (6.47) $$

except for an additional power of $(1 + p^2)$ in the denominator of (6.47). This difference is due to dependence of the NU parameter $p = ER\omega_w$ (or the invariance of $\psi$) with respect to the particle energy $E$.
The gain is positive in the region $-\pi < \zeta < 0$ where the frequency is slightly below $\omega_{res}$, which is the maximum of the spontaneous emission line. Above this maximum the external radiation is preferably absorbed by the beam, giving rise to particle acceleration. The maximum of $f(\zeta)$ corresponds to $\zeta \simeq -1.30$ and is equal to 0.54. The values of $p$ optimal for the gain are close to unity, therefore to obtain an optimal gain for the shorter wavelengths at a fixed particle energy $E$, one should decrease simultaneously the undulator period $\lambda_w$ and the helix radius $R_w$, to preserve the value of $p = E\omega_w R_w$ close to unity. It should be noted that (6.47) gives an upper limit of the gain since it does not take into account inhomogeneous broadening discussed above. Inhomogeneous broadening reduces the gain approximately by a factor $(\Delta\omega_{inh}/\Gamma)^2 \approx (\Delta\omega_{inh}N/\omega)^2$ if $\Delta\omega_{inh}/\omega \gtrsim 1/N$. Thus the effective for NUR FEL number of periods is limited above by $\omega/\Delta\omega_{inh}$.

Noteworthy, since NUR takes place in condensed medium the gain should be large enough compared to the X-ray absorption coefficient, which demands high enough current. As mentioned above, electron beams are probably not suitable for NUR due to their very short dechanneling length, whereas available currents of positron beams are usually much lower than for electrons.

## 7. Miscellaneous effects

Besides CR, CB, PXR and NUR other types of X-ray electromagnetic radiation from ultra-relativistic particles can exist in carbon nanotubes and fullerites. As shown in [107] theoretically and measured in [108], near the K-edge of carbon there is a relatively narrow frequency interval where the dielectric susceptibility is positive and Cherenkov X-ray radiation (CXR) can be emitted by fast enough charged particles. Cherenkov X-ray radiation was detected at normal [109] and grazing [110] incidence of electron beam on carbon materials. Due to the high monochromaticity and narrow angular distribution CXR was considered [111] as a candidate for soft X-ray free-electron laser (FEL), however since the particle beam must move inside a dense medium, multiple scattering substantially reduces the beam quality (angular divergence) needed for any kind of FEL. In this connection for this purpose it may be interesting to use particle beam channeled in nanotubes. From the results of Section 5.2 it follows that the angular divergence of the channeled beam stays within the limits of the Lindhard angle over a long distance (dechanneling length), thus the influence of multiple scattering can be substantially reduced.

Another interesting type of radiation arises when a laser beam collides with a beam of channeled particles [112,113]. The backward Compton scattering of light on ultra-relativistic electron beams is well known as a source of polarized $\gamma$-quanta. The resonance scattering of light on channeled particles have much higher cross section $\sigma(\omega, \mathbf{n}, \omega', \mathbf{n}')$ than the Compton scattering. On the resonance condition $\omega - \mathbf{k} \cdot \mathbf{v} = \omega_{if}$ it can be presented by the Breit-Wigner formula

$$ \sigma(\omega, \mathbf{n}, \omega', \mathbf{n}') = \sigma^{(c)}(\omega, \mathbf{n}) \frac{w(\omega', \mathbf{n}')}{\Gamma_f}, $$

where $\sigma^{(c)}(\omega, \mathbf{n})$ is the cross section of the initial photon capture resulting in the transition of channeled particle from the initial transverse energy level $i$ to the higher final one $f$, $\omega_{if} = \epsilon_i - \epsilon_f$, $\mathbf{n} = \mathbf{k}/k$ is the unit vector in the direction of the initial photon beam, $w(\omega', \mathbf{n}')$ is probability (6.1) of the spontaneous radiation of a photon with energy $\omega'$ per unit of time in the direction of $\mathbf{n}'$, $\Gamma_f$ is the total width of level $f$ determined mostly by non-radiative (dechanneling) processes. Cross section $\sigma^{(c)}(\omega, \mathbf{n})$ is connected with
the spontaneous radiation probability $w(\omega, \mathbf{n})$ in the direction $\mathbf{n}$ by the Einstein relation

$$ \sigma^{(c)}(\omega, \mathbf{n}) = 2\pi\lambda^2 w(\omega, \mathbf{n}) . $$

Since the level width $\Gamma_f$ for positrons in nanotubes is much less than that in ordinary crystals, the backward resonance scattering is easier to observe in nanotubes, however they must be transparent for the initial photons, like diamond.

It is known that rough metal surfaces can emit ultraviolet radiation under electron bombardment at grazing incidence. The reason for such radiation is the excitation of local plasmons (oscillations of the substance electron density) in small metal particles representing the roughness [114]. These particles act like the resonators for the electromagnetic waves with resonance frequencies depending on their shape and size (if this size is larger than the wavelength of radiation). The local plasmons are instantaneous sources of electromagnetic radiation with resonance frequencies. The condition for the excitation of local plasmons is a negative value of the real part of the dielectric permittivity $\epsilon(\omega)$ which can be satisfied for metals in UV and visible range. The excitation of the local plasmons by electrons or an external electromagnetic radiation with the resonance frequency gives rise to a significant enhancement of the local electric field near the roughness and consequently to the enhancement of all electromagnetic processes depending on the local field strength, particularly surface enhanced Raman scattering (SERS) [115] by adsorbed molecules. On the surfaces with broad randomly distributed shapes and sizes of roughness the resonance frequencies occur to have relatively broad distribution too. Nanotube films represent the better surface for the local plasmons due to the uniform shapes of the structures (cylinders with or without hemisphere caps). The field enhancement near the nanotube tips due to the local plasmon excitation can be superimposed on the spike effect (see below) and used for the enhancement of the electron emission from nanotube films under the electromagnetic radiation with the resonance frequency. This possibility should be considered elsewhere with more details.

Let us say some more words about the field emission (FE) from nanotubes and other related phenomena, such as photoemission (PhE) and secondary electron emission (SEE), in conjunction with possible applications of these phenomena in devices for high-energy physics.

Due to the very small curvature radius of nanotube tips very strong electric field (up to $10^7$ V/cm) can be produced near the tips using the well known effect of the enhancement of external electric field near spikes. This effect can give rise to much higher yield of electrons from the tips of single- and multi-wall nanotubes (SWNT and MWNT) compared with other substances used as electron emitters. The FE properties of nanotubes were first discussed in [116]. At present the electron field-emission (FE) experiments have been carried out with both single nanotubes and arrays of many aligned nanotubes (SWNT or MWNT), nanotubes with tips capped by fullerene hemisphere or opened nanotube tips. Stable in time FE electron current was observed from pure, preheated nanotubes free of any adsorbates. There are review articles [117,118] and some articles containing new interesting results [119] devoted to FE from nanotubes.

Summarizing these results, we can formulate the main features of FE from nanotubes as follows: (1) The Fowler-Nordheim law for current-voltage curves of all nanotube emitters holds only at voltages much lower than those in other metallic and semiconductor emitters. With growing voltages this law is violated, and at further rising of the voltage the emission stops with sudden electric breakdown of nanotubes. (2) The maximum FE current which can be obtained from a single MWNT tip is extremely high (about 0.2 mA) and close to the value at which nanotube can be damaged by the thermal effect of the current (Joule heating). On the other hand, the current density from an array of MWNTs occurs to
<figure><img src="image_57.png" /><figcaption>Fig. 53. Schematic view of electron tunneling through the potential barrier near a nanotube cap.</figcaption></figure>

be relatively low (probably due to insufficient alignment and packing of nanotubes used in experiments).
(3) Higher and more stable currents were observed from MWNTs with closed tips rather than opened.
(4) The width of the energy distributions of the emitted electrons occur to be about 0.2 eV, that is substantially less than for metal emitters. Furthermore, sometimes several relatively narrow peaks in the energy spectra were observed. (5) High Joule heating up to 2000 K was observed along the whole length of nanotubes. (6) It was detected that FE can produce a luminescence the spectrum of which is the superposition of a wide incandescence and a narrow peak in the vicinity of 1.77 eV with 0.022 eV FWHM. Only a few luminescence photons are emitted per million electrons.

In order to explain theoretically the above mentioned features, particularly (3), Ispirian and Melikian [120] suggested the resonance tunneling model which was successfully applied previously [121] for the description of FE from metal-semiconductor compounds. According to this model (Fig. 53) the field repels electrons from nanotube metallic body (region I) through the semiconductor tip (region II) into vacuum (region III). When an external field is applied, a one-dimensional double-barrier potential well is formed where substance electrons are in metastable states with energy below Fermi level. The tunneling of electrons through the potential barrier into vacuum is assumed to take place via resonance energy levels and thus have a transmission coefficient *D* close to unity. Solving Schroedinger equation for electrons in potential well and taking into account the equality *D* ≈ 1, one can find the expressions for the energy levels of electrons, the widths of the resonance levels and the current density. This model quantitatively (and in definite cases qualitatively) can explain most of the mentioned features of FE from nanotubes. Unfortunately, some of the parameters used in this model, particularly the width of the conductivity and forbidden zones and the work of exit from the metallic nanotube bodies (well known for other materials) have not been measured yet for nanotubes.

As mentioned in [121], when the external electric field has a pulsed character, FE from nanotubes may attain some new features which facilitate production of intense electron beams with low angular divergence. This is important for advanced charged particle beam acceleration using radio-frequency electron guns.
<figure><img src="image_58.png" /><figcaption>Fig. 54. Photocurrent excitation spectrum for an individual (18, 10) SWNT.</figcaption></figure>

Let us say some words about other possible applications of nanotubes which has not been studied yet. To our knowledge, there are only a few scientific publications devoted to the very close to FE process of photoelectron emission (PhE). Particularly, in [122] PhE from SWNTs and the influence of Cs-intercalation on PhE was investigated. The photocurrent from an individual (18, 10) semiconducting SWNT was recently measured in [123] using Ti/Sapphire laser as photon source with tunable photon energy (and polarization parallel to the nanotube axis). The photocurrent against photon energy is shown in Fig. 54. It is almost a linear function, but with a peak at 1.73 eV which was attributed to the van Hove singularity of the electron state density of the SWNT. For the development of various kinds of particle detectors using PhE, e.g. ring imaging Cherenkov detectors, a good sensitivity to the entrance point of a single photon into a detector is required. The detector sensitivity is strongly dependent on the quantum efficiency of the medium used and since the latter is expected to be relatively high for nanotubes, these objects are good candidates for this purpose.

Another electromagnetic process closely connected with PhE and FE is secondary electron emission (SEE). The structure of nanotubes is similar to the structure of micro-channel plates, commonly used in particle detectors as signal amplifiers. We can hope that some kinds of nanotube structures will have SEE characteristics appropriate for using them in this field.

## 8. Conclusion

It is well known that the unique properties of fullerite crystals and nanotubes can be used in various fields of industry, medicine and science [124]. As shown above, fullerite single crystals and aligned nanotubes [14, 125, 126] can also find various applications in such fields as: production of X-rays with high-energy particle beams, Bragg mirrors for X-rays down to nanometer wavelength which are probably more resistant to extreme irradiation and heating than other soft X-ray mirrors, steering of charged and
neutral particle beams, etc. Recently, some of the discussed possibilities of using nanotubes in high-energy physics were reconsidered in [127–129].

Theoretical calculations and computer simulations clearly indicate that nanocrystals are comparable to or even go beyond the regular targets (crystals) for the production of intense quasi-monochromatic X-ray photons of parametric X-ray radiation (PXR), coherent bremsstrahlung (CB), channeling radiation (ChR), and other types of emission. The intensity of PXR from nanocrystals can be up to three orders higher than in ordinary crystals like diamond; moreover, it becomes possible to extend the PXR spectrum to the soft X-ray region where ordinary crystals cannot emit this type of radiation at all.

Besides PXR, a specific channeled transition radiation (CTR) can arise at grazing incidence of electrons to MWNT axis which is less monochromatic, but more intense than PXR. All these features make PXR a very promising portable source of radiation in various technological and scientific fields wherever monochromatic and wavelength tunable X-ray radiation is needed, while CTR may be used in lithography and X-ray imaging.

Using channeling of high-energy particles in nanotubes it is possible to deflect and focus charged particles more effectively than in ordinary single crystals and maybe even accelerate them using external electromagnetic waves. Moreover, long and wide enough MWNTs can act like waveguides for X-rays and thermal neutrons. On the other hand, like with ordinary crystals, one can use soft X-ray diffraction, PXR and especially ChR in fullerites and nanotube ropes for the determination of the structural parameters of the fullerene and nanotube targets themselves. These possibilities should be analyzed more carefully in the case of relatively thin MWNT films.

Most of the theoretical predictions reviewed above have been discussed at a number of International Symposiums, e.g. "Radiation of Relativistic Electrons in Periodical Structures (RREPS)", Baikal 1999 and Altai 2001, Russia; NATO sponsored Workshop "Interaction of Electrons and Photons in Dense Media", Nor-Hamberd, Armenia, 2001; Workshop "Application of Carbon Nanotubes and Fullerites in High Energy Physics, X-ray and Neutron Optics", IASA, Athens, Greece, 2002, International Workshop on Interactions between Nano-structure and Particle Beams, Shanghai, China, 2004, Workshop on Charged and Neutral Particles "Channelling 2004", Frascati, Italy. The discussions indicated the evident need for the experimental study of the high-energy beam interactions with nanocrystals in order to verify these predictions and thus to give the stimulus to the further development of this field of science which represents an interplay between nanotechnology, X-ray optics and high-energy physics.

The preparation of perfect enough fullerites and aligned nanotubes arrays is crucial for the kinds of experiments mentioned above. Concerning fullerites, they can be probably produced, using the existing methods [130], in sample suitable for all kinds of experiments with high-energy electrons and X-rays mentioned above. However, for experiments at very high energies of the beam particles (say > 1 GeV) it may become necessary to improve these methods for producing better fullerite single crystals with controllable mosaic spread, dislocations and impurities. More difficult task is the production of films of aligned nanotubes. For this purpose various methods are rapidly developing. Among them the most promising are the catalytic vapor deposition method (CVD) of growth of nanotubes from hydrocarbons on surfaces of porous substrates or crystal surfaces [123, 131, 132] and the method of extraction of carbon nanotube targets from cathode deposits formed in arc evaporation of graphite rods [133]. At present the CVD method is developed better for MWNT than for SWNT production. The achieved angular spread (15° or worse) of nanotubes in grown films is unsatisfactory for the channeling experiments mentioned, since it is two orders above the Lindhard critical angle, even in the case of moderate (20 MeV) electron energies. Fortunately, PXR and CB are less sensitive to the nanotube alignment, but 15° misalignment is
still too large. Recently, a spinning process was developed [134] at Centre de Recherche Paul-Pascal, in
Bordeaux, to assemble SWNTs into macroscopic fibers, with diameters ranging from 10 to 100 µm and
lengths up to several tens of cm. MWNTs with wall thickness from several to more than 100 regularly
spaced carbon layers, which was about 10% of the large inner diameter of 20–800 nm, were produced
using alternative method of hydrothermal synthesis [135]. This kind of MWNTs is very promising for
experiments with X-ray channeling.

Hence, the existing methods of nanotube alignment should be improved to the degree at which at
least the study of PXR (and probably ChR) becomes possible, using electron beams of low energy (up to
10 MeV). Foreseeing the difficulties in the preparations of perfect enough structures, it becomes necessary
to modify the already developed theory in order to estimate the influence of imperfections of the targets
and electron beam quality on the electromagnetic processes mentioned above.

The investigations of soft X-ray diffraction, channeling of high-energy charged particles, different
kinds of electromagnetic radiation accompanying the passage of the particles through nanocrystals, and
other low-energy electromagnetic processes taking place in FC, SWNTs and MWNTs are innovative
since they bring together the problems of chemical physics, solid state physics and high-energy physics.
They will lead to a further development of the theory of electromagnetic interactions of X-rays and
electrons with nanocrystals, and help to test the theoretical predictions of promising new sources of
intense monochromatic X-ray radiation with photon energies from 0.1 to 30 keV, which can be used in
industry, science and medicine. The experimental and theoretical investigations of FE, PhE and SEE from
nanotubes may result in the production of novel pulsed electron emitters which will provide short electron
pulses with low emittance for radio-frequency guns and in the construction of nanotube-based photo-
detectors comparable to or even better than the existing micro-channel plate (MCP) detectors. At the same
time, the investigations of fast particle interactions with nanocrystals and associated phenomena may give
rise to new methods of structure analysis of nanotubes and fullerites themselves. The expected results
will promote advances in many interdisciplinary fields of science and may have industrial applications.

## Acknowledgements

To study the advantages of nanocrystals a collaboration supported by INTAS contract No 97-30392 was
created which gathered teams from Institut de Physique Nucléaire de Lyon (IPNL), Kharkov Institute of
Physics and Technology (KIPT), Russian Research Centre “Kurchatov Institute” (Moscow) and Yerevan
Physics Institute (YerPhi), Institute for Physical Research (IPR, Armenia) and Institute of Accelerating
Systems and Applications (IASA, Athens). The present report is mainly based on the result obtained by this
collaboration. Here we would like to express our pain for the death of two of our friends and collaborators,
who contributed in an essential way to our project: Academician Mikhael L. Ter-Mikhaelyan, from IPR,
and Anatoly A. Greenenko, from KIPT.

## Appendix A

Here we give the analytical representation of the exact (not averaged along the axis) nanotube potential
based on Doyle-Turner approximation to the atomic potential. Let (r, φ, z) denote the cylinder coordinates
attached to the nanotube. Since unit cell of a graphite plane shown in Fig. 1a contains two carbon atoms,
we present the total potential of a nanotube $U(r, \varphi, z)$ as a sum $U(r, \varphi, z) = U^{(1)}(r, \varphi, z) + U^{(2)}(r, \varphi, z)$, where $U^{(1)}$ and $U^{(2)}$ are the potentials generated by all the atoms which are in the positions $(\mathbf{a} + \mathbf{b})/3$ and $2(\mathbf{a} + \mathbf{b})/3$ of a graphite unit cell, respectively. Since the exact nanotube potential is a periodical function of $z$ with period $d_R = (3l/q)(n^2 + nm + m^2)^{1/2}$ we may expand it in the following series:

$$U(r, \varphi, z) = \sum_{v=-\infty}^{\infty} \sum_{\mu=-\infty}^{\infty} \mathfrak{U}(r, v, \mu) \exp \left[ i v \varphi + i \frac{2\pi\mu}{d_R} z \right]. \qquad (A.1)$$

The expansion coefficients is determined by equalities $\mathfrak{U}(r, v, \mu) = \mathfrak{U}^{(1)}(r, v, \mu) + \mathfrak{U}^{(2)}(r, v, \mu)$

$$\mathfrak{U}^{(1)}(r, v, \mu) = (2\pi d_R)^{-1} \int_0^{2\pi} \int_0^{d_R} U^{(1)}(r, \varphi, z) \exp \left[ -i v \varphi - i \frac{2\pi\mu}{d_R} z \right] dz \, d\varphi \qquad (A.2)$$

and similar for $\mathfrak{U}^{(2)}(r, v, \mu)$.

Integration in (A.2) is performed over the unit cell of a nanotube considered as 1D crystal. It is not difficult to see that the number of atoms $N' = r_0 d_R \sigma$ in such a cell coincides with the number $2N$ of atomic rows ($N$ given by (2.6)). Expansion coefficients $\mathfrak{U}^{(1)}(r, v, \mu)$ may be written as a product $\mathfrak{U}^{(1)}(r, v, \mu) = u(r, v, \mu)S^{(1)}(v, \mu)$ (and similar for $\mathfrak{U}^{(2)}(r, v, \mu)$), where

$$u(r, v, \mu) = (4Ze^2/d_R) \sum_{j=1}^{4} a_j b_j^2 \exp[-b_j^2(r^2 + R^2)] I_v(2b_j^2 R r) \exp\left[-\left(\frac{\pi\mu}{d_R b_j}\right)^2\right] \qquad (A.3)$$

refers to a single carbon atom with Fourier transformed potential (2.12) and

$$S^{(1,2)}(v, \mu) = \sum_{k=0}^{N-1} \exp \left[ -i v \varphi_k - i \frac{2\pi\mu}{d_R} z_k \right] \qquad (A.4)$$

is the geometrical structure factor, which is determined by coordinates $\varphi_k, z_k, (\varphi_0 = z_0 = 0)$ of $N$ carbon atoms belonging to the first (second) sequence of rows. $S^{(1)}$ and $S^{(2)}$ are related by the following relation:

$$S^{(2)}(v, \mu) = S^{(1)}(v, \mu) \exp \left[ \frac{2\pi i}{N} \left( \frac{n+m}{q} v + \frac{n-m}{3} \mu \right) \right]. \qquad (A.5)$$

Thermal vibrations may be taken into account by inserting the Debye-Waller factor $e^{-W}$, with

$$W = (1/2)(k_r^2 u_r^2 + k_\tau^2 u_\tau^2 + k_z^2 u_z^2),$$

in (A.3). As further calculations show, $S^{(1)}(v, \mu)$ is either $N$ or zero depending upon the relations between nanotube indices $n, m$ and harmonics $v$ and $\mu$. Particularly, in two specific cases:

$$S^{(1)}(v, 0) = N \delta_{v,sN}, \quad S^{(1)}(0, \mu) = N \delta_{v,sN^*}.$$

Here $\delta$ is the Kronecker symbol, $N^* = N/q^*$, $q^*$ is the largest common denominator of $n$ and $m$, and $s=0, \pm 1, \pm 2, \dots$. The case where $\mu=0$ corresponds to continuum potential (2.14), and $v=0$ corresponds to the actual potential of a nanotube averaged over the azimuthal angle $\varphi$. The above results may be useful, for example, in calculations of resonant excitation of channeled ions (Okorokov effect) or electron density in nanotubes in the next approximation.
Appendix B

The solution $\Psi(x)$ of the Dirac equation for an electron in an external field can be presented in the form

$$ \Psi(x) = (\gamma^{\mu} P_{\mu} + m)F(x), \qquad (B.1) $$

where we use the relativistic units, $\gamma^\mu$ are the Dirac matrices, $P_\mu = -i\partial/\partial x_\mu - eA_\mu$, and $x = \{x, y, z, t\}$ denote space and time coordinates of an electron. Particularly, when particle moves in the continuum potential of a nanotube, only the scalar component of the 4-potential $A_\mu$ is non-zero, therefore we obtain the following equation for $F(x)$:

$$ \left[ \left( i \frac{\partial}{\partial t} - U(\mathbf{r}) \right)^2 + \Delta - m^2 + i\alpha \cdot \nabla U(\mathbf{r}) \right] F(x) = 0. \qquad (B.2) $$

Let $E$ be the energy of the relativistic particle and $U_0$ the characteristic potential energy of the particle, then we may neglect the terms $U^2$ and $\alpha \cdot \nabla U$ (the latter describes the interaction of the electron spin with the continuum potential) as small compared to $U\partial/\partial t \sim UE$. Representing $F(x)$ as

$$ F(x) = \exp[i(p_z z - Et)]\psi(\mathbf{r}) \begin{pmatrix} \phi \\ 0 \end{pmatrix}, \qquad (B.3) $$

where $p_z$ is the longitudinal momentum and $\phi$ is the Pauli spinor describing the spin state of the electron, we find the equation for the scalar transverse wave function $\psi$

$$ \left[ -\frac{1}{2E_{\parallel}} \Delta + U(\mathbf{r}) \right] \psi(\mathbf{r}) = \varepsilon \psi(\mathbf{r}). \qquad (B.4) $$

Here we have introduced the following notations: $E_\parallel = (p_z^2 + m^2)^{1/2}$, $\varepsilon = E - E_\parallel$ is the so-called transverse energy. Substitution of (B.3) into (B.1) gives

$$ \Psi(x) = \left( \frac{2E_{\parallel}}{E_{\parallel} + m} \right)^{-1/2} \begin{pmatrix} \phi \\ -i\sigma_j \phi \nabla_j / (E_{\parallel} + m) \end{pmatrix} \exp[i(p_z z - Et)]\psi(\mathbf{r}), \qquad (B.5) $$

where $\sigma$ are the Pauli matrices. The spinor structure of electron wave function (B.5) may be of importance only if radiation of hard photon (with energy $\omega$ comparable to $E$) by the electron is under consideration. In this case we must take into account the spin flip effects. In other cases the more simple scalar function (B.3) is sufficient.

In the case of a curved channel the above equations should be modified in the following way. We choose orthogonal curvilinear spatial coordinates $(x', y', s)$, $s$ being the curvilinear abscissa along the axis of one particular channel (in a family of neighboring channels) and the $(x', y')$ surface being orthogonal to the channel axes and write $F(x, y, z, t) = F'(x', y', s, t)$. We have approximately

$$ \Delta F(x, y, z, t) = \Delta_{\perp} F'(x', y', s, t) + (1 + 2\mathbf{C} \cdot \mathbf{X}') \frac{\partial^2}{\partial s^2} F'(x', y', s, t), $$

where $\mathbf{C}$ is the curvature vector and $\mathbf{X}' = (x', y')$ is the particle location vector in the transverse plane. Writing, instead of (B.3),

$$ F'(x', y', s, t) = \exp(iPs - Et)\psi(x', y', s), $$
where $P = \sqrt{E^2 - m^2}$ and keeping the dependence in $s$, instead of (B.4) one gets

$$i \frac{\partial}{\partial s} \psi(x', y', s) = \left[ -\frac{1}{2P} \Delta_{\perp} + U(x', y') + P \mathbf{C} \cdot \mathbf{X}' \right] \psi(x', y', s) . \qquad (B.6)$$

## Appendix C

Gauged with zero scalar potential $\phi$ the electric field $E(\mathbf{r}, t)$ in a non-magnetic medium is connected with the vector potential $A(\mathbf{r}, t)$ by the relation

$$E(\mathbf{r}, t) = -\partial A(\mathbf{r}, t) / \partial t . \qquad (C.1)$$

Using (C.1) for the corresponding Fourier time expansion coefficients we find

$$E(\mathbf{r}, \omega) = i\omega A(\mathbf{r}, \omega) . \qquad (C.2)$$

From the Maxwell equations it follows the wave equation for $A(\mathbf{r}, \omega)$:

$$[\varepsilon(\mathbf{r}, \omega)\omega^2 - \nabla \times \nabla]A(\mathbf{r}, \omega) = -4\pi\mathbf{j}(\mathbf{r}, \omega), \qquad (C.3)$$

where $\varepsilon(\mathbf{r}, \omega) = 1 + \chi(\mathbf{r}, \omega)$ and $\mathbf{j}(\mathbf{r}, \omega)$ are the dielectric permittivity of the inhomogeneous medium and the Fourier expansion coefficient of the current created by a charged particle.

## Appendix D

It is well known [81] that the spectral-angular distribution of the electromagnetic energy $W$ radiated by a particle with charge $e$ over the total time of its motion with velocity $\mathbf{v}(t)$ along a trajectory $\mathbf{r}(t)$ in vacuum can be presented as

$$\frac{d^2 W}{d\omega d\Omega} = \frac{\omega^2}{(2\pi)^2} |\mathbf{e} \cdot \mathbf{j}(\mathbf{k}, \omega)|^2, \qquad (D.1)$$

where $\omega$ and $\mathbf{k}$ is the radiation frequency and the wavevector respectively, $\mathbf{e}$ is the orthogonal to $\mathbf{k}$ unit vector which determines the radiation polarization (direction of the electric field in the electromagnetic wave), $\Omega$ is the solid angle,

$$\mathbf{j}(\mathbf{k}, \omega) = e \int_{-\infty}^{\infty} \mathbf{v}(t) \exp[i(\omega t - \mathbf{k}\mathbf{r}(t))] dt \qquad (D.2)$$

is the Fourier expansion of the electric current created by the charged particle. The summation over all possible polarizations $\mathbf{e}$ in (D.1) gives the following result

$$\frac{d^2 W}{d\omega d\Omega} = \frac{\omega^2}{(2\pi)^2} |\mathbf{n} \times \mathbf{j}(\mathbf{k}, \omega)|^2, \qquad (D.3)$$

where $\mathbf{n} = \mathbf{k}/k$ is the unit vector along the radiation direction. Usually, the above formulae are derived as a result of the calculation of the electromagnetic energy flux through an infinitely distant sphere surrounding particle trajectory. Practically the distance must be much greater than the coherence length of radiation.
The standard formulae (D.1)–(D.3) should be modified in the case of radiation in a medium. Furthermore, in an absorptive medium the flux turns to zero. In the general case of radiation in a media with the complex dielectric permittivity $\epsilon(\omega) = 1 + \chi(\omega)$ it is more reasonable to calculate the total electromagnetic energy loss of the particle which coincides with the work of the electromagnetic field generated by the particle upon the particle itself

$$W = - \int \int_{-\infty}^{\infty} \mathbf{j}^*(\mathbf{r}, t) \cdot \mathbf{E}(\mathbf{r}, t) \, dt \, d\mathbf{r}, \qquad (D.4)$$

where $\mathbf{j}(\mathbf{r}, t) = e\mathbf{v}(t)\delta(\mathbf{r} - \mathbf{r}(t))$ and $\mathbf{E}(\mathbf{r}, t)$ are the electric current of the particle and the strength of the electric field at particle locus, respectively. Using the Fourier expansion of the kind

$$\mathbf{E}(\mathbf{r}, t) = (2\pi)^{-4} \int \int_{-\infty}^{\infty} \mathbf{E}(\mathbf{k}, \omega) \exp[i(\mathbf{k}\mathbf{r} - \omega t)] d\omega \, d^3\mathbf{k} \qquad (D.5)$$

and relations $\mathbf{j}^*(-\mathbf{k}, -\omega) = \mathbf{j}(\mathbf{k}, \omega)$, $\mathbf{E}(-\mathbf{k}, -\omega) = \mathbf{E}^*(\mathbf{k}, \omega)$, we can rewrite (D.4) as

$$W = -\frac{1}{8\pi^4} \operatorname{Re} \int_0^\infty \int \mathbf{j}^*(\mathbf{k}, \omega) \cdot \mathbf{E}(\mathbf{k}, \omega) \, d\omega \, d^3\mathbf{k}. \qquad (D.6)$$

Then from the Maxwell equations (Appendix C) we find the three Cartesian components ($i = 1, 2, 3$) of the electric field generated by the particle

$$E_i(\mathbf{k}, \omega) = i\omega \frac{4\pi}{k^2 - \omega^2\varepsilon(\omega)} \sum_{l=1}^{3} \left( \delta_{il} - \frac{k_i k_l}{\omega^2\varepsilon(\omega)} \right) j_l(\mathbf{k}, \omega). \qquad (D.7)$$

Substitution of (D.7) into (D.6) gives us the spectral-angular distribution of the energy loss in the form

$$W = \frac{\omega}{2\pi^3} \operatorname{Im} \int \int_0^\infty \left[ |\mathbf{j}(\mathbf{k}, \omega)|^2 - \frac{|\mathbf{k} \cdot \mathbf{j}(\mathbf{k}, \omega)|^2}{\omega^2 \varepsilon(\omega)} \right] \frac{d\omega \, d^3\mathbf{k}}{k^2 - \omega^2 \varepsilon(\omega)}. \qquad (D.8)$$

For a transparent medium $\varepsilon''(\omega) = 0$ we can use the relation

$$\lim_{\varepsilon''(\omega) \to 0} \operatorname{Im} \frac{1}{k^2 - \omega^2 \varepsilon(\omega)} = \frac{\pi}{2k} \delta\left(k - \omega \sqrt{\varepsilon'(\omega)}\right).$$

Then the integration in (D.8) over the wavevector modulus $k$ gives the spectral-angular distribution of radiation

$$\frac{d^2 W}{d\omega d\Omega} = \frac{\omega k}{(2\pi)^2} |\mathbf{n} \times \mathbf{j}(\mathbf{k}, \omega)|^2, \qquad (D.9)$$

where $k = \omega\sqrt{\varepsilon'(\omega)}$. For vacuum $\varepsilon'(\omega) = 1$ (D.9) coincides with the standard expression (D.3).

Noteworthy that even in a transparent dielectric medium ($\varepsilon''(\omega) = 0$) the particle can also lose energy due to the radiation of longitudinally polarized waves (plasmon excitation). This becomes possible at frequencies $\omega$ where the real part of $\varepsilon(\omega)$ tends to zero, e.g. in the optical region in metals. Moreover, in absorptive media ($\varepsilon''(\omega) \neq 0$) the particle can lose energy without any real electromagnetic radiation at all. For example, in the $\omega$-region where $\varepsilon''(\omega)$ is determined by the photo production of $e^+e^-$ pairs, (D.8) describes the energy loss due to the direct production of $e^+e^-$ pairs by the charged particle.
In all cases (except PXR) considered in this Report where ultrarelativistic particle moves predominantly along z-direction and radiates X-rays (|χ| ≪ 1) at low polar angles, the phase angle $\phi = \omega t - kr(t)$ in (D.2) can be rewritten as

$$ \phi \approx \omega\theta\mathbf{n}_{\perp} \cdot \rho(t) - (\omega/2) \left[ \theta^2 + E^{-2} - \chi(\omega) \right]t + \int_{0}^{t} \mathbf{v}_{\rho}^2(\tau) \, d\tau, \qquad (D.10) $$

where $\mathbf{n}_{\perp} = \{\cos \phi, \sin \phi\}$ is the unit vector of the radiation azimuth, $\rho(t)$ and $\mathbf{v}_{\rho}$ are the transverse components of the corresponding vectors. The presence of $\chi(\omega)$ in the phase angle reflects possible influence of the polarization of the medium on the radiation spectrum. Furthermore, in the region where $\chi(\omega)$ is positive this term gives the Cherenkov radiation mentioned in Section 7. In order of magnitude $\mathbf{n}_{\perp} \cdot \rho(t)$ is proportional to the deviation angle of the particle from z-direction, while the integrand in the brackets is the squared deviation angle. If the deviation angle is smaller than the effective angle of radiation $\theta_{\text{eff}} \sim E^{-1}$, then $\phi \approx -(\omega/2)[\theta^2 + E^{-2} - \chi(\omega)]t$. This is the case of the dipole radiation.

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